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GATE QUESTION BANK for

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GATE QUESTION BANK

Contents

Contents #1.

#2.

Subject Name Mathematics

Topic Name

Page No. 1-148

1 2 3 4 5 6 7

Linear Algebra Probability & Distribution Numerical Methods Calculus Differential Equations Complex Variables Laplace Transform

1 – 28 29 – 57 58 – 73 74 – 112 113 – 131 132 – 143 144 – 148

Network Theory 8 9 10 11 12 13

#3.

19

Introduction to Signals & Systems Linear Time Invariant (LTI) systems Fourier Representation of Signals Z-Transform Laplace Transform Frequency response of LTI systems and Diversified Topics

168 – 185 186 – 203 204 – 206 207 – 214 215 – 216

217 – 223 224 – 238 239 – 250 251 – 256 257 – 261 262 – 275

276 – 340 Basics of Control System Time Domain Analysis Stability & Routh Hurwitz Criterion Root Locus Technique Frequency Response Analysis using Nyquist plot Frequency Response Analysis using Bode Plot Compensators & Controllers State Variable Analysis

Analog Circuits 28 29 30 31

149 – 167

217 – 275

Control Systems 20 21 22 23 24 25 26 27

#5.

Network Solution Methodology Transient/Steady State Analysis of RLC Circuits to DC Input Sinusoidal Steady State Analysis Laplace Transform Two Port Networks Network Topology

Signals & systems 14 15 16 17 18

#4.

149 – 216

276 – 282 283 – 294 295 – 300 301 – 308 309 – 316 317 – 322 323 – 329 330 – 340

341 – 421 Diode Circuits - Analysis and Application AC & DC Biasing-BJT and FET Small Signal Modeling Of BJT and FET BJT and JFET Frequency Response th

th

341 – 353 354 – 363 364 – 372 373 – 375 th

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GATE QUESTION BANK

32 33 34

#6.

422 – 472 Number Systems & Code Conversions Boolean Algebra & Karnaugh Maps Logic Gates Logic Gate Families Combinational and Sequential Digital Circuits AD/DA Convertor Semiconductor Memory Introduction to Microprocessors

422 – 424 425 – 430 431 – 435 436 – 438 439 – 456 457 – 462 463 – 464 465 – 472

473 – 504 Amplitude Modulation (AM) DSBSC, SSB and VSB Modulation Angle Modulation Receivers Noise in Analog Modulation Digital Communications

473 – 476 477 – 479 480 – 481 482 – 483 484 – 489 490 – 504

EDC 49 50 51 52 53

#9.

376 – 381 382 – 420 421

Communication 43 44 45 46 47 48

#8.

Feedback and Oscillator Circuits Operational Amplifiers and Its Applications Power Amplifiers

Digital Circuits 35 36 37 38 39 40 41 42

#7.

Contents

505 – 525 Semiconductor Theory P - N Junction Theory and Characteristics Transistor Theory (BJT, FET) Basics of IC Bipolar, MOS & CMOS Types Basics of OPTO Electronics

505 – 510 511 – 513 514 – 520 521 – 524 525

EMT 54 55 56 57 58

526 – 557 Electromagnetic Field EM Wave Propagation Transmission Lines Guided E.M Waves Antennas

526 – 536 537 – 543 544 – 550 551 – 554 555 – 557

th

th

th

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GATE QUESTION BANK

Mathematics

Linear Algebra ME – 2005 1. Which one of the following is an Eigenvector of the matrix[

(A) [

]

(B) [ ]

2.

5.

]?

(C) [

]

(D) [

]

A is a 3 4 real matrix and Ax=B is an inconsistent system of equations. The highest possible rank of A is (A) 1 (C) 3 (B) 2 (D) 4

ME – 2006 3. Multiplication of matrices E and F is G. Matrices E and G are os sin E [ sin ] and os G

4.

[

sin os

sin (B) [ os

os sin

]

os (C) [ sin

sin os

]

sin (D) [ os

os sin

7.

Eigenvectors of 0

1 is

(A) 0 (B) 1

(C) 2 (D) Infinite

If a square matrix A is real and symmetric, then the Eigenvalues (A) are always real (B) are always real and positive (C) are always real and non-negative (D) occur in complex conjugate pairs

]

ME – 2008 8.

The Eigenvectors of the matrix 0

1 are

written in the form 0 1 and 0 1. What is a + b? (A) 0 (B) 1/2

]

Eigen values of a matrix 0

ME – 2007 6. The number of linearly independent

]. What is the matrix F?

os (A) [ sin

S

Match the items in columns I and II. Column I Column II P. Singular 1. Determinant is not matrix defined Q. Non-square 2. Determinant is matrix always one R. Real 3. Determinant is symmetric zero matrix S. Orthogonal 4. Eigen values are matrix always real 5. Eigen values are not defined (A) P - 3 Q - 1 R - 4 S - 2 (B) P - 2 Q - 3 R - 4 S - 1 (C) P - 3 Q - 2 R - 5 S - 4 (D) P - 3 Q - 4 R - 2 S - 1

9.

(C) 1 (D) 2

The matrix [

] has one Eigenvalue p equal to 3. The sum of the other two Eigenvalues is (A) p (C) p – 2 (B) p – 1 (D) p – 3

1are 5 and 1. What are the

Eigenvalues of the matrix = SS? (A) 1 and 25 (C) 5 and 1 (B) 6 and 4 (D) 2 and 10 th

th

th

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GATE QUESTION BANK

10.

For what value of a, if any, will the following system of equations in x, y and z have a solution x y x y z x y z (A) Any real number (B) 0 (C) 1 (D) There is no such value

11.

ME – 2012 15.

For a matrix,M-

*

x

√

(B) (√ )

1 is

(A) 2 (B) 2 3

3

(C) 2 3 (D) 2

3

ME – 2011 13. Consider the following system equations: x x x x x x x The system has (A) A unique solution (B) No solution (C) Infinite number of solutions (D) Five solutions 14.

of

Eigen values of a real symmetric matrix are always (A) Positive (C) Negative (B) Real (D) Complex

(D) ( ) √

√

of the matrix is equal to the inverse of the ,M- . The value of x is matrix ,Mgiven by ) (A) ( (C) ⁄ ( ⁄ ) (B) (D) ⁄

0

1 , one of the

(C) (√ )

(A) (√ )

+, the transpose

ME – 2010 12. One of the Eigenvectors of the matrix

For the matrix A=0

normalized Eigenvectors is given as

16.

ME – 2009

Mathematics

x + 2y + z =4 2x + y + 2z =5 x–y+z=1 The system of algebraic equations given above has (A) a unique algebraic equation of x = 1, y = 1 and z = 1 (B) only the two solutions of ( x = 1, y = 1, z = 1) and ( x = 2, y = 1, z = 0) (C) infinite number of solutions. (D) No feasible solution.

ME – 2013 17. The Eigenvalues of a symmetric matrix are all (A) Complex with non –zero positive imaginary part. (B) Complex with non – zero negative imaginary part. (C) Real (D) Pure imaginary. 18.

Choose correct set of functions, which are linearly dependent. (A) sin x sin x n os x (B) os x sin x n t n x (C) os x sin x n os x (D) os x sin x n os x

ME – 2014 19. Given that the determinant of the matrix [

] is

12 , the determinant of

the matrix [ (A) th

] is (B)

th

(C) th

(D)

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GATE QUESTION BANK

20.

One of the Eigenvectors of the matrix 0

21.

22.

2.

Consider a non-homogeneous system of linear equations representing mathematically an over-determined system. Such a system will be (A) consistent having a unique solution (B) consistent having many solutions (C) inconsistent having a unique solution (D) inconsistent having no solution

3.

Consider the matrices , - . The order of , (

1 is

(A) {– }

(C) 2

(B) {– }

(D) 2 3

3

Consider a 3×3 real symmetric matrix S such that two of its Eigenvalues are with respective Eigenvectors x y [x ] [y ] If then x y + x y +x y x y equals (A) a (C) ab (B) b (D) 0 Which one of the following equations is a correct identity for arbitrary 3×3 real matrices P, Q and R? (A) ( ) ) (B) ( ( ) (C) et et et ) (D) (

CE – 2005 1. Consider the system of equations ( ) is s l r Let ( ) ( ) where ( ) e n Eigen -pair of an Eigenvalue and its corresponding Eigenvector for real matrix A. Let I be a (n × n) unit matrix. Which one of the following statement is NOT correct? (A) For a homogeneous n × n system of linear equations,(A ) X = 0 having a nontrivial solution the rank of (A ) is less than n. (B) For matrix , m being a positive integer, ( ) will be the Eigen pair for all i. (C) If = then | | = 1 for all i. (D) If = A then is real for all i.

Mathematics

,

-

,

-

and

- will be ) (C) (4 × 3) (D) (3 × 4

(A) (2 × 2) (B) (3 × 3

CE – 2006 4. Solution for the system defined by the set of equations 4y + 3z = 8; 2x – z = 2 and 3x + 2y = 5 is (A) x = 0; y =1; z = ⁄ (B) x = 0; y = ⁄ ; z = 2 (C) x = 1; y = ⁄ ; z = 2 (D) non – existent

5.

For the given matrix A = [

],

one of the Eigen values is 3. The other two Eigen values are (A) (C) (B) (D) CE – 2007 6. The minimum and the maximum Eigenvalue of the matrix [

]are 2

and 6, respectively. What is the other Eigenvalue? (A) (C) (B) (D) 7.

For what values of and the following simultaneous equations have an infinite of solutions? X + Y + Z = 5; X + 3Y + 3Z = 9; X+2Y+ Z (A) 2, 7 (C) 8, 3 (B) 3, 8 (D) 7, 2 th

th

th

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GATE QUESTION BANK

8.

The inverse of the (A)

0

(B)

0

1 1

m trix 0

(A) (B)

1

is

(D)

0

1

( )

0

( )

0

( )

0

( )

0

11.

is

15. (C) (D)

i

i

i

i i

i

i

i i

i

i

i i

i i

1

1

1

1

i

i

i i

i

1

CE – 2012

1 are and 8 and 5

The inverse of the matrix 0

0

The Eigenvalue of the matrix [P] = 0

14.

(C)

CE – 2008 9. The product of matrices ( ) (A) (C) (B) (D) PQ 10.

1 is

Mathematics

n n

The following simultaneous equation x+y+z=3 x + 2y + 3z = 4 x + 4y + kz = 6 will NOT have a unique solution for k equal to (A) 0 (C) 6 (B) 5 (D) 7

CE – 2009 12. A square matrix B is skew-symmetric if (C) (A) (D) (B) CE – 2011 13. [A] is square matrix which is neither symmetric nor skew-symmetric and , is its transpose. The sum and difference of these matrices are defined as [S] = [A] + , - and [D] = [A] , - , respectively. Which of the following statements is TRUE? (A) Both [S] and [D] are symmetric (B) Both [S] and [D] are skew-symmetric (C) [S] is skew-symmetric and [D] is symmetric (D) [S] is symmetric and [D] is skew symmetric

The Eigenvalues of matrix 0 (A) (B) (C) (D)

1 are

2.42 and 6.86 3.48 and 13.53 4.70 and 6.86 6.86 and 9.50

CE – 2013 16. There is no value of x that can simultaneously satisfy both the given equations. Therefore, find the ‘le st squares error’ solution to the two equations, i.e., find the value of x that minimizes the sum of squares of the errors in the two equations. 2x = 3 and 4x = 1 17.

What is the minimum number of multiplications involved in computing the matrix product PQR? Matrix P has 4 rows and 2 columns, matrix Q has 2 rows and 4 columns, and matrix R has 4 rows and 1 column. __________

CE – 2014 18.

Given the matrices J = [ K

19.

[

] n

], the product K JK is

The sum of Eigenvalues of the matrix, [M] is, where [M] = [

]

(A) 915 (B) 1355 th

th

(C) 1640 (D) 2180 th

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GATE QUESTION BANK

4. 20.

The determinant of matrix [

Let A be a 4x4 matrix with Eigenvalues –5, –2, 1, 4. Which of the following is an I Eigenvalue of 0 1, where I is the 4x4 I identity matrix? (A) (C) (B) (D)

]

is ____________ 21.

The

rank

[

of

the

matrix

] is ________________

CS – 2005 1. Consider the following system of equations in three real variables x x n x x x x x x x x x x This system of equation has (A) no solution (B) a unique solution (C) more than one but a finite number of solutions (D) an infinite number of solutions 2.

What are the Eigenvalues of the following 2 2 matrix? 0 (A) (B)

1 n n

(C) (D)

n n

CS – 2006 3. F is an n x n real matrix. b is an n real vector. Suppose there are two nx1 vectors, u and v such that u  v , and Fu=b, Fv=b. Which one of the following statement is false? (A) Determinant of F is zero (B) There are infinite number of solutions to Fx=b (C) There is an x  0 such that Fx=0 (D) F must have two identical rows

Mathematics

CS – 2007 5. Consider the set of (column) vectors defined by X={xR3 x1+x2+x3=0, where XT =[x1, x2, x3]T }. Which of the following is TRUE? (A) {[1, 1, 0]T, [1, 0, 1]T} is a basis for the subspace X. (B) {[1, 1, 0]T, [1, 0, 1]T} is a linearly independent set, but it does not span X and therefore, is not a basis of X. (C) X is not the subspace for R3 (D) None of the above CS – 2008 6. The following system of x x x x x x x x x Has unique solution. The only possible value (s) for is/ are (A) 0 (B) either 0 or 1 (C) one of 0,1, 1 (D) any real number except 5 7.

How many of the following matrices have an Eigenvalue 1? 0

1 0

1 n 0

1 0

(A) One (B) two

1

(C) three (D) four

CS – 2010 8. Consider the following matrix A=[

] x y If the Eigen values of A are 4 and 8, then (A) x = 4, y = 10 (C) x = 3, y = 9 (B) x = 5, y = 8 (D) x = 4, y = 10 th

th

th

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GATE QUESTION BANK

CS – 2011 9. Consider the matrix as given below [

13.

The value of the dot product of the Eigenvectors corresponding to any pair of different Eigenvalues of a 4-by-4 symmetric positive definite matrix is __________.

14.

If the matrix A is such that

]

Which one of the following options provides the CORRECT values of the Eigenvalues of the matrix? (A) 1, 4, 3 (C) 7, 3, 2 (B) 3, 7, 3 (D) 1, 2, 3

[

CS – 2013 11. Which one of x x equal [ y y z z x(x y(y (A) | z(z x (B) | y z x y (C) | y z z x y (D) | y z z

15.

-

The product of the non – zero Eigenvalues of the matrix

is __________. [ 16.

the following does NOT ] ) x ) y | ) z x | y z x y y z | z x y y z | z

],

Then the determinant of A is equal to __________.

CS – 2012 10. Let A be the 2

2 matrix with elements and . Then the Eigenvalues of the matrix are (A) 1024 and (B) 1024√ and √ (C) √ n √ (D) √ n √

Mathematics

]

Which one of the following statements is TRUE about every n n matrix with only real eigenvalues? (A) If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative. (B) If the trace of the matrix is positive, all its eigenvalues are positive. (C) If the determinant of the matrix is positive, all its eigenvalues are positive. (D) If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.

ECE – 2005 1. Given an orthogonal matrix

CS – 2014 12. Consider the following system of equations: x y x z x y z x y z The number of solutions for this system is __________.

A= [

]. ,

-

is

⁄ (A) [

⁄

]

⁄ ⁄

th

th

th

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GATE QUESTION BANK

Mathematics

⁄ ⁄

(B) [

6.

]

⁄

The rank of the matrix [

⁄ (C) [

(A) 0 (B) 1

] ⁄ ⁄

(D) [

]

⁄ ⁄

2.

Let,

A=0

1 and

Then (a + b)= (A) ⁄ (B) ⁄ 3.

= 0

1.

⁄ ⁄

(C) (D)

Given the matrix 0

⁄

Eigenvector is (C) 0

1

(B) 0 1

(D) 0

1

ECE – 2006 4.

For the matrix 0 corresponding 0

the

ECE – 2007 7. It is given that X1 , X2 …… M are M nonzero, orthogonal vectors. The dimension of the vector space spanned by the 2M vector X1 , X2 … XM , X1 , X2 … XM is (A) 2M (B) M+1 (C) M (D) dependent on the choice of X1 , X2 … XM.

9.

All the four entries of the 2 x 2 matrix p p P = 0p p 1 are non-zero, and one of its Eigenvalues is zero. Which of the following statements is true? (A) p p p p (B) p p p p (C) p p p p (D) p p p p

Eigenvector

1 is

(A) 2 (B) 4 5.

1 , the Eigenvalue to

(C) 6 (D) 8

The Eigenvalues and the corresponding Eigenvectors of a 2 2 matrix are given by Eigenvalue Eigenvector =8

v =0 1

=4

(C) 2 (D) 3

ECE – 2008 8. The system of linear equations 4x + 2y = 7, 2x + y = 6 has (A) a unique solution (B) no solution (C) an infinite number of solutions (D) exactly two distinct solutions

1 the

(A) 0 1

]

v =0

1

ECE – 2009 10. The Eigen values of the following matrix are [

The matrix is (A) 0

1

(C) 0

1

(B) 0

1

(D) 0

1

]

(A) 3, 3 + 5j, 6 j (B) 6 + 5j, 3 + j, 3 j (C) 3 + j, 3 j, 5 + j (D) 3, 1 + 3j, 1 3j

th

th

th

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GATE QUESTION BANK

ECE – 2010 11. The Eigenvalues of a skew-symmetric matrix are (A) Always zero (B) Always pure imaginary (C) Either zero or pure imaginary (D) Always real ECE – 2011 12. The system of equations x y z x y z x y z has NO solution for values of given by (A) (C) (B) (D)

Mathematics

ECE – 2014 16. For matrices of same dimension M, N and scalar c, which one of these properties DOES NOT ALWAYS hold? (A) (M ) M (M) (B) ( M ) (C) (M N) M N (D) MN NM 17.

A real (4 × 4) matrix A satisfies the equation I where 𝐼 is the (4 × 4) identity matrix. The positive Eigenvalue of A is _____.

18.

Consider the matrix

n

J ECE\EE\IN – 2012 13.

Given that A = 0

1 and I = 0

the value of A3 is (A) 15 A + 12 I (B) 19A + 30

(C) 17 A + 15 I (D) 17A +21

ECE – 2013 14. The minimum Eigenvalue of the following matrix is [

19.

The determinant of matrix A is 5 and the determinant of matrix B is 40. The determinant of matrix AB is ________.

20.

The system of linear equations

]

(A) 0 (B) 1 15.

[ ] Which is obtained by reversing the order of the columns of the identity matrix I . Let I J where is a nonnegative real number. The value of for which det(P) = 0 is _____.

1,

(C) 2 (D) 3

(

Let A be a m n matrix and B be a n m matrix. It is given that ) determinant Determinant(I (I ) where I is the k k identity matrix. Using the above property, the determinant of the matrix given below is

(A) 2 (B) 5

(

)h s

(A) a unique solution (B) infinitely many solutions (C) no solution (D) exactly two solutions 21.

[

)4 5

] (C) 8 (D) 16

th

Which one of the following statements is NOT true for a square matrix A? (A) If A is upper triangular, the Eigenvalues of A are the diagonal elements of it (B) If A is real symmetric, the Eigenvalues of A are always real and positive th

th

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GATE QUESTION BANK

(C) If A is real, the Eigenvalues of A and are always the same (D) If all the principal minors of A are positive, all the Eigenvalues of A are also positive 22.

The maximum value of the determinant among all 2×2 real symmetric matrices with trace 14 is ___.

EE – 2005 1.

5.

If R = [

] , then top row of

(A) , (B) ,

2.

-

(C) , (D) ,

(B) [

] [

] [

]

(C) [

] [

] [

]

(D) [

] [

] [

]

-

(A) [ ] (B) [

]

(C) [

(B) [

]

(D) [ ]

]

In the matrix equation Px = q, which of the following is necessary condition for the existence of at least one solution for the unknown vector x (A) Augmented matrix [P/Q] must have the same rank as matrix P (B) Vector q must have only non-zero elements (C) Matrix P must be singular (D) Matrix P must be square

] ,R=[

(C) [ ] ]

(D) [

]

EE – 2007 6. X = [x , x . . . . x - is an n-tuple non-zero vector. The n n matrix V = X (A) Has rank zero (C) Is orthogonal (B) Has rank 1 (D) Has rank n 7.

The linear operation L(x) is defined by the cross product L(x) = b x, where b =[0 1 0- and x =[x x x - are three dimensional vectors. The matrix M of this operation satisfies x L(x) = M [ x ] x Then the Eigenvalues of M are (A) 0, +1, 1 (C) i, i, 1 (B) 1, 1, 1 (D) i, i, 0

8.

Let x and y be two vectors in a 3 dimensional space and denote their dot product. Then the determinant xx xy det 0 y x yy 1 (A) is zero when x and y are linearly independent (B) is positive when x and y are linearly independent (C) is non-zero for all non-zero x and y (D) is zero only when either x or y is zero

EE – 2006 Statement for Linked Answer Questions 4 and 5.

4.

]

is

] , one of

(A) [

] ,Q=[

] [

-

For the matrix p = [

P=[

(A) [

The following vector is linearly dependent upon the solution to the previous problem

the Eigenvalues is equal to 2 . Which of the following is an Eigenvector?

3.

Mathematics

] are

three vectors An orthogonal set of vectors having a span that contains P,Q, R is th

th

th

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GATE QUESTION BANK

Statement for Linked Questions 9 and 10. Cayley-Hamilton Theorem states that a square matrix satisfies its own characteristic equation. Consider a matrix. A=0

A satisfies the relation (A) A + 3 + 2 =0 2 (B) A + 2A + 2 = 0 (C) (A+ ) (A 2) = 0 (D) exp (A) = 0

10.

equals (A) 511 A + 510  (B) 309 A + 104  (C) 154 A + 155  (D) exp (9A)

EE – 2008 11. If the rank of a ( ) matrix Q is 4, then which one of the following statements is correct? (A) Q will have four linearly independent rows and four linearly independent columns (B) Q will have four linearly independent rows and five linearly independent columns (C) Q will be invertible (D) Q will be invertible 12.

13.

(A) A A+ A = A (B) (AA+ ) = A A+ 14.

The characteristic equation of a ( ) matrix P is defined as () = | P| =    =0 If I denotes identity matrix, then the inverse of matrix P will be (A) ( I) (B) ( I) (C) ( I) (D) ( I)

(C) A+ A =  (D) A A+ A = A+

Let P be a real orthogonal matrix. x⃗ is a real vector [x x - with length ⃗x (x x ) . Then, which one of the following statements is correct? (A) x⃗ x⃗ where at least one vector satisfies x⃗ x⃗ (B) x⃗ x⃗ for all vectors x⃗ (C) x⃗ x⃗ where at least one vector satisfies x⃗ x⃗ (D) No relationship can be established between x⃗ and x⃗

1

9.

Mathematics

EE – 2009 15. The trace and determinant of a matrix are known to be –2 and –35 respe tively It’s Eigenv lues re (A) –30 and –5 (C) –7 and 5 (B) –37 and –1 (D) 17.5 and –2 EE – 2010 16. For the set of equations x x x x =2 x x x x =6 The following statement is true (A) Only the trivial solution x x x x = 0 exists (B) There are no solutions (C) A unique non-trivial solution exists (D) Multiple non-trivial solutions exist

17.

An Eigenvector of

[

(A) , (B) ,

(C) , (D) ,

-

] is -

EE – 2011 18.

The matrix[A] = 0

1 is decomposed

into a product of a lower triangular matrix [L] and an upper triangular matrix [U]. The properly decomposed [L] and [U] matrices respectively are

A is m n full rank matrix with m > n and  is an identity matrix. Let matrix A+ = ( ) , then, which one of the following statements is FALSE?

(A) 0 th

th

1 and 0

1 th

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GATE QUESTION BANK

(B) 0 (C) 0 (D) 0

1 and 0 1 and 0 1 and 0

23.

1 1 1

EE – 2013 19.

The equation 0

x 1 0x 1

0 1 has

0 1.

Eigenvector of the matrix A = 0

(C) Non – zero unique solution (D) Multiple solution 20.

(A) [ 1 1]T (B) [3 1]T

A matrix has Eigenvalues – 1 and – 2. The corresponding Eigenvectors are 0 0

1 respectively. The matrix is

(A) 0

1

(C) 0

1

(B) 0

1

(D) 0

1

Which one of the following statements is true for all real symmetric matrices? (A) All the eigenvalues are real. (B) All the eigenvalues are positive. (C) All the eigenvalues are distinct. (D) Sum of all the eigenvalues is zero.

1?

(C) [1 1]T (D) [ 2 1]T

Let A be a 3 3 matrix with rank 2. Then AX = 0 has (A) only the trivial solution X = 0 (B) one independent solution (C) two independent solutions (D) three independent solutions

2.

1 and

EE – 2014 21. Given a system of equations: x y z x y z Which of the following is true regarding its solutions? (A) The system has a unique solution for any given and (B) The system will have infinitely many solutions for any given and (C) Whether or not a solution exists depends on the given and (D) The system would have no solution for any values of and 22.

Two matrices A and B are given below: p q pr qs p q [ ] 0 1 r s pr qs r s If the rank of matrix A is N, then the rank of matrix B is (A) N (C) N (B) N (D) N

IN – 2005 1. Identify which one of the following is an

(A) No solution x (B) Only one solution 0x 1

Mathematics

IN – 2006 Statement for Linked Answer Questions 3 and 4 A system of linear simultaneous equations is given as Ax=B where [

] n

[ ]

3.

The rank of matrix A is (A) 1 (C) 3 (B) 2 (D) 4

4.

Which of the following statements is true? (A) x is a null vector (B) x is unique (C) x does not exist (D) x has infinitely many values

5.

For a given that 0

1

matrix A, it is observed 0

1 n

0

1

0

1

Then matrix A is

th

th

th

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GATE QUESTION BANK

 2 1  1 0   1 1     1 1  0 2 1 2

10.

(A) A  

1

1  1 0  2

1

1  1

1

(B) A      1 2 0 2 1 1 (C) A    1 2  0

02 1 2 1 1

0 2

(D) A    1 3 IN – 2007 6. Let A = [ ] i j n with n = i. j. Then the rank of A is (A) (C) n (B) (D) n 7.

n

Let A be an n×n real matrix such that = I and y be an n- dimensional vector. Then the linear system of equations Ax=Y has (A) no solution (B) a unique solution (C) more than one but finitely many independent solutions (D) Infinitely many independent solutions

The matrix P =[

12.

9.

The Eigenvalues of a (2 2) matrix X are 2 and 3. The Eigenvalues of matrix ( I) ( I) are (A) (C) (B) (D)

(

)(

)

(D) n IN – 2011 13.

The matrix M = [

] has

Eigenvalues . An Eigenvector corresponding to the Eigenvalue 5 is , - . One of the Eigenvectors of the matrix M is (A) , (C) , √ (B) , (D) ,

] rotates a vector

(C) (D)

A real n × n matrix A = [ ] is defined as i i j follows: { otherwise The summation of all n Eigenvalues of A is (A) n(n ) (B) n(n ) (C)

about the axis[ ] by an angle of (A) (B)

Let P 0 be a 3 3 real matrix. There exist linearly independent vectors x and y such that Px = 0 and Py = 0. The dimension of the range space of P is (A) 0 (C) 2 (B) 1 (D) 3

IN – 2010 11. X and Y are non-zero square matrices of size n n. If then (A) |X| = 0 and |Y| 0 (B) |X| 0 and |Y| = 0 (C) |X| = 0 and |Y| = 0 (D) |X| 0 and |Y| 0

IN – 2009 8.

Mathematics

IN – 2013 14. The dimension of the null space of the

15.

matrix [

] is

(A) 0 (B) 1

(C) 2 (D) 3

One of Eigenvectors corresponding to the two Eigenvalues of the matrix 0 (A) [

j

] 0

(B) 0 1 0 th

th

j

1 is

(C) [ ] 0 1 j j (D) [ ] 0 1 j

1 1 th

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GATE QUESTION BANK

Mathematics

IN – 2014 16. For the matrix A satisfying the equation given below, the eigenvalues are , -[

]

[

(A) ( 𝑗,𝑗) (B) (1,1,0)

] (C) ( ) (D) (1,0,0)

Answer Keys and Explanations ME 1.

and G = [

[Ans. A] [

Now E × F = G

]

h r teristi equ tions is | I| ( )( )( ) ∴ Real eigenvalues are 5, 5 other two are complex Eigenvector corresponding to is ( I) (or) →( ) Verify the options which satisfies relation (1) Option (A) satisfies. [Ans. B] Given

n

in onsistent

4.

5.

[Ans. A]

6.

[Ans. B] 1 Eigenv lues re 2, 2 I)

(

I)

No. of L.I Eigenvectors ( (no of v ri les)

( ⁄ )

7.

matrix be A = 0 sin os

.

/ I)

[Ans. A] ( I) . olving for , Let the symmetric and real

[Ans. C] os Given , E = [ sin

]

matrix, if Eigenvalues are … … … … … then for matrix, the Eigenvalues will be , , ……… For S matrix, if Eigenvalues are 1 and 5 then for matrix, the Eigenvalues are 1 and 25.

No (

3.

sin os

[Ans. A] For S

0

( ) n ( ⁄ ) ( ( ) minimum of m n) For inconsistence ( ) ( ⁄ ) ∴ he highest possi le r nk of is

os [ sin

,E-

∴

2.

]

]

th

1

Now |

|

Which gives ( ⟹

)

th

th

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GATE QUESTION BANK

⟹ Hence real Eigen value. 8.

Mathematics

x [

][

]

0

1

x

[Ans. B] Let

0

eigenv lues re

1

12.

Eigen vector corresponding to is ( I) x . / .y/ . / By simplifying K . / . / y t king K

⁄

Equating the elements x

n

[Ans. A] 0

1 → Eigenv lues re

Eigenve tor is x 13.

Eigen vector corresponding to =2 is ( I) x . / .y/ . / K By simplifying ( ) 4 5 by ⁄ K

[Ans. C] [

]

[

→

[

]

→

taking K

x verify the options

( )

[

]

]

infinite m ny solutions

⁄ ⁄ 9.

10.

[Ans. C] Sum of the diagonal elements = Sum of the Eigenvalues ⟹ 1 + 0 + p = 3+S ⟹ S= p 2

[Ans. B] Eigenvalues of a real symmetric matrix are always real

15.

[Ans. B] 0

If

1 eigenv lues v lue

Eigen vector will be .

/

Norm lize ve tor

[Ans. B] ( ⁄ )

11.

14.

[

]

√( )

(

)

[

]

[√( )

(

) ]

→ →

[

→

[

*

]

]

16.

system will h ve solution

[Ans. A] iven M

M

→ MM

I

th

⁄ √ + ⁄ √

[Ans. C] The given system is x y z x y z x y z Use Gauss elimination method as follows Augmented matrix is

th

th

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GATE QUESTION BANK

, | -

[

→

→

| ] [

So, |

|

[

|

|

[Ans. C] Suppose the Eigenvalue of matrix A is ( i )(s y) and the Eigenvector is ‘x’ where s the onjug te p ir of Eigenvalue and Eigenvector is ̅ n x̅. So Ax = x … ① and x̅ ̅x̅……② king tr nspose of equ tion ② x̅ x̅ ̅ … ③ [( ) n ̅ is s l r ] ̅ x̅ x x̅ x x̅ x x̅ ̅x … , x̅ x x̅ ̅ x ̅ (x̅ x) ( ̅ re s l r ) (x̅ x) ̅

20.

[Ans. C] We know that os x os x sin x ( ) os x sin x ( ) os x Hence 1, 1 and 1 are coefficients. They are linearly dependent.

1 eigen v lues

Eigenve tor is

verify for oth n

21.

[Ans. D] We know that the Eigenvectors corresponding to distinct Eigenvalues of real symmetric matrix are orthogonal. x y x y [ ][ ] x y x y x y y x

22.

[Ans. D] ( ) In case of matrix PQ

CE 1.

QP (generally)

[Ans. C] If = i.e. A is orthogonal, we can only s y th t if is n Eigenv lue of then

also will be an Eigenvalue of A,

which does not necessarily imply that | | = 1 for all i. 2.

[Ans. A] In an over determined system having more equations than variables, it is necessary to have consistent unique solution, by definition

3.

[Ans. A] With the given order we can say that order of matrices are as follows:  3×4 Y  4×3  3×3

[Ans. A] |

[Ans. D] 0

nnot e zero )

Hence Eigenvalue of a symmetric matrix are real

19.

|

(Taking 2 common from each row) ( )

]

( x x̅ re Eigenve tors they i i i 0

18.

|

]

nk ( ) nk ( | ) So, Rank (A) = Rank (A|B) = 2 < n (no. of variables) So, we have infinite number of solutions 17.

Mathematics

|

th

th

th

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GATE QUESTION BANK

( )  3×3 P 2×3  3×2 P( )  (2×3) (3×3) (3×2) 2×2 ( ( ) )  2×2

Using Gauss elimination we reduce this to an upper triangular matrix to find its rank | ]→

[

→ 4.

[Ans. D] The augmented matrix for given system is [

| ]→

[

| ]→

| ]

[

8.

| ]

[

[

|

[

|

]

( ⁄ ) ( ) ( ) ( ⁄ ) ∴ olution is non – existent for above system. 5.

6.

7.

[Ans. B] ∑ = Trace (A) + + = Trace (A) = 2 + ( 1) + 0 = 1 Now = 3 ∴3+ + =1  Only choice (B) satisfies this condition. [Ans. B] ∑ = Trace (A) + + =1+5+1=7 Now = 2, = 6 ∴ 2+6+ =7 =3

0

1

∴0

]

1 is (

1

)

(

) 0

9.

10.

0

1

0

1 1

[Ans. B] ( ) P=( ( )( ) =( ) (I) =

)P

[Ans. B] A=0

1

Characteristic equation of A is |

|=0

 (4 )( 5 ) 2 × 5 =0  + 30 = 0 6, 5 11.

[Ans. A] The augmented matrix for given system is [

]

[Ans. A] Inverse of 0

→

|

Now for infinite solution last row must be completely zero ie –2=0 n –7=0  n

Then by Gauss elimination procedure [

Mathematics

| ]

th

[Ans. D] The augmented matrix for given system is x [ | ] 6y7 [ ] z k Using Gauss elimination we reduce this to an upper triangular matrix to find its rank

th

th

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GATE QUESTION BANK

| ]→

[

17.

[Ans. 16] , , M trix , The product of matrix PQR is , - , - , The minimum number of multiplications involves in computing the matrix product PQR is 16

18.

[Ans. 23]

k [

| ]

[

| ]

→

Now if k Rank (A) = rank (A|B) = 3 ∴ Unique solution If k = 7, rank (A) = rank (A|B) = 2 which is less than number of variables ∴ When K = 7, unique solution is not possible and only infinite solution is possible 12.

[Ans. A] A square matrix B is defined as skewsymmetric if and only if = B

13.

[Ans. D] By definition A + is always symmetric is symmetri is lw ys skew symmetri is skew symmetri

Mathematics

[

][

]

[

,

K JK

]

-[ ,

]

,

[

] -

-

19.

[Ans. A] Sum of Eigenvalues = Sum of trace/main diagonal elements = 215 + 150 + 550 = 915

20.

[Ans. 88] The determinant of matrix is [

]

→

14.

[Ans. B] 1 =(

0

i

∴ 0

15.

i

i

i

,( =

0

0

)

i)( i i

[

1

→

1 i -

i) i i

0

i i

i i

1

[

1

[

]

1 Interchanging Column 1& Column 2 and taking transpose

Sum of the Eigenvalues = 17 Product of the Eigenvalues = From options, 3.48 + 13.53 = 17 (3.48)(13.53) = 47 16.

]

→

[Ans. B] 0

]

[

[Ans. 0.5] 0.5

]

|

th

th

|

th

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GATE QUESTION BANK * (

)

= ( 21.

(

(

)+

= 1, 6 ∴ The Eigenvalues of A are 1 and 6

[Ans. 2] ]

3.

[Ans. D] Given that Fu =b and Fv =b If F is non singular, then it has a unique inverse. Now, u = b and v= b Since is unique, u = v but it is given th t u v his is contradiction. So F must be singular. This means that (A) Determinant of F is zero is true. Also (B) There are infinite number of solution to Fx= b is true since |F| = 0 (C) here is n su h the is also true, since X has infinite number of solutions., including the X = 0 solution (D) F must have 2 identical rows is false, since a determinant may become zero, even if two identical columns are present. It is not necessary that 2 identical rows must be present for |F| to become zero.

4.

[Ans. C] It is given that Eigenvalues of A is 5, 2, 1, 4 I Let P = 0 1 I Eigenvalues of P : | I| I | | I ( ) I I I Eigenvalue of P is ( 5 +1 ), ( 2+ 1), (1+ 1), (4+1 ), ( 5 1 ), ( 2 1 ),(1 1), (4 1) = 4, 1, 2, 5, 6, 3,0,3

5.

[Ans. B] |x X= {x x x = ,x x x - then,

→ [ ( )

(

)

( ) ]

( )

( )

[

] ( )

no. of non zero rows = 2

[Ans. B] The augmented matrix for the given system is [

| ]

Using elementary transformation on above matrix we get, [

| ]

→

⁄ | ] ⁄ ⁄

[

→

[

|

]

Rank ([A B]) = 3 Rank ([A]) = 3 Since Rank ([A B]) = Rank ([A]) = number of variables, the system has unique solution. 2.

[Ans. B] 0

1

The characteristic equation of this matrix is given by | I| |

)

)

[

CS 1.

)(

Mathematics

+

| th

th

th

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GATE QUESTION BANK

{ [1, 1, 0]T , [1,0, 1 ]T } is a linearly independent set because one cannot be obtained from another by scalar multiplication. However (1, 1, 0) and (1,0, 1) do not span X, since all such combinations (x1, x2, x3) such that x1+ x2+ x3 =0 cannot be expressed as linear combination of (1, 1,0) and (1,0, 1) 6.

7.

Only one matrix has an Eigenvalue of 1 which is 0

| ] →

→

[

[

1

Correct choice is (A) 8.

[Ans. D] |

| x y ( )( y) When ( y) x y x When ( y) x y x x y Solving (1) & (2) x y

[Ans. D] The augmented matrix for above system is [

Mathematics

| ] | ]

x

( )

( )

Now as long as – 5 0, rank (A) =rank (A|B) =3 ∴ can be any real value except 5. Closest correct answer is (D).

9.

[Ans. A] The Eigenvalues of a upper triangular matrix are given by its diagonal entries. ∴ Eigenvalues are 1, 4, 3 only

[Ans. A]

10.

[Ans. D]

Eigenvalues of 0 |

1

0

| =0

Eigenvalues of 0 |

Eigenvalues of the matrix (A) are the roots of the characteristic polynomial given below.

=0,1 1

|

| =0 =0

1

|

(√ )

)( ) =0 = –1, 1

n

√

n ( √ ) n

1

n

| =0

( (

) )

√ Eigenvalues of A are √ respectively So Eigenvalues of

) =0 ) = i or 1 = 1 –i or 1 + i

Eigenvalues of 0

)( )(

(

|= 0

( (

|

(

= 0, 0

Eigenvalues of 0 |

1

√

) =0

th

th

n

√

th

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GATE QUESTION BANK

11.

12.

[Ans. A] → p q nd Since 2 & 3rd columns have been swapped which introduces a –ve sign Hence (A) is not equal to the problem

[

]

→

[

] →

→

[

] →

16.

[

]

( ) ( ) no of v ri ∴ nique solution exists

14.

[ ] x x Let X = x e eigen ve tor x [x ] By the definition of eigenvector, AX = x x x x x x x x [ ] [x ] [x ] x x x x x x x x x x x x x x x x x x x x x x n x x x x x x (I) If s yx x x x x x x x x x (2) If Eigenv lue ∴ Three distinct eigenvalues are 0, 2, 3 Product of non zero eigenvalues = 2 × 3 = 6

]

→ →

les

[Ans. 0] The Eigenvectors corresponding to distinct Eigenvalues of real symmetric matrix are orthogonal

ECE 1.

2.

[Ans. A] If the trace or determinant of matrix is positive then it is not necessary that all eigenvalues are positive. So, option (B), (C), (D) are not correct

[Ans. C] Since, ,

] (

=I

16

0

[

-

[Ans. A] We know,

[Ans. 0]

| |

[Ans. 6] Let A =

[Ans. 1] x y x z x y z x y z ugmente m trix is [

13.

15.

Mathematics

0

7=0 1

0

1 1

1 b ,  a 60 10 1 1 21 7   a+b =   3 60 60 20

)

Or 2a 0.1b=0,  2a 

th

th

th

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GATE QUESTION BANK

3.

[Ans. C]

8.

0

[Ans. B] Approach 1: Given 4x + 2y =7 and 2x + y =6

1

 (A I)=0  ( 4 ) (3 ) 2 4=0  2 + 20=0  = 5, 4

4 2 x  7 2 1y   6      0 0 x  5 2 1y   6      

 x1   x2 

Putting = 5, 0

1   =0

 x + 2x = 0 x = 2x

On comparing LHS and RHS 0= 5, which is irrelevant and so no solution. Approach 2: 4x + 2y =7

x x  1= 2 2 1 Hence, 0 4.

1 is Eigenvector.

[Ans. C]

Then Eigenvector is x Verify the options (C) 5.

or 2x  y=

1 We know th t it is Eigenvalue

0

We know

0

1

|I A|=0

|

|

 2 –I2 +32 =0  = 4, 8 (Eigenvalues) For

= 4, ( I

)=0

1

)=0

1

9.

[Ans. C] Matrix will be singular if any of the Eigenvalues are zero. |  |= 0 For = 0, P = 0 p p  |p p | =0  p p p p

10.

[Ans. D] Approach1: Eigenvalues exists as complex conjugate or real Approach 2: Eigenvalues are given by

v =0 1 For

= 8, ( I

v =0 6.

1

[Ans. C] [

] [

|

]

[Ans. C] There are M non-zero, orthogonal vectors, so there is required M dimension to represent them ’

| =0

( 

( ) 7.

7 2

2x+y=6 Since both the linear equation represent parallel set of straight lines, therefore no solution exists. Approach 3: Rank (A)=1; rank (C)=2, As Rank (A) rank (C) therefore no solution exists.

x

[Ans. A] or m trix

Mathematics

11.

th

)(( ,

)=0

) j

j

[Ans. C] Eigenvalue of skew – symmetric matrix is either zero or pure imaginary. th

th

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GATE QUESTION BANK

12.

13.

[Ans. B] Given equations are x y z x y z and x y z If and , then x y z have Infinite solution If and , then x y z ( ) no solution x y z If n x y z will have solution x y z and will also give solution

et of , -

et of [

]

16.

[Ans. D] Matrix multiplication is not commutative in general.

17.

[Ans. *] Range 0.99 to 1.01 Let ‘ ’ e Eigenv lue of ‘ ’ hen ‘ e Eigenv lue of ‘ ’ A. =I= Using Cauchey Hamilton Theorem,

[Ans. B] 0

Mathematics

’ will

1

Characteristic Equations is 18. By Cayley Hamilton theorem I ∴ ( I) I 14.

I | | [

[Ans. A] [

]

→

(

[

[Ans. *] Range 199 to 201 From matrix properties we know that the determinant of the product is equal to the product of the determinants. That is if A and B are two matrix with determinant | | n | | respectively, then | | | | | | ∴| | | | | |

20.

[Ans. B]

) ]

]

19.

| |

| | Product of Eigenvalues = 0 ∴ Minimum Eigenv lue h s to e ‘ ’ 15.

[Ans. *] Range 0.99 to 1.01 I J I J

[Ans. B] ,

Let

-

[ ]

[

I

I

[

Then AB = [4]; BA Here m = 1, n = 4 ) And et(I

]

→

[

→ →

[

[

]

th

les

[Ans. B] onsi er

)

]

]

( ) ( | ) no of v r Infinitely many solutions 21.

et(I

]

th

0

1 th

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GATE QUESTION BANK

whi h is re l symmetri m trix h r teristi equ tion is | I| ( ) ∴ (not positive) ( ) is not true (A), (C), (D) are true using properties of Eigenvalues 22.

EE 1.

2.

[Ans. B] ] j( ) | |

=[

]

∴ Top row of

=,

-

[Ans. D] Since matrix is triangular, the Eigenvalues are the diagonal elements themselves namely  = 3, 2 & 1. Corresponding to Eigenvalue  = 2, let us find the Eigenvector [A - ] x̂ = 0 x [ ][x ] [ ] x Putting in above equation we get, x [ ][x ] [ ] x Which gives the equations, 5x x x =0 . . . . . (i) x =0 . . . . . (ii) 3x = 0 . . . . . (iii) Since eqa (ii) and (iii) are same we have 5x x x =0 . . . . . (i) x =0 . . . . . (ii) Putting x = k, we get x = 0, x = k and 5x k =0

[Ans. *] Range 48.9 to 49.1 Real symmetric matrices are diagnosable Let the matrix be x 0 1 s tr e is x So determinant is product of diagonal entries So | | x x ∴ M ximum v lue of etermin nt x x ∴| |

R= [

Mathematics

, of tor( )| |

x = k | |=|

|

∴ Eigenvectorss are of the form x k x [ ] * k + x

= 1(2 + 3) – 0(4 + 2) – 1 (6 – 2) = 1 Since we need only the top row of , we need to find only first column of (R) which after transpose will become first row adj(A). cof. (1, 1) = + |

|=2+3=5

cof. (2, 1) =

|= 3

|

cof. (2, 1) = + |

i.e. x x x = k : k : 0 = :1:0 =2:5:0 x x ∴ [ ]=[ ] is an Eigenvector of matrix p. x

|= +1 3.

∴ cof. (A) = [

[Ans. A] Rank [P|Q] = Rank [P] is necessary for existence of at least one solution to x q.

]

Adj (A) =, of ( )=[

]

Dividing by |R| = 1 gives th

th

th

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GATE QUESTION BANK

4.

  (

[Ans. A] We need to find orthogonal vectors, verify the options. Option (A) is orthogonal vectors (

)(

[Ans. B] The vector ( ) is linearly dependent upon the solution obtained in - and , Q. No. 4 namely , We can easily verify the linearly dependence as |

6.

7.

i

[Ans. B] xy xx | yx

xy xx x n xy yx xy x xy y y | |y x y | (x y) x y = Positive when x and y are linearly independent.

Option (B), (C), (D) are not orthogonal 5.

) i

8.

)

Mathematics

9.

[Ans. A] A=0

1

|A – | = 0  |

|

[Ans. B] hen n n m trix xx x x x x x x x x x x x x * + x x x x x x Take x common from 1st row, x common from 2nd row …… x common from nth row. It h s r nk ‘ ’

| =0  A will satisfy this equation according to Cayley Hamilton theorem i.e. I=0 Multiplying by on oth si es we get I=0 I =0 10.

[Ans. A] To calculate Start from derived above

I = 0 which has I

[Ans. D] ⃗ k L(x) = |

(

| x

x

= (x )

I)(

x (

⃗( k

)

(

x )

I) I

x = x

⃗ =[ x k

(

x L(x) = M [x ] x Comparing both , we get,

(

|

)

I

I) (

I) I

| (

I)

I) I

(

Hence Eigenvalue of M : | M



I)( I

]

|

I

] x

M=[

I) I

(

) th

th

th

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GATE QUESTION BANK

11.

12.

13.

[Ans. A] If rank of (5 6 ) matrix is 4,then surely it must have exactly 4 linearly independent rows as well as 4 linearly independent columns.

= A is correct =A[( ) -A = A[( ) Put =P Then A [ ] = A. = A Choice (C) =  is also correct since =( ) = I 14.

os

x in )

|| x⃗ || = √x

(x in

x

x

→

[Ans. C] Trace = Sum of Principle diagonal elements.

16.

[Ans. D] On writing the equation in the form of AX =B

+

, *

+

nk ( ) nk( ) Number of variables = 4 Since, Rank (A) = Rank(C) < Number of variables Hence, system of equations are consistent and there is multiple non-trivial solution exists. 17.

[Ans. B] Characteristic equation | |

I|

|

 (1 ) ( )( )  Eigenve tors orrespon ing to ( I) x [ ] [x ] [ ] x 2x x  x x At x x x x  x x At x ,x

is

Eigenvectors = c[ ]{Here c is a constant}

os ) 18.

[Ans. D] , - ,L-, - ⟹ Options D is correct

19.

[Ans. D] x x … (i) } (i) n (ii) re s me x x … (ii) ∴x x So it has multiple solutions.

|| x⃗ || = || x̅|| for any vector x̅ 15.

* +

Argument matrix C =*

[Ans. B] Let orthogonal matrix be os in P=0 1 in os By Property of orthogonal matrix A I x os x in So, x⃗ = [ ] x in x os || x⃗ || = √(x

x x + *x + x

*

[Ans. D] If characteristic equation is    =0 Then by Cayley – Hamilton theorem, I=0 = Multiplying by on both sides, = I = ( I) [Ans. D] Choice (A) Since

Mathematics

th

th

th

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GATE QUESTION BANK

20.

[Ans. D] Eigen value

|A

Eigenvectors 0

1 n 0

Let matrix 0 x

1

x 10

1

0

1

0

10

1

0

1

I|= |

|

i.e., (1 ) (2 ) 2 Thus the Eigenvalue are 1, 2. If x, y, be the component of Eigenvectors corresponding to the Eigenv lues we have x [A- I- 0 1 0y1=0

1

0

Mathematics

For =1, we get the Eigenvector as 0 Hence, the answer will be ,

21.

22.

23.

IN 1.

1

0

[Ans. B] AX=0 and (A) = 2 n=3 No. of linearly independent solutions = n r = 3 =1

3.

[Ans. C] There are 3 non-zero rows and hence rank (A) = 3

4.

[Ans. C] Rank (A) = 3 (This is Co-efficient matrix) Rank (A:b) =4(This is Augmented matrix) s r nk( ) r nk ( ) olution oes not exist.

5.

[Ans. C] We know Hen e from the given problem, Eigenvalue & Eigenvector is known.

1

[Ans. B] Since there are 2 equations and 3 variables (unknowns), there will be infinitely many solutions. If if then x y z x y z x z y For any x and z, there will be a value of y. ∴ Infinitely many solutions [Ans. A] For all real symmetric matrices, the Eigenvalues are real (property), they may be either ve or ve and also may be same. The sum of Eigenvalues necessarily not be zero. [Ans. C] p q 0 1 r s ( pplying → p q →r s element ry tr nsform tions) p q pr qs [ ] pr qs r s ∴ hey h ve s me r nk N

1 X1    , X2  1

1 2 , 1  1, 2  2  

We also know that

, where

1 1

P  X1 X2     1 2

1 0  1 0     0 2   0 2

[Ans. B] Given:

-

2. Solving 0

1

& D=  0

1 Hence

Characteristic equation is,

th

th

th

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GATE QUESTION BANK

 1 1  1 0   2 1  A       1 2  0 2 1 1 6.

12.

[Ans. B] A= [

]=[

A=[

[Ans. B] Given I Hence rank (A) = n Hence AX= Y will have unique solution

8.

[Ans. C]

9.

[Ans. C] Approach 1:

13.

14.

Assume,

0

(

∴A

(

0

1

0

10

Now | I

0

[Ans. B] Dim of null space [A]= nullity of A.

0

[

]

1

| )(

]

Apply row operations 1

1

- is also vector

For given A = [

1

I) 0

[Ans. B] If AX = → From this result [1, 2, for M

|

| (

I

1

I)

]

n For diagonal matrix Eigenvalues are diagonal elements itself. n(n ) ∴ n

]

Hence, rank (A) =1 7.

[Ans. A] A=[ ] i if i j = 0 otherwise. For n n matrix

]

Using elementary transformation [

Mathematics

)=0

[Ans. D]

11.

[Ans. C] A null matrix can be obtained by multiplying either with one null matrix or two singular matrices.

[

→

[

] ]

∴ ( ) By rank – nullity theorem Rank [A]+ nullity [A]= no. of columns[A] Nullity [A]= 3 ∴ Nullity , -

Approach 2: Eigenvalues of ( I) is = 1, 1/2 Eigenvalues of (X+5I) is = 3, 2 Eigenvalues of ( I) (X+5I) is = , 10.

→

15.

[Ans. A] A=|

|

Characteristics equation | |

I|

| j j

th

th

th

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GATE QUESTION BANK

j

[

j

x ] 0x 1

Mathematics

0 1

x x

j j

[

j j

x ] 0x 1

x

0 1

j

x 16.

[Ans. C]

A[

]=[

→| | |

|

] |

|

→| | (

|

|

|

| two rows ounter lose thus | |

| |) =Product of eigenvalues Verify options Options (C) correct answer

th

th

th

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GATE QUESTION BANK

Mathematics

Probability and Distribution ME - 2005 1. A single die is thrown twice. What is the probability that the sum is neither 8 nor 9? (A) ⁄ (C) ⁄ ⁄ (B) ⁄ (D)

ME - 2008 6. A coin is tossed 4 times. What is the probability of getting heads exactly 3 times? (A) ⁄ (C) ⁄ (B) ⁄ (D) ⁄

2.

ME - 2009 7. The standard deviation of a uniformly distributed random variable between 0 and 1 is (A) (C) ⁄√ √ (B) (D) √ √

A lot has 10% defective items. Ten items are chosen randomly from this lot. The probability that exactly 2 of the chosen items are defective is (A) 0.0036 (C) 0.2234 (B) 0.1937 (D) 0.3874

ME - 2006 3. Consider a continuous random variable with probability density function f(t) = 1 + t for 1  t  0 = 1 t for 0  t  1 The standard deviation of the random variable is: (C) ⁄ (A) ⁄√ (D) ⁄ (B) ⁄√ 4.

A box contains 20 defective items and 80 non-defective items. If two items are selected at random without replacement, what will be the probability that both items are defective? ⁄ (A) ⁄ (C) ⁄ ⁄ (B) (D)

ME - 2007 5. Let X and Y be two independent random variables. Which one of the relations between expectation (E), variance (Var) and covariance (Cov) given below is FALSE? (A) E (XY) = E (X) E (Y) (B) Cov (X, Y) = 0 (C) Var (X + Y) = Var (X) + Var (Y) (D)

(X Y )

( (X)) ( (Y))

8.

If three coins are tossed simultaneously, the probability of getting at least one head is (A) 1/8 (C) 1/2 (B) 3/8 (D) 7/8

ME - 2010 9. A box contains 2 washers, 3 nuts and 4 bolts. Items are drawn from the box at random one at a time without replacement. The probability of drawing 2 washers first followed by 3 nuts and subsequently the 4 bolts is (A) 2/315 (C) 1/1260 (B) 1/630 (D) 1/2520 ME - 2011 10. An unbiased coin is tossed five times. The outcome of each toss is either a head or a tail. The probability of getting at least one head is________ ⁄ (A) ⁄ (C) ⁄ ⁄ (B) (D) ME - 2012 11. A box contains 4 red balls and 6 black balls. Three balls are selected randomly from the box one after another, without replacement. The probability that the selected set has one red ball and two black balls is (A) 1/20 (C) 3/10 (B) 1/12 (D) 1/2 th

th

th

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GATE QUESTION BANK

ME - 2013 12. Let X be a normal random variable with mean 1 and variance 4. The probability (X ) is (A) 0.5 (B) Greater than zero and less than 0.5 (C) Greater than 0.5 and less than 1.0 (D) 1.0 13.

The probability that a student knows the correct answer to a multiple choice

the probability of obtaining red colour on top face of the dice at least twice is _______ 17.

A group consists of equal number of men and women. Of this group 20% of the men and 50% of the women are unemployed. If a person is selected at random from this group, the probability of the selected person being employed is _______

18.

A machine produces 0, 1 or 2 defective pieces in a day with associated probability of 1/6, 2/3 and 1/6, respectively. The mean value and the variance of the number of defective pieces produced by the machine in a day, respectively, are (A) 1 and 1/3 (C) 1 and 4/3 (B) 1/3 and 1 (D) 1/3 and 4/3

19.

A nationalized bank has found that the daily balance available in its savings accounts follows a normal distribution with a mean of Rs. 500 and a standard deviation of Rs. 50. The percentage of savings account holders, who maintain an average daily balance more than Rs. 500 is _______

20.

The number of accidents occurring in a plant in a month follows Poisson distribution with mean as 5.2. The probability of occurrence of less than 2 accidents in the plant during a randomly selected month is (A) 0.029 (C) 0.039 (B) 0.034 (D) 0.044

question is . If the student dose not know the answer, then the student guesses the answer. The probability of the guessed answer being correct is . Given that the student has answered the questions correctly, the conditional probability that the student knows the correct answer is (A) ⁄ (C) ⁄ (B) ⁄ (D) ⁄ ME - 2014 14. In the following table x is a discrete random variable and P(x) is the probability density. The standard deviation of x is x 1 2 3 P(x) 0.3 0.6 0.1 (A) 0.18 (C) 0.54 (B) 0.3 (D) 0.6 15.

16.

Box contains 25 parts of which 10 are defective. Two parts are being drawn simultaneously in a random manner from the box. The probability of both the parts being good is ( )

( )

( )

( )

Consider an unbiased cubic dice with opposite faces coloured identically and each face coloured red, blue or green such that each colour appears only two times on the dice. If the dice is thrown thrice,

Mathematics

CE - 2005 1. Which one of the following statements is NOT true? (A) The measure of skewness is dependent upon the amount of dispersion

th

th

th

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GATE QUESTION BANK

(B) In a symmetric distribution the value of mean, mode and median are the same (C) In a positively skewed distribution mean > median > mode (D) In a negatively skewed distribution mode > mean > median CE - 2006 2. A class of first years B. Tech students is composed of four batches A, B, C and D each consisting of 30 students. It is found that the sessional marks of students in Engineering Drawing in batch C have a mean of 6.6 and standard deviation of 2.3. The mean and standard deviation of the marks for the entire class are 5.5 and 4.2 respectively. It is decided by the course instruction to normalize the marks of the students of all batches to have the same mean and standard deviation as that of the entire class. Due to this, the marks of a student in batch C are changed from 8.5 to (A) 6.0 (C) 8.0 (B) 7.0 (D) 9.0 3.

There are 25 calculators in a box. Two of them are defective. Suppose 5 calculators are randomly picked for inspection (i.e. each has the same chance of being selected). What is the probability that only one of the defective calculators will be included in the inspection? (A) ⁄ (C) ⁄ (B) ⁄ (D) ⁄

CE - 2007 4. If the standard deviation of the spot speed of vehicles in a highway is 8.8 kmph and the mean speed of the vehicles is 33 kmph, the coefficient of variation in speed is (A) 0.1517 (C) 0.2666 (B) 0.1867 (D) 0.3646

Mathematics

CE - 2008 5. If probability density function of a random variable x is x for x nd f(x) { for ny other v lue of x Then, the percentage probability P.

x

/ is

(A) 0.247 (B) 2.47 6.

(C) 24.7 (D) 247

A person on a trip has a choice between private car and public transport. The probability of using a private car is 0.45. While using the public transport, further choices available are bus and metro out of which the probability of commuting by a bus is 0.55. In such a situation, the probability, (rounded upto two decimals) of using a car, bus and metro, respectively would be (A) 0.45, 0.30 and 0.25 (B) 0.45, 0.25 and 0.30 (C) 0.45, 0.55 and 0.00 (D) 0.45, 0.35 and 0.20

CE - 2009 7. The standard normal probability function can be approximated as (x )

|x | ) exp( Where x = standard normal deviate. If mean and standard deviation of annual precipitation are 102 cm and 27 cm respectively, the probability that the annual precipitation will be between 90 cm and 102 cm is (A) 66.7% (C) 33.3% (B) 50.0% (D) 16.7% CE - 2010 8. Two coins are simultaneously tossed. The probability of two heads simultaneously appearing is (A) 1/8 (C) 1/4 (B) 1/6 (D) 1/2

th

th

th

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GATE QUESTION BANK

CE - 2011 9. There are two containers with one containing 4 red and 3 green balls and the other containing 3 blue and 4 green balls. One ball is drawn at random from each container. The probability that one of the balls is red and the other is blue will be (A) 1/7 (C) 12/49 (B) 9/49 (D) 3/7 CE - 2012 10. The annual precipitation data of a city is normally distributed with mean and standard deviation as 1000mm and 200 mm, respectively. The probability that the annual precipitation will be more than 1200 mm is (A) < 50 % (C) 75 % (B) 50 % (D) 100 % 11.

14.

A traffic office imposes on an average 5 number of penalties daily on traffic violators. Assume that the number of penalties on different days is independent and follows a poisson distribution. The probability that there will be less than 4 penalties in a day is ____.

15.

A fair (unbiased) coin was tossed four times in succession and resulted in the following outcomes: (i) Head (iii) Head (ii) Head (iv) Head The prob bility of obt ining ‘T il’ when the coin is tossed again is (A) 0 (C) ⁄ (B) ⁄ (D) ⁄

16.

An observer counts 240 veh/h at a specific highway location. Assume that the vehicle arrival at the location is Poisson distributed, the probability of having one vehicle arriving over a 30-second time interval is ____________

In an experiment, positive and negative values are equally likely to occur. The probability of obtaining at most one negative value in five trials is (A)

(C)

(B)

(D)

CE - 2013 12. Find the value of such that the function f(x) is a valid probability density function ____________________ (x )( f(x) x) for x otherwise CE - 2014 13. The probability density function of evaporation E on any day during a year in a watershed is given by f( )

{

mm d y

Mathematics

CS - 2005 1. Box P has 2 red balls and 3 blue balls and box Q has 3 red balls and 1 blue ball. A ball is selected as follows: (i) select a box (ii) choose a ball from the selected box such that each ball in the box is equally likely to be chosen. The probabilities of selecting boxes P and Q are 1/3 and 2/3 respectively. Given that a ball selected in the above process is red, the probability that it comes from box P is (A) 4/19 (C) 2/9 (B) 5/19 (D) 19/30 2.

Let f(x) be the continuous probability density function of a random variable X. The probability that a  X  b , is (A) f(b a) (C) ∫ f(x)dx

otherwise The probability that E lies in between 2 and 4 mm/day in a day in the watershed is (in decimal) ______

(B) f(b)

th

th

f( )

(D) ∫ x f(x)dx

th

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GATE QUESTION BANK

CS - 2006 3. For each element in a set of size 2n, an unbiased coin is tossed. The 2n coin tosses are independent. An element is chosen if the corresponding coin toss were head. The probability that exactly n elements are chosen is (A) ( n ⁄ ) (C) ( ⁄ n ) (D) ⁄ (B) ( n ⁄ ) CS - 2007 Linked Data for Q4 & Q5 are given below. Solve the problems and choose the correct answers. Suppose that robot is placed on the Cartesian plane. At each step it is easy to move either one unit up or one unit right, i.e if it is at (i,j) then it can move to either (i+1,j) or (i,j+1) 4. How many distinct paths are there for the robot to reach the point (10,10) starting from the initial position (0,0)? (C) 210 (A) 20 (D) None of these (B) 2 5.

Suppose that the robot is not allowed to traverse the line segment from (4,4) to (5,4). With this constraint, how many distinct paths are there for the robot to reach (10,10) starting from (0,0)? (A) 29 (B) 219 (C) . / . (D) .

6.

/

/ . / .

/

Suppose we uniformly and randomly select a permutation from the 20! ermut tions of ………… Wh t is the probability that 2 appears at an earlier position than any other even number in the selected permutation? (A) ⁄ (C) ⁄ (B) ⁄ (D) none of these

Mathematics

CS - 2008 7. Let X be a random variable following normal distribution with mean +1 and variance 4. Let Y be another normal variable with mean of 1 and variance unknown If (X ) (Y≥ ) the standard deviation of Y is (A) 3 (C) √ (B) 2 (D) 1 8.

Aishwarya studies either computer science or mathematics every day. If she studies computer science on a day, then the probability that she studies mathematics the next day is 0.6. If she studies mathematics on a day, then the probability that she studies computer science the next day is 0.4. Given that Aishwarya studies computer science on Monday, what is the probability that she studies computer science on Wednesday? (A) 0.24 (C) 0.4 (B) 0.36 (D) 0.6

CS - 2009 9. An unbalanced dice (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is 90% of the probability that the face value is even. The probability of getting any even numbered face is the same. If the probability that the face is even given that it is greater than 3 is 0.75, which one of the following options is closest to the probability that the face value exceeds 3? (A) 0.453 (C) 0.485 (B) 0.468 (D) 0.492 CS - 2010 10. Consider a company that assembles computers. The probability of a faulty assembly of any computer is p. The company therefore subjects each computer to a testing process. This testing process gives the correct result for any computer with a probability of q. What is the probability of a computer being declared faulty? th

th

th

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GATE QUESTION BANK

(A) (B) (C) (D) 11.

12.

pq+(1 – p)(1 – q) (1 – q)p (1 – p)q pq

What is the probability that a divisor of is a multiple of ? (A) 1/625 (C) 12/625 (B) 4/625 (D) 16/625 If the difference between the expectation of the square if a random variable ( ,x -) and the square if the exopectation of the random variable ( ,x-) is denoted by R, then (A) R = 0 (C) R≥ (B) R< 0 (D) R > 0

CS - 2011 13. A deck of 5 cards (each carrying a distinct number from 1 to 5) is shuffled thoroughly. Two cards are then removed one at a time from the deck. What is the probability that the two cards are selected with the number on the first card being one higher than the number on the second card? (A) ⁄ (C) ⁄ (B) ⁄ (D) ⁄ 14.

Consider a finite sequence of random values X = [x1, x2 … xn].Let be the me n nd σx be the standard deviation of X. Let another finite sequence Y of equal length be derived from this as yi, a*xi+b, where a and b are positive constants. Let μy be the me n nd σy be the standard deviation of this sequence. Which one of the following statements is INCORRECT? (A) Index position of mode of X in X is the same as the index position of mode of Y in Y. (B) Index position of median of X in X is the same as the index position of median of Y in Y. (C) μy μx + b (D) σy σx + b

15.

Mathematics

If two fair coins flipped and at least one of the outcomes is known to be a head, what is the probability that both outcomes are heads? (A) 1/3 (C) 1/4 (B) 1/2 (D) 2/3

CS - 2012 16. Suppose a fair six – sided die is rolled once. If the value on the die is 1,2, or 3 the die is rolled a second time. What is the probability that the some total of value that turn up is at least 6? (A) 10/21 (C) 2/3 (B) 5/12 (D) 1/6 17.

Consider a random variable X that takes values +1 and 1 with probability 0.5 each. The values of the cumulative distribution function F(x) at x = and +1 are (A) 0 and 0.5 (C) 0.5 and 1 (B) 0 and 1 (D) 0.25 and 0.75

CS - 2013 18. Suppose p is the number of cars per minute passing through a certain road junction between 5 PM and 6 PM, and p has a Poisson distribution with mean 3. What is the probability of observing fewer than 3 cars during any given minute in this interval? ⁄ e (A) ⁄ e (C) ⁄ e (B) ⁄ e (D) CS - 2014 19. Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________ . 20.

th

Four fair six – sided dice are rolled. The probability that the sum of the results being 22 is x/1296. The value of x is ____________

th

th

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GATE QUESTION BANK

21.

The security system at an IT office is composed of 10 computers of which exactly four are working. To check whether the system is functional, the officials inspect four of the computers picked at random (without replacement). The system is deemed functional if at least three of the four computers inspected are working. Let the probability that the system is deemed functional be denoted by p. Then 100p = _____________.

22.

Each of the nine words in the sentence “The quick brown fox jumps over the l zy dog” is written on sep r te piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The expected length of the word drawn is _____________. (The answer should be rounded to one decimal place.)

23.

The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is __________.

24.

Let S be a sample space and two mutually exclusive events A and B be such that ∪ S If ( ) denotes the prob bility of the event, the maximum value of P(A) P(B) is _______

ECE - 2006 3. A probability density function is of the ). form (x) e || x ( The value of K is (A) 0.5 (C) 0.5a (B) 1 (D) A 4.

Three Companies X, Y and Z supply computers to a university. The percentage of computers supplied by them and the probability of those being defective are tabulated below Company % of Probability computers of being supplied defective X 60% 0.01 Y 30% 0.02 Z 10% 0.03 Given that a computer is defective, the probability that it was supplied by Y is (A) 0.1 (C) 0.3 (B) 0.2 (D) 0.4

ECE - 2007 5. If E denotes expectation, the variance of a random variable X is given by (A) E[X2] E2[X] (C) E[X2] (B) E[X2] + E2[X] (D) E2[X] 6.

An examination consists of two papers, Paper1 and Paper2. The probability of failing in Paper1 is 0.3 and that in Paper2 is 0.2.Given that a student has failed in Paper2, the probability of failing in paper1 is 0.6. The probability of a student failing in both the papers is (A) 0.5 (C) 0.12 (B) 0.18 (D) 0.06

ECE - 2005 1. A fair dice is rolled twice. The probability that an odd number will follow an even number is

2.

( )

( )

( )

( )

The value of the integral

I

Mathematics

ECE - 2008 7. The probability density function (PDF) of a random variable X is as shown below.

 x2  1  exp   dx is 2 0  8

(A) 1 (B)

(C) 2 (D) th

th

th

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GATE QUESTION BANK

1

(x) exp( |x|) exp( |x|) is the probability density function for the real random variable X, over the entire x axis. M and N are both positive real numbers. The equation relating M and N is

8.

PDF PDF

1

0

1 x 1

(A)

The corresponding 0 cumulative 1 distribution function (CDF) has the form

(A)

(B) 2M

1

0

ECE - 2009 9. Consider two independent random variables X and Y with identical distributions. The variables X and Y take value 0, 1 and 2 with probabilities

x

and respectively. What is the conditional probability (x y ) |x y| (A) 0 (C) ⁄ ⁄ (B) (D) 1

CD F C

1

D F

1

10. 0

1 -1

1

x

1

A fair coin is tossed 10 times. What is the probability that ONLY the first two tosses will yield heads? 2

10

1  2

CDF 1

0

(C)

N=1

(C) M + N = 1 (D) M + N = 3

x

(B)

1

CDF

1

1

Mathematics

1  2

(A)  

0

(C)   2

(B)

1

0

x

11.

1

(D)

1 1 1

0 0 1

CDF

1 1

x

th

10

10

1 C2    2

(D)

10

1 C2    2

A discrete random variable X takes values from 1 to 5 with probabilities as shown in the table. A student calculates the mean of X as 3.5 and her teacher calculates the variance of X as 1.5. Which of the following statements is true? k P(X=k) 1 0.1 2 0.2 3 0.4 4 0.2 5 0.1

th

th

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GATE QUESTION BANK

(A) Both the student and the teacher are right (B) Both the student and the teacher are wrong (C) The student is wrong but the teacher is right (D) The student is right but the teacher is wrong ECE - 2010 12. A fair coin is tossed independently four times. The prob bility of the event “the number of times heads show up is more th n the number of times t ils show up” is (A) ⁄ (C) ⁄ (B) ⁄ (D) ⁄ ECE - 2011 13. A fair dice is tossed two times. The probability that the second toss results in a value that is higher than the first toss is (A) 2/36 (C) 5/12 (B) 2/6 (D) 1/2 ECE\EE\IN - 2012 14. A fair coin is tossed till a head appears for the first time probability that the number of required tosses is odd , is (A) 1/3 (C) 2/3 (B) 1/2 (D) 3/4 ECE - 2013 15. Let U and V be two independent zero mean Gaussian random variables of variances ⁄ and ⁄ respectively. The probability ( V ≥ U) is (A) 4/9 (C) 2/3 (B) 1/2 (D) 5/9 16.

Consider two identically distributed zeromean random variables U and V . Let the cumulative distribution functions of U and 2V be F(x) and G(x) respectively. Then, for all values of x (x)) (A) ( (x) (B) ( (x)

(C) ( (x) (D) ( (x)

Mathematics

(x)) x (x)) x ≥

ECE - 2014 17. In a housing society, half of the families have a single child per family, while the remaining half have two children per family. The probability that a child picked at random, has a sibling is _____ 18.

Let X X nd X , be independent and identically distributed random variables with the uniform distribution on [0, 1]. The probability P{X is the largest} is _____

19.

Let X be a random variable which is uniformly chosen from the set of positive odd numbers less than 100. The expectation, E[X], is __________.

20.

An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is (A) 0.067 (C) 0.082 (B) 0.073 (D) 0.091

21.

A fair coin is tossed repeatedly till both head and tail appear at least once. The average number of tosses required is _______.

22.

Let X X and X be independent and identically distributed random variables with the uniform distribution on [0, 1]. The probability P{X X X } is ______.

23.

Let X be a zero mean unit variance Gaussian random variable. ,|x|- is equal to __________

24.

Parcels from sender S to receiver R pass sequentially through two post-offices. Each post-office has a probability

of

losing an incoming parcel, independently of all other parcels. Given that a parcel is lost, the probability that it was lost by the second post-office is ____________.

(x)) ≥ th

th

th

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GATE QUESTION BANK

EE - 2005 1. If P and Q are two random events, then the following is TRUE (A) Independence of P and Q implies that probability (P Q) = 0 (B) Probability (P ∪ Q)≥ Probability (P) +Probability (Q) (C) If P and Q are mutually exclusive, then they must be independent (D) Probability (P Q) Probability (P) 2.

A fair coin is tossed three times in succession. If the first toss produces a head, then the probability of getting exactly two heads in three tosses is (A) ⁄ (C) ⁄ (B) ⁄ (D) ⁄

EE - 2006 3. Two f ir dice re rolled nd the sum “ r ” of the numbers turned up is considered (A) Pr (r > 6) = (B) Pr (r/3 is an integer) = (C) Pr (r = 8|r/4 is an integer) = (D) Pr (r = 6|r/5 is an integer) = EE - 2007 4. A loaded dice has following probability distribution of occurrences Dice Value Probability ⁄ 1 2

⁄

3

⁄

4

⁄

5

⁄

⁄ 6 If three identical dice as the above are thrown, the probability of occurrence of values, 1, 5 and 6 on the three dice is (A) same as that of occurrence of 3, 4, 5 (B) same as that of occurrence of 1, 2, 5 (C) 1/128 (D) 5/8

Mathematics

EE - 2008 5. X is a uniformly distributed random variable that takes values between 0 and 1. The value of E{X } will be (A) 0 (C) 1/4 (B) 1/8 (D) 1/2 EE - 2009 6. Assume for simplicity that N people, all born in April (a month of 30 days), are collected in a room. Consider the event of atleast two people in the room being born on the same date of the month, even if in different years, e.g. 1980 and 1985. What is the smallest N so that the probability of this event exceeds 0.5? (A) 20 (C) 15 (B) 7 (D) 16 EE - 2010 7. A box contains 4 white balls and 3 red balls. In succession, two balls are randomly selected and removed from the box. Given that the first removed ball is white, the probability that the second removed ball is red is (A) 1/3 (C) 1/2 (B) 3/7 (D) 4/7 ECE\EE\IN - 2012 8. Two independent random variables X and Y are uniformly distributed in the interval , -. The probability that max , - is less than 1/2 is (A) 3/4 (C) 1/4 (B) 9/16 (D) 2/3 EE - 2013 9. A continuous random variable x has a probability density function + is f(x) e x . Then *x (A) 0.368 (C) 0.632 (B) 0.5 (D) 1.0

th

th

th

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GATE QUESTION BANK

EE - 2014 10. A fair coin is tossed n times. The probability that the difference between the number of heads and tails is (n – 3) is (C) (A) (B) (D) 11.

12.

13.

14.

IN - 2005 1. The probability that there are 53 Sundays in a randomly chosen leap year is (A) ⁄ (C) ⁄ (B) ⁄ (D) ⁄ 2.

A mass of 10 kg is measured with an instrument and the readings are normally distributed with respect to the mean of 10 kg. Given that

Consider a dice with the property that the probability of a face with n dots showing up is proportional to n. The probability of the face with three dots showing up is _______________ Let x be a random variable with probability density function for |x| f(x) { |x| for otherwise The probability P(0.5 < x < 5) is_________ Lifetime of an electric bulb is a random variable with density f(x) kx , where x is measured in years. If the minimum and maximum lifetimes of bulb are 1 and 2 years respectively, then the value of k is__________ The mean thickness and variance of silicon steel laminations are 0.2 mm and 0.02 respectively. The varnish insulation is applied on both the sides of the laminations. The mean thickness of one side insulation and its variance are 0.1 mm and 0.01 respectively. If the transformer core is made using 100 such varnish coated laminations, the mean thickness and variance of the core respectively are (A) 30 mm and 0.22 (B) 30 mm and 2.44 (C) 40 mm and 2.44 (D) 40 mm and 0.24

Mathematics

exp .

∫

√

/ d =0.6

and that 60per cent of the readings are found to be within 0.05 kg from the mean, the standard deviation of the data is (A) 0.02 (C) 0.06 (B) 0.04 (D) 0.08 3.

The measurements of a source voltage are 5.9V, 5.7V and 6.1V. The sample standard deviation of the readings is (A) 0.013 (C) 0.115 (B) 0.04 (D) 0.2

IN - 2006 4. You have gone to a cyber-cafe with a friend. You found that the cyber-café has only three terminals. All terminals are unoccupied. You and your friend have to make a random choice of selecting a terminal. What is the probability that both of you will NOT select the same terminal? (A) ⁄ (C) ⁄ (B) ⁄ (D) 1 5.

Probability density function p(x) of a random variable x is as shown below. The value of  is p(x) α

0

th

α

α b

α c

(A)

c

(C)

(B)

c

(D)

th

th

(

)

(

)

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GATE QUESTION BANK

6.

Mathematics

Two dices are rolled simultaneously. The probability that the sum of digits on the top surface of the two dices is even is (A) 0.5 (C) 0.167 (B) 0.25 (D) 0.125

measurements, it can be expected that the number of measurement more than 10.15 mm will be (A) 230 (C) 15 (B) 115 (D) 2

IN - 2007 7. Assume that the duration in minutes of a telephone conversation follows the

IN - 2011 12. The box 1 contains chips numbered 3, 6, 9, 12 and 15. The box 2 contains chips numbered 6, 11, 16, 21 and 26. Two chips, one from each box, are drawn at random. The numbers written on these chips are multiplied. The probability for the product to be an even number is (A) ⁄ (C) ⁄ ⁄ (B) ⁄ (D)

exponential distribution f(x) =

e ,x≥

The probability that the conversation will exceed five minutes is (A) e (C) (B) e (D) e IN - 2008 8. Consider a Gaussian distributed random variable with zero mean and standard deviation . The value of its cummulative distribution function at the origin will be (A) 0 (C) 1 (B) 0.5 (D) σ 9.

A random variable is uniformly distributed over the interval 2 to 10. Its variance will be ⁄ ⁄ (A) (C) (B) 6 (D) 36

IN - 2013 13. A continuous random variable X has probability density f(x) = . Then P(X > 1) is (A) 0.368 (C) 0.632 (B) 0.5 (D) 1.0 IN - 2014 14. Given that x is a random variable in the r nge , - with prob bility density function

the value of the constant k is

___________________ IN - 2009 10. A screening test is carried out to detect a certain disease. It is found that 12% of the positive reports and 15% of the negative reports are incorrect. Assuming that the probability of a person getting a positive report is 0.01, the probability that a person tested gets an incorrect report is (A) 0.0027 (C) 0.1497 (B) 0.0173 (D) 0.2100

15.

IN - 2010 11. The diameters of 10000 ball bearings were measured. The mean diameter and standard deviation were found to be 10 mm and 0.05mm respectively. Assuming Gaussian distribution of

The figure shows the schematic of production process with machines A,B and C. An input job needs to be preprocessed either by A or by B before it is fed to C, from which the final finished product comes out. The probabilities of failure of the machines are given as:

Assuming independence of failures of the machines, the probability that a given job is successfully processed (up to third decimal place)is ______________

th

th

th

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GATE QUESTION BANK

Mathematics

Answer Keys and Explanations ME 1.

4. [Ans. D] The number of ways coming 8 and 9 are (2,6),(3,5),(4,4),(5,3),(6,2),(3,6),(4,5), (5,4),(6,3) Total ways =9 So Probability of coming 8 and 9 are

[Ans. D]

5.

[Ans. D] X and Y are independent ∴ ( ) ( ) ( ) re true Only (D) is odd one

6.

[Ans. A] Number of favourable cases are given by HHHT HHTH HTHH THHH Total number of cases = 2C1 2C1 2C1 2C1 =16

So probability of not coming these

2.

[Ans. B] Probability of defective item = Probability of not defective item = 1 0.1 = 0.9 So, Probability that exactly 2 of the chosen items are defective = ( ) ( )

3.

[Ans. B]

∴ Probability = 7.

[Ans. A] A uniform function

Mean (t)̅ = ∫ t f(t) dt ∫ t( t 6

t)dt t

t 6

7

[

]

∫ t(

[

t

t)dt

t)dt

=∫ (t =0

t )dt 1

0

7

density

Density function

 1  f(x)   b  a  0

∫ t (

t)dt

a,x  b a  x,x  b

 Mean E(x)=

t)dt

b

x(F(x))  x a

ab 2

 Variance = F(x)2  f(x)

2

1

2

b b   x2F(x)  xF(x) x a  x a 

= Standard deviation = √v ri nce =

and

0,x  a x  a  f(x)   f  x dx   , axb 0 b  a  xb 0,

]

∫ t (

distribution

x

Variance = ∫ t f(t)dt =∫ t (

( oth defective) S mple sp ce

( oth defective)

Put the value of F(x), we get √

2

1 1 b  dx  x. dx  Variance  x ba x a  x a b  a  b

th

th

2

th

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GATE QUESTION BANK b

 x3   xL      3(b  a) a  2 b  a  

Mathematics 3

2

1 7  (3  3  1)    2 8



b3  a3 (b2  a2 )2  3(b  a) 4 b  a 2



(b  a)(b2  ab  a2 ) (b  a)2(b  a)2  2 3(b  a) 4 b  a



4b2  4ab  4a2  3a2  3b2  6ab 12



b2  a2  2ab 12

9.

[Ans. C] Probability of drawing 2 washers, first followed by 3 nuts, and subsequently the 4 bolts

10.

[Ans. D] Required probability =

(b  a)2  12

. / . /

Standard deviation = √v ri nce

(b  a)2 12 (b  a)  12 

11.

[Ans. D] Given 4R and 6B , -

12.

[Ans. C]

Given: b=1, a=0

 Standard deviation =

8.

10 1  12 12

[Ans. D] Let probability of getting atleast one head = P(H) then P (at least one head) = 1 P(no head)  P(H)=1 P(all tails) But in all cases, 23=8

1 7 8 8

X=0

 P (H) = 1  

(X ) is Below X (X ) has to be less than 0.5 but greater than zero

Alternately Probability of getting at least one head ( ) ( )

13.

1 7 1   8 8 Alternately From Binomial theorem Probability of getting at least one head pq ( )

( )

X=1

[Ans. D] A event that he knows the correct answer B event that student answered correctly the question P(B) = ? ( )

(

)

( )

th

th

th

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GATE QUESTION BANK

( )

he knows correct nswer

(

) (

( ⁄ )

14.

No. of employed men = 80% of men = 80 No. of employed women = 50% of women = 50 Probability if the selected one person being employed = probability of one employed women +probability of one employed man

⏟

⏟

e does not know correct nswer so he guesses

( ) ( ⁄ ) ) ⁄ ( ) ⁄

[Ans. D] x 1 2 P(x) 0.3 0.6 (x) (x) x

18.

V(x) x (

σ

(x )

[Ans. A]

3 0.1 So from figure Mean value = 1 V ri nce : μ me n x defective pieces (x μ) σ ) n(n ( ) ( ) ( ) ( )

(x) x (x) σ

Mathematics

, (x)-

(x) ( x (x)) ) ( )

√ ( )

15.

[Ans. A] 19.

16.

[Ans. *](Range 49 to 51)

[Ans. *] Range 0.25 to 0.27 p orm l distribution

q

Given that μ σ x μ x z σ ere x μ , s x gre ter th n z ) ence prob bility (z

Using Binomial distribution (x ≥ )

17.

( ) ( )

( ) ( )

-

∫ e dz σ√ ∴ of s ving ccount holder

[Ans. *] Range 0.64 to 0.66 Let number of men = 100 Number of women = 100 th

th

th

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GATE QUESTION BANK

20.

[Ans. B] Mean m = np = 5.2 me (x ) e

25 Calculators

m

23 Non-defective

2 Defective

e

5 Calculators

e (x

Mathematics

) 4 Non-defective

1 Defective

CE 1.

2.

[Ans D] A, B, C are true (D) is not true. Since in a negatively skewed distribution mode > median > mean [Ans. D] Let the mean and standard deviation of the students of batch C be μ and σ respectively and the mean and standard deviation of entire class of first year students be μ and σ respectively Now given, μ σ and μ σ In order to normalise batch C to entire class, the normalize score must be equated since Z = Z =

=

Now Z =

p( defective in c lcul tors)

4.

[Ans. C] σ μ

5.

[Ans. B] Given f(x) = x for x = 0 else where (

)

∫ f(x)dx

∫ x dx

=0 1 The probability expressed in percentage P= = 2.469% = 2.47% 6.

[Ans. A] Given P(private car) = 0.45 P(bus 1 public transport) = 0.55 Since a person has a choice between private car and public transport P(public transport) = 1 – P(private car) = 1 – 0.45 =0.55 P(bus) = P(bus public transport) (bus public tr nsport) (public tr nsport) = 0.55 × 0.55 = 0.3025 ≃ 0.30 Now P(metro) = 1 [P(private car) + P(bus)] = 1 (0.45 + 0.30) = 0.25

=

Equation these two and solving, we get =  x = 8.969 ≃ 9.0 3.

x

[Ans. B] Since population is finite, hypergeometric distribution is applicable

th

th

th

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GATE QUESTION BANK

∴ P(private car) = 0.45 P(bus) = 0.30 and P(metro) = 0.25 7.

12.

[Ans. D] ere μ cm; σ ( x 102) =P.

[Ans. 6] ∫ f(x)dx ( x

f(x)

{

∴∫

( x

cm 6

x

Mathematics

)

x

)dx

x

x

x

(

)

x otherwise

x7

/ [

=P( x ) This area is shown below:

[

(

(

)]

]

[

-0.44

)

]

The shades area in above figure is given by F(0) –F ( 0.44) =

( )

(

(

)(

)

= 0.5 – 0.3345 = 1.1655 ≃ 16.55% Closest answer is 16.7% 8.

)

13.

[Ans. 0.4] (

)

∫ f( )d

[Ans. C] ( )|

P(2 heads) = 9.

[Ans. C] P(one ball is Red & another is blue) = P(first is Red and second is Blue)

14.

= 10.

[Ans. A] Given μ = 1000, σ = 200 We know that Z When X= 1200, Z Req. Prob = P (X (Z ) ( Z Less than 50%

11.

∫ d

[Ans. D] (X ) ( )

(X

)

(

[Ans. *] Range 0.26 to 0.27 Avg= 5 Let x denote penalty (x ) (x ) (x ) (x ) (x ) e ew (x n) x e e e ) p(x e

[

)

e

]

)

)

(X

15.

[Ans. B] S * T+ n( ) ( ) n(S)

16.

[Ans. *] Range 0.25 to 0.28 ( t) e (n t) n

)

( )

th

th

th

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GATE QUESTION BANK

no of vehicles (

veh km

e

)

.

m ke ex ctly ‘ ’ moves nd ‘U’ moves in any order. Similarly to reach (10, 10) from (0,0) the robot h s to m ke ‘ ’ moves nd ‘U’ moves in any order. The number of ways this can be done is same as number of permutations of a word consisting of 10 ‘ ’ s nd ’U’s Applying formula of permutation with limited repetitions we get the answer as

/

= 2.e = 0.2707 CS 1.

[Ans. A] P: Event of selecting Box P, Q: Event of selecting Box P P(P)=1/3, P(Q)=2/3 P(R/P)=2/5, P(R/Q)=3/4

P(R/P).P(P) P(R/P).P(P)  P(R/Q)P(Q) 2/51/3   4/19 2/51/3  3/ 4  2/3 P(P/R)=

2.

5.

[Ans. D] The robot can reach (4,4) from (0,0) in 8C ways as argued in previous problem. 4 Now after reaching (4,4) robot is not allowed to go to (5,4) Let us count how many paths are there from (0,0) to (10,10) if robot goes from (4,4) to (5,4) and then we can subtract this from total number of ways to get the answer. Now there are 8C4 ways for robot to reach (4,4) from (0,0) and then robot takes the ‘U’ move from ( ) to ( ) ow from (5,4) to (10,10) the robot has to make 5 ‘U’ moves nd ‘ ’ moves in ny order which can be done in 11! ways = 11C5 ways Therefore, the number of ways robot can move from (0,0) (10,10) via (4,4) – (5,4) move is

[Ans. C] If f (x) is the continuous probability density function of a random variable X then, ( x b) P( x b) b

= f  x  dx

 a

3.

4.

[Ans. A] The probability that exactly n elements are chosen =The probability of getting n heads out of 2n tosses =

(

) . /

= =

(

) (

Mathematics

(Binomial formula) )

8C 4

[Ans. A] Consider the following diagram (3,3)

11C 5

 8   11   4 5 

=   

No. of ways robot can move from (0,0) to (10,10) without using (4,4) to (5,4) move is

 20  8   11        ways  10   4   5 

(0,0) The robot can move only right or up as defined in problem. Let us denote right move by ‘ ’ nd up move by ‘U’ ow to reach (3, 3), from (0,0) , the robot has to

which is choice (D)

th

th

th

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GATE QUESTION BANK

6.

[Ans. D] umber of permut tions with ‘ ’ in the first position =19! Number of permutations with ‘ ’ in the second position = 10 18! (Fill the first space with any of the 10 odd numbers and the 18 spaces after the 2 with 18 of the remaining numbers in 18! ways) umber of permut tions with ‘ ’ in rd position =10 9 17! (Fill the first 2 place with 2 of the 10 odd numbers and then the remaining 17 places with remaining 17 numbers) nd so on until ‘ ’ is in th place. After that it is not possible to satisfy the given condition, since there are only 10 odd numbers v il ble to fill before the ‘ ’ So the desired number of permutations which satisfies the given condition is

8.

[Ans. C] Let C denote computes science study and M denotes maths study. P(C on Monday and C on Wednesday) = P(C on Monday, M on Tuesday and C on Wednesday) + P(C on Monday, C on Tuesday and C on Wednesday) =1 0.6 0.4+ 1 0.4 0.4 = 0.24 + 0.16 = 0.40

9.

[Ans. B] It is given that P (odd) = 0.9 P (even) Now since 𝜮P(x) = 1 ∴ P (odd) + P (even) = 1 0.9 P (even) + P (even) = 1 P(even) =

…

∴ P(2) = P(4) = P(6) = P(even) )

=

Which are clearly not choices (A), (B) or (C) 7.

/ = P (z ≥

.z

/ = P (z ≥

(z

) = P (z ≥

(

)

)

10. _____(i)

)

(

)

(

)

P(f ce )

) (

) (

(0.5263)

= 0.1754 It is given that P(even | face > 3) = 0.75

[Ans. A] Given μ = 1, σ = 4 σ =2 and μ = 1, σ is unknown Given, P(X ) = P (Y ≥ 2 ) Converting into standard normal variates, .z

= 0.5263

Now, it is given that P(any even face) is same i.e. P(2) = P(4) = P(6) Now since, P(even) = P(2) or P(4) or P(6) = P(2) + P(4) + P(6)

… Now the probability of this happening is given by = (

Mathematics

= 0.75

= 0.75 ( )

)

( )

=1

decl red f ulty

f ulty

p

q p

σ =3

not f ulty

decl red not f ulty decl red f ulty

q

q

decl red not f ulty

From above tree (decl red f ulty) th

=0.468

[Ans. A] The tree diagram of probabilities is shown below q

Now since we know that in standard normal distribution P (z ) = P (z ≥ 1) _____(ii) Comparing (i) and (ii) we can say that

=

th

pq th

(

q)(

p)

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GATE QUESTION BANK

11.

[Ans. A] If b c … Then, no. of divisors of (x )(y )(z )… iven ∴ o of ivisors of ( )( ( )( )

(

(

which are multiples

13.

14.

15.

[Ans. C] (x ) , (x)V(x) Where V(x) is the variance of x, Since variance is σ and hence never negative, ≥

( t le st one he d)

TT )

So required prob bility

)

∴ equired rob bility 12.

)

( ∪ )

16. No. of divisors of of o of divisors of ( )( )

Mathematics

[Ans. B] Required Probability = P (getting 6 in the first time) + P (getting 1 in the first time and getting 5 or 6 in the second time) + P (getting 2 in the first time and getting 4 or 5 or 6 in the second time) + P (getting 3 in the first time and getting 3 or 4 or 5 or 6 in the second time) ( )

( )

( )

17.

[Ans. C] The p.d.f of the random variable is x +1 P(x) 0.5 0.5 The cumulative distribution function F(x) is the probability upto x as given below x +1 F(x) 0.5 1.0 So correct option is (C)

18.

[Ans. C] e (k)

[Ans. A] + The five cards are * Sample space ordered pairs st nd P (1 card = 2 card + 1) )( )( )( )+ *(

k P is no. of cars per minute travelling.

[Ans. D] 𝛔y = a 𝛔x is the correct expression Since variance of constant is zero.

For no cars. (i.e. k = 0) For no cars. P(0) e So P can be either 0,1,2. (i.e. k = 0,1,2) For k = 1, p(1)=e

[Ans. A] Let A be the event of head in one coin. B be the event of head in second coin. The required probability is * ) ( ∪ )+ ( )| ∪ ) ( ∪ ) ( ) ( ∪ ) ( ) (both coin he ds)

For k = 2 , P(2)= Hence ( ) e e th

( )

( )

e 4

th

e 5

th

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GATE QUESTION BANK

e (

(

20.

(

)

∴ equired prob bility is ( ∪ ( )

∪ )

⁄ )

(

( ) (

)

e

[Ans. *] Range 0.24 to 0.27 The smaller sticks, therefore, will range in length from almost 0 meters upto a maximum of 0.5 meters, with each length equally possible. Thus, the average length will be about 0.25 meters, or about a quarter of the stick.

24.

(

)

[Ans. 10] 22 occurs in following ways 6 6 6 4 w ys 6 6 5 5 w ys

[Ans. 0.25] ( ∪ ) P(S) = 1 ( ) ( ) ( ) utu lly exclusive ( ) ( ) ( ) et ( ) x; ( ) x P(A) P(B) = x( x) Maximum value of y = x ( x) dy ( x) x dx = 2x = 1 x

equired prob bility

(max)

x 21.

( )

) e

19.

( )

Mathematics

ximum v lue of y

[Ans. *] Range 11.85 to 11.95 For functioning 3 need to be working (function)

ECE 1.

(

)

[Ans. D]

3 1  6 2 3 1 P(even number )   6 2 Since events are independent, therefore 1 1 1  P(odd/even)    2 2 4 P(Odd number) 

p 22.

[Ans. *] Range 3.8 to 3.9 Expected length = Average length of all words

2.

[Ans. A] I

(

∫ e √ omp ring with (

23.

σ√ ut μ

[Ans. *] Range 0.259 to 0.261 Let A = divisible by 2, B = divisible by 3 and C = divisible by 5, then n(A) = 50, n(B) = 33, n(C) = 20 n( ) n( ) n( ) n( ) P( ∪ ∪ )

∫

∫ rom x σ σ th

th

dx

)

e

dx

σ

√

)

e

dx

…

nd x

th

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)

GATE QUESTION BANK

Put σ ∫

3.

in

Probability of failing in paper 2, P (B) = 0.2 Probability of failing in paper 1, when

equ tion

e

√

A   0.6 B  A  P A  B We know that, P   PB  B

student has failed in paper 2, P

[Ans. C] 

 P x.dx  1





 Ke



Mathematics

(

∴

.dx  1

ax

)

e dx

∫

e

dx

 x x,for x  0      x for x  0 K K   1 a a

( )

= 0.6 0.2 = 0.12



or ∫

( )

7.

[Ans. A] CDF: F x  

x

 PDF

dx



For x0, F x   F0 

x

 x  1

dx

0

[Ans. A] var[x]=σ =E[(x x)2] Where, x=E[x] x= expected or mean value of X defining

 1  x2    x  concave downwards 2  2  Hence the CDF is, shown in the figure (A).



E[X] =

 xf  xdx x



8.

[Ans. A]

Given: Px  x   Me2|x|  Ne3|x|



    x P xi   x  xi dx    i



 P  xdx  1 x

 xiP xi 

 



i

Variance σ is a measure of the spread of the values of X from its mean x. Using relation , E[X+Y]= E[X]+E[Y] And E[CX]=CE[X] On var[x]= σ =E[(x x)2] σ = ,Xx2 = E[X2] [ ,X-] 6.











Me2|x|  Ne3|x| dx  2 Me2|x|  Ne3|x| dx  1 0

By simplifying

2 3

 M N 1 9.

[Ans. B] x+y=2 x y=0 => x =1, y = 1 P(x=1,y=1) = ¼

[Ans. C] Probability of failing in paper 1, P (A) = 0.3 th

th

¼ = 1/16 th

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GATE QUESTION BANK

10.

[Ans. C]

14.

Probability of getting head in first toss = Probability of getting head in second toss =

[Ans. C] P(no. of tosses is odd) (no of tosses is

…)

P(no. of toss is 1) = P(Head in 1st toss)

and in all other 8 tosses there should

P(no. of toss is 3) = P(tail in first toss, tail in second toss and head in third toss)

be tail always. So probability of getting head in first two tosses ( )( )( )…………… ( ) = (1/2)10

P(no. of toss is 5) = P(T,T,T,TH) . /

11.

Mathematics

[Ans. B] Both the teacher and student are wrong Mean = ∑ k = 0.1 + 0.4 + 1.2 + 0.8 + 0.5 = 3.0 E(x2) = ∑ k

… etc

So, P(no. of tosses in odd)

⁄ ⁄ ⁄ ⁄

Variance(x)= E(x2) – * (x)+ =10.2 9=1.2 12.

[Ans. D] P(H, H, H, T) +P (H, H, H, H ) =

13.

. /

. /

15.

[Ans. B] ( V ≥ V) ( V V≥ ) *z v v+ Linear combination of Gaussian random variable is Gaussian ∴ (z ≥ ) and not mean till zero because both random variables has mean zero hence ( ) Hence Option B is correct

16.

[Ans. D] F(x) = P{X x} (x) * X x+ x 2X 3

. / =

[Ans. C] Total number of cases = 36 Favorable cases: (1, 1) (2, 1) (3, 1) (4, 1) (5, 1) (6, 1) (1, 2) (2, 2) (3, 2) (4, 2) (5, 2) (6, 2) (1, 3) (2, 3) (3, 3) (4, 3) (5, 3) (6, 3) (1, 4) (2, 4) (3, 4) (4, 4) (5, 4) (6, 4) (1, 5) (2, 5) (3, 5) (4, 5) (5, 5) (6, 5) (1, 6) (2, 6) (3, 6) (4, 6) (5, 6) (6, 6) Total number of favorable cases

For positive value of x, (x) (x) is always greater than zero For negative value of x (x) (x)is ve ut , (x) (x)- x ≥

Then probability th

th

th

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GATE QUESTION BANK

17.

18.

[Ans. *] Range 0.65 to 0.68 et ‘ ’ be different types of f milies nd ‘S’ be there siblings. S S S (siblings) Probability that child chosen at random having sibling is 2/3

(x)

et S

,

∑x f(x)

[Ans. C]

21.

[Ans. *] Range 2.9 to 3.1 Let the first toss be head. Let x denotes the number of tosses(after getting first head) to get first tail. We can summarize the even as Event(After x Probability getting first H) T 1 1/2 HT 2 1/2 1/2=1/4 HHT 3 1/8 nd so on…

II)gives

(

)S

……

(x) i.e. The expected number of tosses (after first head) to get first tail is 2 and same can be applicable if first toss results in tail. Hence the average number of tosses is

22.

-

20.

(I

(II)

S

∑x …

……

(I)

S

f(x) ∴ (x)

……

S

[Ans. *] Range 0.32 to 0.34 This is a tricky question, here, X X X independent and identically distributed random variables with the uniform distribution , -. So, they are equiprobable. So X X or X have chances being largest are equiprobable. So, [P {X is largest}] or [P {X is largest}] or [P {X is largest}] =1 and P {X is largest} = P {X is largest} = P {X is largest}

[Ans. *] Range 49.9 to 50.1 Set of positive odd number less than 100 is 50. As it is a uniform distribution

∑ x (x) …

∴ *X is l rgest + 19.

Mathematics

[Ans. *] Range 0.15 to 0.18 X X X X X X et z X X X (X ) X X (z ) Pdf of z we need to determine. It is the convolution of three pdf

(z

23.

th

)

∫

z

dz

z

|

[Ans. *] Range 0.79 to 0.81 |x| ,|x|- ∫ e dx √

th

th

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GATE QUESTION BANK

√ √ √ √ √

∫ |x| exp 4

x x

∫ x exp 4

∴pr(P ∪ Q) pr(P) + pr(Q) (D) is true since P Q P n(P Q) n(P) pr(P Q) pr(P)

5 dx 5 dx

5 dx 2.

x

∫ x exp 4

x

∫ x exp 4

√ [ exp (

√ , 24.

x

∫ |x| exp 4

-

[Ans. B] P(A|B) =

5 dx 5 dx

x ) dx]

( he d in tosses nd first toss in he d) = P(HHT) + P(HTH)

4/5

Parcel is sent to R

∴ Required probability = R

3.

1/5

S

Parcel is lost Parcel is lost

parcel

is

=

[Ans. C] If two fair dices are rolles the probability distribution of r where r is the sum of the numbers on each die is given by r P(r)

4/5

that

) ( )

Also, P(first toss is head) =

√

Parcel is sent to

Probability

(

∴ P(2 heads in 3 tosses | first toss is head) ( he ds in tosses nd first toss in he d) (first toss is he d)

[Ans. *] Range 0.43 to 0.45 Pre flow diagram is

1/5

Mathematics

lost

2 Probability that parcel is lost by 3 Probability that parcel is lost by provided that the parcel is lost

4 5

EE 1.

6 [Ans. D] (A) is false since of P & Q are independent pr(P Q) = pr(P) pr(Q) which need not be zero. (B) is false since pr(P ∪ Q) = pr(P) + pr(Q) – pr(P Q) (C) is false since independence and mutually exclusion are unrelated properties.

7 8 9 10 11 12 th

th

th

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GATE QUESTION BANK

The above table has been obtained by taking all different all different ways of obtaining a particular sum. For example, a sum of 5 can be obtained by (1, 4), (2, 3), (3, 2) and (4, 1). ∴ P(x = 5) = 4/36 Now let us consider choice (A) Pr(r > 6) = Pr(r ≥ 7)

P(1, 5, 6) =

=

P(3, 4, 5) =

=

P(1, 2, 5) =

=

∴ Choice (C) P(1, 5 and 6) = 5.

is correct.

[Ans. C] x is uniformly distributes in [0, 1] ∴ Probability density function

= =

Mathematics

=

f(x) =

∴ Choice (A) i.e. pr(r > 6) = 1/6 is wrong. Consider choice (B) pr(r/3 is an integer) = pr(r = 3) + pr (r = 6) + pr (r = 9) + pr (r = 1)

=

=1

∴ f(x) = 1 0 < x < 1 Now E(x ) = ∫ x f(x)dx ∫x

dx

= =

6.

=

[Ans. B] Let N people in room. So no. of events that at least two people in room born at same date

∴ Choice (B) i.e. pr (r/3) is an integer = 5/6 is wrong. Consider choice (C) Now, pr(r/4 is an integer) = pr(r = 4) + pr (r = 8) + pr (r = 12) = =

≥

N

Solving, we get N = 7

+

7.

[Ans. C] (II is red|I is white) (II is red nd I is white) (I is white) (I is white nd II is red) (I is white)

8.

[Ans. B]

=

pr(r = 8) = ∴ pr(r = 8 | r/4 is an integer) =

…

=

= ∴ Choice (C) is correct. 4.

[Ans. C] Dice value 1

Probability

2

and

is the entrie

3

rectangle The region in which maximum of {x, y} is

4

less than ⁄

is shown below as shaded

region inside this rectangle

5 6

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GATE QUESTION BANK y (

)

(

12.

)

Mathematics

[Ans. *] Range 0.35 to 0.45 (

)

x ∫

dx

∫

x|

(

(

)

dx

∫

dx

x|

)

13. p .m x,x y-

∫ f(x)dx

[Ans. *] Range 0.4 to 0.5 ∫ f(x) dx

/

by property

∴ ∫ kx dx k 9.

14.

[Ans. A] (x

) ,e

10.

∫ e dx e -

, e -

e

[Ans. B] Let number of heads = x, ∴ Number of tails n x ∴ ifference x (n x)or (n x n or n x If x n n x n x

n

If n

x

n

IN 1.

x

x

|

[Ans. D]

=52 weeks and 2 days are extra. Out of x)

x

7, (SUN MON) (or) (SAT SUN) are favorable. So, Probability of this event= 2.

or x

[Ans. C] Since the reading taken by the instrument is normally distributed, hence P(x

x )

Where, [Ans. *] Range 0.13 to 0.15 Let proportionality constant = k ∴ ( dot) k ( dots) k ( dots) k ( dots) k ( dots) k ( dots) k ∴k k k k k k k ∴k ∴ rob bility of showing dots

∴k

[Ans. D] Since leap chosen will be random, so, we assume it being the case of uniform probability distribution function. Number of days in a leap year=366 days

As x and n are integers, this is not possible ∴ Probability 0 11.

k

Now

√

√

∫ e

(

)

.dx

μ e n of the distribution σ St nd rd devi tion of the distribution. ∫

exp(

)dx

where, n=x 10 (∴μ kg) and from the data given in question √

∫

e

dx

On equating, we get 0.05=0.84 σ k

σ

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GATE QUESTION BANK

3.

[Ans. D] Mean=

8.

[Ans. B] By definition of Gaussian distribution, total area under the curve =1. Hence half of the area =0.5

9.

[Ans. A]

=5.9 V. (

√

S

̅)

(

̅)

(

̅)

V (closest answer is 0.2)

P(x)= 4.

[Ans. C] ( )

=

Mean = μ ( )

∫ x (x)dx = ∫

Var(x)= ∫ (x

1 2  3 3 5.

Mathematics

=∫

[Ans. A] ]

(x

(x)dx

μ)

) dx =

10.

[Ans. C] Probability of incorrect report

11.

[Ans. C] σ mm μ mm Then probability



 P(x)dx  1

x dx = 6



 Area under triangle =

c 1 2

α 6.

[Ans. A] Probability that the sum of digits of two dices is even is same either both dices shows even numbers or odd numbers on the top of the surface ( ) ∴ ( ) ( ) Where ( ) Probability of occurring even number of both the dices ( ) Probability of occurring odd number of both the dices (

(X where x

(

X

μ

σ mm

(

)

)

( )

e

√ e

√

So, number of measurement more than 10.15mm P Total number of measurement

)

nd (

)

) ≃

∴ ( ) 12. 7.

[Ans. A] ∫ f(x) dx=P or ∫

e

or e

|

.dx =P

[Ans. D] For the product to be even, the numbers from both the boxes should not turn out to be odd simultaneously. ∴ ( )

( )( )

or P = .

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GATE QUESTION BANK

13.

[Ans. A] ∫ f(x)dx e |

14.

15.

∫ e dx

e

[Ans. 2] For valid pdf ∫ ∫

Mathematics

dx

pdf dx

;

;k

[Ans. *] Range 0.890 to 0.899 Probability that job is successfully processed = ( )( )

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GATE QUESTION BANK

Mathematics

Numerical Methods ME – 2005 1. Starting from x = 1, one step of Newton – Raphson method in solving the equation x³ +3x 7=0 gives the next value (x₁) as (A) x₁=0.5 (C) x₁ = .5 (B) x₁= . 0 (D) x₁=2 2.

With a 1 unit change in b, what is the change in x in the solution of the system of equation = 2 .0 0. = (A) Zero (C) 50 units (B) 2 units (D) 100 units

ME – 2006 3. Match the items in columns I and II. Column I Column II (P) Gauss-Seidel (1) Interpolation method (Q) Forward (2) Non-linear Newton-Gauss differential method equations (R) Runge-Kutta (3) Numerical method integration (S) Trapezoidal (4) Linear algebraic Rule equation (A) 2 (B) 2 (C) 2 (D) 2 4.

Equation of the line normal to function ) f(x) = (x (A) y = x 5 (B) y = x 5

at (0 5) is (C) y = x (D) y = x

5 5

ME – 2007 5. A calculator has accuracy up to 8 digits 2

after decimal place. The value of

 sinxdx 0

when evaluated using this calculator by trapezoidal method with 8 equal intervals, to 5 significant digits is (A) 0.00000 (C) 0.00500 (B) 1.0000 (D) 0.00025

ME – 2010 6. Torque exerted on a flywheel over a cycle is listed in the table. Flywheel energy (in J per unit cycle) using impson’s rule is Angle (degree) Torque (N-m) 0 0 60 1066 120 323 180 0 240 323 300 55 360 0 (A) 542 (C) 1444 (B) 992.7 (D) 1986 ME – 2011 7.

The integral ∫

dx, when evaluated by

using impson’s / rule on two equal subintervals each of length 1, equals (A) 1.000 (C) 1.111 (B) 1.098 (D) 1.120 ME – 2013 8. Match the correct pairs. Numerical Order of Fitting Integration Scheme Polynomial . impson’s / 1. First Rule Q. Trapezoidal Rule 2. Second . impson’s / 3. Third Rule (A) P – 2 , Q – 1, R – 3 (B) P – 3, Q – 2 , R – 1 (C) P – 1, Q – 2 , R – 3 (D) P – 3, Q – 1 , R – 2 ME – 2014 9. Using the trapezoidal rule, and dividing the interval of integration into three equal sub intervals, the definite integral ∫ |x|dx is ____________

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GATE QUESTION BANK

10.

The value of ∫ .

( )

“value approximate estimate?

calculated using

the Trapezoidal rule with five sub intervals is _______ 11.

The definite integral ∫

12.

The real root of the equation 5x 2cosx = 0 (up to two decimal accuracy) is _______

13.

Consider

an

equation

= t

.If x =x at t = 0 , the

CE – 2005 Linked Answer Question 1 and 2 Give a>0, we wish to calculate its reciprocal value 1/a by using Newton Raphson Method for f(x) = 0.

2.

The Newton Raphson algorithm for the function will be (A) x

= (x

)

(B) x

= (x

x )

(C) x

= 2x

ax

(D) x

=x

x

in

the

(C) – (D)

CE – 2007 4. The following equation needs to be numerically solved using the NewtonRaphson method x3 + 4x – 9 = 0 the iterative equation for the purpose is (k indicates the iteration level)

differential

increment in x calculated using RungeKutta fourth order multi-step method with a step size of Δt = 0.2 is (A) 0.22 (C) 0.66 (B) 0.44 (D) 0.88

1.

(A) – (B) 0

value”)

is evaluated

using Trapezoidal rule with a step size of 1. The correct answer is _______

ordinary

Mathematics

For a = 7 and starting with x = 0.2 the first two iteration will be (A) 0.11, 0.1299 (C) 0.12, 0.1416 (B) 0.12, 0.1392 (D) 0.13, 0.1428

CE – 2006 3. A 2nd degree polynomial f(x) has values of 1, 4 and 15 at x = 0, 1 and 2 respectively.

5.

(A) x

=

(B) x

=

(C) x

=x

(D) x

=

x

Given that one root of the equation x 10x + 31x – 30 = 0 is 5, the other two roots are (A) 2 and (C) and (B) 2 and (D) 2 and

CE – 2008 6. Three values of x and y are to be fitted in a straight line in the form y = a + bx by the method of least squares. Given ∑x = 6, ∑y = 2 ∑x = and ∑xy = the values of a and b are respectively (A) 2 and 3 (C) 2 and 1 (B) 1 and 2 (D) 3 and 2 CE – 2009 7. In the solution of the following set of linear equation by Gauss elimination using partial pivoting 5x + y + 2z = 34; 4y – 3z = 12; and 10x – 2y + z = 4; the pivots for elimination of x and y are (A) 10 and 4 (C) 5 and 4 (B) 10 and 2 (D) 5 and 4

The integral ∫ f(x) dx is to be estimated by applying the trapezoidal rule to this data. What is the error (define as true th

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GATE QUESTION BANK

CE – 2010 8. The table below given values of a function F(x) obtained for values of x at intervals of 0.25. x 0 0.25 0.5 0.75 1.0 F(x) 1 0.9412 0.8 0.64 0.50 The value of the integral of the function between the limits 0 to using impson’s rule is (A) 0.7854 (C) 3.1416 (B) 2.3562 (D) 7.5000 CE 9.

2011 The square root of a number N is to be obtained by applying the Newton Raphson iterations to the equation x = 0. If i denotes the iteration index, the correct iteration scheme will be (A) x (B) x

= (x

)

= (x

CE – 2013 12. Find the magnitude of the error (correct to two decimal places) in the estimation of following integral using impson’s ⁄ Rule. Take the step length as 1.___________ ∫ (x

1.

Consider

)

(D) x

= (x

)

he error in

 xe dx

=

)

(

)

is 2

1 R xn1   xn   can be used to compute 2 xn 

.

the (A) square of R (B) reciprocal of R (C) square root of R (D) logarithm of R

0 .

CE – 2012 The estimate of ∫ .

1 3

to an accuracy of at least 106

The Newton-Raphson iteration

for a continuous

The values of and ( ) are 19.78 and 500.01, respectively. The corresponding error in the central difference estimate for h = 0.02 is approximately (A) . 0 (C) .5 0 (B) .0 0 (D) .0 0

11.

x

1

function estimated with h=0.03 using the central difference formula f(x)|

=

using the trapezoidal rule is (A) 1000e (C) 100e (B) 1000 (D) 100

f(x)|

(

series

CS – 2008 2. The minimum number of equal length subintervals needed to approximate

3. 10.

the

= 0.5 obtained from the NewtonRaphson method. The series converges to (A) 1.5 (C) 1.6 (D) 1.4 (B) √2

)

= (x

0) dx

CS – 2007

2

(C) x

Mathematics

obtained using

impson’s rule with three – point function evaluation exceeds the exact value by (A) 0.235 (C) 0.024 (B) 0.068 (D) 0.012

CS – 2010 4. Newton-Raphson method is used to compute a root of the equation x 13 = 0 with 3.5 as the initial value. The approximation after one iteration is (A) 3.575 (C) 3.667 (B) 3.677 (D) 3.607

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GATE QUESTION BANK

CS – 2012 5. The bisection method is applied to compute a zero of the function f(x) = x x x in the interval [1,9]. The method converges to a solution after ___________ iterations. (A) 1 (C) 5 (B) 3 (D) 7 CS – 2013 6. Function f is known at the following points: x f(x) 0 0 0.3 0.09 0.6 0.36 0.9 0.81 1.2 1.44 1.5 2.25 1.8 3.24 2.1 4.41 2.4 5.76 2.7 7.29 3.0 9.00 he value of ∫ f(x)dx computed using the trapezpidal rule is (A) 8.983 (C) 9.017 (B) 9.003 (D) 9.045 CS – 2014 7. The function f(x) = x sin x satisfied the following equation: ( ) + f(x) + t cos x = 0. The value of t is _________. 8.

In the Newton-Raphson method, an initial guess of = 2 made and the sequence x x x .. is obtained for the function 0.75x 2x 2x =0 Consider the statements (I) x = 0. (II) The method converges to a solution in a finite number of iterations. Which of the following is TRUE? (A) Only I

Mathematics

(B) Only II (C) Both I and II (D) Neither I nor II 9.

With respect to the numerical evaluation of the definite integral,

= ∫ x dx where a and b are given, which of the following statements is/are TRUE? (I) The value of K obtained using the trapezoidal rule is always greater then or equal to the exact value of the defined integral (II) The value of K obtained using the impson’s rule is always equal to the exact value of the definite integral (A) I only (B) II only (C) Both I and II (D) Neither I nor II ECE– 2005 1. Match the following and choose the correct combination Group I Group II (A) Newton1. Solving nonRaphson linear equations method (B) Runge-Kutta 2. Solving linear method simultaneous equations (C) impson’s 3. Solving ordinary Rule differential equations (D) Gauss 4. Numerical elimination integration method 5. Interpolation 6. Calculation of Eigen values (A) A-6, B-1, C-5, D-3 (B) A-1, B-6, C-4, D-3 (C) A-1, B-3, C-4, D-2 (D) A-5, B-3, C-4, D-1

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GATE QUESTION BANK

ECE– 2007 2. The equation x3 x2+4x 4=0 is to be solved using the Newton-Raphson method. If x=2 is taken as the initial approximation of the solution, then the next approximation using this method will be (A) 2/3 (C) 1 (B) 4/3 (D) 3/2

(A) 2

x

x

.

sin x ..

2 cos x

x

.

..

(C) 2

x

x

.

..

(D) 2

x

x

.

..

8.

The series ∑

eXn 1  eXn X2  eXn 1  Xn   1 (D) Xn1  n Xn -eXn

ECE– 2014 6. The Taylor expansion of is

x

Match the application to appropriate numerical method. Application Numerical Method P1:Numerical M1:Newtonintegration Raphson Method P2:Solution to a M2:Runge-Kutta transcendental Method equation P3:Solution to a M : impson’s system of linear 1/3-rule equations P4:Solution to a M4:Gauss differential equation Elimination Method (A) P1—M3, P2—M2, P3—M4, P4—M1 (B) P1—M3, P2—M1, P3—M4, P4—M2 (C) P1—M4, P2—M1, P3—M3, P4—M2 (D) P1—M2, P2—M1, P3—M3, P4—M4

(C) Xn1  1  Xn 

ECE– 2013 5. A polynomial f(x) = a x a x a x a x a with all coefficients positive has (A) No real root (B) No negative real root (C) Odd number of real roots (D) At least one positive and one negative real root

(B) 2

7.

ECE– 2008 3. The recursion relation to solve x= using Newton-Raphson method is (A) =e (B) = e

ECE– 2011 4. A numerical solution of the equation f(x) = x √x = 0 can be obtained using Newton – Raphson method. If the starting value is x = 2 for the iteration, the value of x that is to be used in the next step is (A) 0.306 (C) 1.694 (B) 0.739 (D) 2.306

Mathematics

converges to

(A) 2 ln 2 (B) √2

(C) 2 (D) e

EE– 2007 1.

The differential equation

=

is

discretised using Euler’s numerical integration method with a time step T > 0. What is the maximum permissible value of T to ensure stability of the solution of the corresponding discrete time equation? (A) 1 (C) (B) /2 (D) 2 EE– 2008 2. Equation e = 0 is required to be solved using ewton’s method with an initial guess x = . Then, after one

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GATE QUESTION BANK

step of ewton’s method estimate x of the solution will be given by (A) 0.71828 (C) 0.20587 (B) 0.36784 (D) 0.00000 3.

A differential equation dx/dt = e u(t) has to be solved using trapezoidal rule of integration with a step size h = 0.01 sec. Function u(t) indicates a unit step function. If x(0)= 0, then value of x at t = 0.01 s will be given by (A) 0.00099 (C) 0.0099 (B) 0.00495 (D) 0.0198

EE– 2009 4. Let x 7 = 0. The iterative steps for the solution using Newton – aphson’s method is given by (A) x

= (x

(B) x

=x

(C) x

=x

(D) x

=x

)

EE– 2013 7. When the Newton – Raphson method is applied to solve the equation f(x) = x 2x = 0 the solution at the end of the first iteration with the initial guess value as x = .2 is (A) 0.82 (C) 0.705 (B) 0.49 (D) 1.69 EE– 2014 8. The function ( ) = is to be solved using Newton-Raphson method. If the initial value of is taken as 1.0, then the absolute error observed at 2nd iteration is ___________ IN– 2006 1. For k = 0 2 . the steps of Newton-Raphson method for solving a non-linear equation is given as

2 5 xk 1  xk  xK2 . 3 3 (x

Starting from a suitable initial choice as k tends to , the iterate tends to (A) 1.7099 (C) 3.1251 (B) 2.2361 (D) 5.0000

)

EE– 2011 5. Solution of the variables and for the following equations is to be obtained by employing the Newton-Raphson iterative method equation(i) 0x inx 0. = 0 equation(ii) 0x 0x cosx 0. = 0 Assuming the initial values = 0.0 and = .0 the jacobian matrix is 0 0. 0 0. (A) * (C) * + + 0 0. 0 0. 0 0 0 0 (B) * (D) * + + 0 0 0 0 6.

Mathematics

IN– 2007 2. Identify the Newton-Raphson iteration scheme for finding the square root of 2.

3.

(A) x

=

(x

)

(B) x

= (x

)

(C) x

=

(D) x

= √2

x

The polynomial p(x) = x + x + 2 has (A) all real roots (B) 3 real and 2 complex roots (C) 1 real and 4 complex roots (D) all complex roots

Roots of the algebraic equation x x x = 0 are ) (A) ( (C) (0 0 0) (B) ( j j) (D) ( j j)

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GATE QUESTION BANK

IN– 2008 4. It is known that two roots of the nonlinear equation x3 – 6x2 +11x 6 = 0 are 1 and 3. The third root will be (A) j (C) 2 (B) j (D) 4

IN– 2013 8. While numerically solving the differential equation

The differential equation

=

with

x(0) = 0 and the constant 0 is to be numerically integrated using the forward Euler method with a constant integration time step T. The maximum value of T such that the numerical solution of x converges is (C) (A) (B)

2xy = 0 y(0) =

using

Euler’s predictor – corrector (improved Euler – Cauchy )method with a step size of 0.2, the value of y after the first step is (A) 1.00 (C) 0.97 (B) 1.03 (D) 0.96

IN – 2009 5.

Mathematics

IN– 2014 9. The iteration step in order to solve for the cube roots of a given number N using the Newton- aphson’s method is

(D) 2

(A) x

=x

(B) x

= (2x

(C) x

=x

(D) x

= (2x

(

x ) )

(

x ) )

IN– 2010 6. The velocity v (in m/s) of a moving mass, starting from rest, is given as

=v

t.

Using Euler’s forward difference method (also known as Cauchy-Euler method) with a step size of 0.1s, the velocity at 0.2s evaluates to (A) 0.01 m/s (C) 0.2 m/s (B) 0.1 m/s (D) 1 m/s IN– 2011 7. The extremum (minimum or maximum) point of a function f(x) is to be determined by solving

( )

= 0 using the

Newton-Raphson method. Let f(x) = x x and x = 1 be the initial guess of x. The value of x after two iterations (x ) is (A) 0.0141 (C) 1.4167 (B) 1.4142 (D) 1.5000

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GATE QUESTION BANK

Mathematics

Answer Keys and Explanations ME 1.

y = sin ( ) = 2

[Ans. C] By N-R method ,

=x –

f(x) = x  f( ) =

x

x =x

(

)

(

)

( ) ( )

y = sin (

x =

y = sin( ) = 0 5 y = sin ( ) =

7

y = sin (

 f (x) = x f ( )= , 

=1

(

)

) = 0.70 0

)=

7 y = sin ( ) =

[Ans. C] Given x y = 2 (i) .0 x 0.0 y = b (ii) Multiply 0.99 is equation (i) and subtract from equation (ii); we get ( .0 0. )x = b (2 0. ) 0.02x = b . 0.02Δx = Δb Δx =

0.02

[Ans. D]

4.

[Ans. B] Given f(x) = (x 2 ) f (x) = (x

f(x)dx = [(y

∫ y ∫

0.70 0

6.

y )

[(0

0)

0.70 0

0.70 0

[(0

y )]

7.

[Ans. C] x y= ∫

( 0

0)

2( 2.7 /unit cycle.

=

Slope of normal = 3 (∵ roduct of slopes = 1) Slope of normal at point (0, 5) y 5 = (x 0) y= x 5 [Ans. A] b a 2 0 h= = = n y = sin(0) = 0

0=0

[Ans. B] ower = ω = Area under the curve. h (y = [(y y ) y y )

/

Slope of tangent at point (0, 5) 2 ) / = m = (0

y

2(0.70 0

2(y

)

2(y

)]

sinx dx =

=

5.

)=0

Trapezoidal rule

= 50 units

3.

0.70 0

(0.5) = .5

=

y = sin ( 2.

0.70 0

1 1

x dx = (y

= (

55)

2 )]

3

2 h

x

2

2

0

y 2

y )

)

= . 8.

[Ans. D] By the definition only

y = sin ( ) = 0.70 0

th

th

th

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GATE QUESTION BANK

9.

[Ans. *] Range 1.10 to 1.12

By intermediate value theorem roots lie be between 0 and 1. et x = rad = 57. 2 By Newton Raphson method f(x ) x =x f (x ) 2x sin x 2 cos x x = 5 2 sin x x = 0.5 2 x = 0.5 25 x = 0.5 2

∫ |x|dx is h ∫ ydx = [y 2

2(y

y

y

x

1

y

1

0.33

2

y ]

y

0.33

2

∫ |x|dx =

.)

y

[

y

0.33

1

0.333

1

2(0.

0.

)]

13.

= . 0 10.

[Ans. *] Range 1.74 to 1.76 2.5 h= = 0. 5 y 2y 2y ∫ . ln (x)dx = [ 2y y =

.

[ln(2.5)

2ln( . ) = .75 11.

2(ln2. )

.

2y

]

2 ln( . )

CE 1.

[Ans. *]Range 1.1 to 1.2

t|

Δx = 0.0

0. = 0.

= 2t

t|

Set up the equation as x = i.e. = a

∫ f(x)dx = [y

y

2(y

iven in question 0 1 1 2 1 0.5

∫ dx = [y x 2

.

.

t Δx = 2

To calculate using N-R method

rapezoidal rule

x y

)dt

[Ans. C]

∫ dx by trapezoidal rule x

h=

[Ans. D] The variation in options are much, so it can be solved by integrating directly dx = t dt ∫ dx = ∫ ( t

ln( )]

2ln( .7)

Mathematics

y

y

..y

)]

a=0 i.e. f(x) =

2 3 0.33

a=0

Now f (x) = f(x ) =

a

f (x ) =

2(y )]

For N-R method = [ 2 = .

0.

2

0.5]

x

=x x

12.

[Ans. *] Range 0.53 to 0.56 Let f(x) = 5x 2 cos x f (x) = 5 2 sin x f(0) = f( ) = 2.

=x

(

)

(

) (

)

Simplifying which we get x = 2x ax

th

th

th

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GATE QUESTION BANK

2.

[Ans. B] For a = 7 iteration equation Becomes x = 2x 7x with x = 0.2 x = 2x 7x = 2× 0.2 – 7(0.2) = 0.12 and x = 2x 7x = 2× 0.12 7(0. 2) = 0.1392

x

[Ans. A] f(x) = 1, 4, 15 at x = 0, 1 and 2 respectively ∫ f(x)dx = (f

2f

=x x

=

5.

f )

∫ f(x)dx = (1 + 2(4) + 15) = 12

and α β

4.

αβ

(

βγ =

)

= 30

βγ

γα = 5β

βγ

5γ =

=

 5 (β γ) βγ = ince βγ = from (i) 5 (β γ) = β γ=5 βγ = olving for β and γ β (5 β) = β 5β =0  β = 2 and γ = Alternative method 5 1 0 31 0 0 5 25 30 1 5 6 0 (x 5)(x )=0 5x (x 5)(x 2)(x )=0 x=2 5

x )dx + =

2=

[Ans. A ] Given f(x) = x x =0 f (x) = x Newton – Raphson formula is

γα =

 βγ = (i) Also

Error = exact – Approximate value =

βγ

α βγ = 5

Now exact value ∫ f(x)dx

= *x

2x x

αβγ=

Approximate value by rapezoidal ule = 12 Since f(x) is second degree polynomial, let f(x) = a0 + a x + a x f(0) = 1 a 0 0= a = f(1) = 4 a a a =  1+ a a = a a =  f(2) = 15 a 2a a = 5  2a a = 5  2a a = Solving (i) and (ii) a = and a = f(x) = 1 – x + 4 x x

=

[Ans. A] Given x – 10 x + 31x 30 = 0 One root = 5 Let the roots be α β and γ of equation ax + bx + cx + d = 0

(3 points Trapezoidal Rule) Here h = 1

=∫ (

f(x ) f (x ) (x x ) ( x ) x x x ( x )

=x

x 3.

Mathematics

6.

[Ans. D] Y = a + bx Given n= ∑x = and ∑xy = th

th

∑y = 2 ∑x = 14

th

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GATE QUESTION BANK

Normal equations are ∑y = na b∑x ∑xy = a∑x b∑x Substitute the values and simply a= b=2 7.

9.

5 [0 0

2

(

)

| 2] → 2

0 [0 5

2

=

f(x ) f(x )

=x

=x

(

x

)

2x

x

=

2x

2

[x

x

]

10.

[Ans. D] Error in central difference formula is (h) This means, error If error for h = 0.03 is 2 0 then Error for h = 0.02 is approximately (0.02) 2 0 0 (0.0 )

11.

[Ans. D] Exact value of ∫ .

| 2] 2

So the pivot for eliminating x is a = 10 Now we eliminate x using this pivot as follows : 0 2 [0 | 2] 5 2 5 0 2 0 2] → [0 0 2 /2 Now to eliminate y, we need to compare the elements in second column at and below the diagonal element Since a = 4 is already larger in absolute value compares to a = 2 The pivot element for eliminating y is a = 4 itself. The pivots for eliminating x and y are respectively 10 and 4 8.

[Ans. A] x

[Ans. A] The equation is 5x + y + 2z = 34 0x + 4y – 3z = 12 and 10x – 2y + z = The augmented matrix for gauss elimination is 5 2 [0 | 2] 0 2 Since in the first column maximum element in absolute value is 10 we need to exchange row 1 with row 3

Mathematics

.

dx = .0

Using impson’s rule in three – point form, b a .5 0.5 h= = = 0.5 n 2 So, x 0.5 1 1.5 y 2 1 0.67 ∫

= =

[

0.5

] [2

0. 7

]

= . So, the estimate exceeds the exact value by Approximate value – Exact value = 1.1116 1.0986 =0.012(approximately)

12.

[Ans. *](Range 0.52 to 0.55) Using impson’s ule X 0 1 2 3 Y 10 11 26 91

4 266

[Ans. A] I = h(f = 0. = 0.7 5

f

2f

f

0.25(

0.

0.

0.5)

∫ (x

f ) 2

= [( 0

2

0)dx 2

)

2(2 )

(

)]

= 2 5. The value of integral th

th

th

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GATE QUESTION BANK

∫ (x

0) dx = *

=

0

=2

x 5

=

0x+ 3.

.

x

=

[Ans. A] +

, x = 0.5

)

(α

)

2α =

=x =α

2α = α + R α =R α=√ So this iteration will compute the square root of R

α= + α= 8α = 4α +9 α = 4.

α = = 1.5 [Ans. A] Here, the function being integrated is f(x) = xe f (x) = xe + e = e (x + 1) f’ (x) = xe + e + e = e (x + 2) Since, both are increasing functions of x, maximum value of f ( ) in interval 1 2, occurs at = 2 so ( )| (2 max |f =e 2) = e Truncation Error for trapezoidal rule = TE (bound)

[Ans. D] y=x dy = 2x dx f(x)= x x

= .5

5.

=

(b – a) max |f ( )| 1

=

(2 – 1) [e (2 + 2)]

=

e

Now putting

(

)

=

= 57 7

)=5 f(x ) 2 oot lies between and

x =(

0

0

)=2 f(x ) 0 2 After ' ' interations we get the root

x =(

= max |f ( )|

.

[Ans. B] f( ) = 5 f( ) = 5 72 ) ) f( 0 f( 0

is number of subintervals

=

.

= . 07

max |f ( )|

Where

(x

2α=α+ =

when the series converges x

=

= 1000 e

At convergence x =x =α α=

Given x

2.

)/

[Ans. C]

5 Magnitude of error = 2 5. 2 . = 0.5 CS 1.

(

Mathematics

2

6.

[Ans. D] h ∫ f(x)dx = [f(0) 2 =

0

=

=

0 [

. .

[

0.

f( )

2(0.0 0. . . 7.2 )

2(∑f )] ]

5 . ]

= 9.045

h= Now, No. of intervals,

= th

th

th

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GATE QUESTION BANK

7.

8.

9.

ECE 1.

2.

[Ans. – ] Given (x) + f(x) + t cos x = 0 and f(x) = x sin x f (x) = x cos x + sin x f (x) = x ( sin x) + cos x + cos x = 2 cos x – x sin x = 2 cos x – f (x) 2 cos x – f (x) + f(x) +t cos x = 0 2 cos x = tcos x t = 2 [Ans. A] f(x) = 0.75x 2x 2x f (x) = 2.25x x 2 x =2 f = 2 f = f x =x =0 f f = f = 2 f x =x =2 f f = 2 f = f x =x =0 f Also, root does not lies between 0 and 1 So, the method diverges if x = 2 nly ( )is true.

x1  2  3.

 1  x n  4.

x1  x0 

e e

e xn 1  exn

[Ans. C] x

f(x ) f (x )

=x

f(2) = (2

x

) = √2

√2

f (2) =

√

=2

and

√

=

√ )

(√ √

= .

√

5.

[Ans. D] f(x) = a x a x a x a x a If the above equation have complex roots, then they must be in complex conjugate pair, because it’s given all co-efficients are positive ( they are real ) So if complex roots are even no. (in pair) then real roots will also be even. ption ( )is wrong From the equation ( 0) roduct of roots = As no. of roots = 4, Product of roots < 1 either one root 0 (or) Product of three roots < 0 ption ( )is rong. Now, take option (A), Let us take it is correct . Roots are in complex conjugate pairs = Product of roots 0 | | | | 0 which is not possible ption (A) is wrong orrect answer is option ( )

[Ans. C] By definition (& the application) of various methods

4=0

Next approximation x1  x0 

8 4  12 3

[Ans. C] Given : f(x)= x e By Newton Raphson method, f(x ) x x =x =x f (x )

[Ans. C] For value of K if trapezoidal rule is used then the value is either greater than actual value of definite integral and if impson’s rule is used then value is exact Hence both statements are TRUE

[Ans. B] y(t) =x3 x2 + 4x x0 = 2

Mathematics

f  x0  f ' x0 

x03  x02  4x0  4 3x02  2x0  4 th

th

th

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GATE QUESTION BANK

6.

[Ans. A] sin x 2 cos x x = (x – )

7.

[Ans. B]

8.

[Ans. D] ∑

n

2( –

x 2

.

x

)

Put x = as given, x = [e ( 2) ]/e = 0.71828

)

[Ans. C]

=e

.. = e

u(t)

x

x 2

.

. . x in t

[Ans. D] Here,

x = ∫ e u(t) dt = ∫ f(t) dt At t = 0.01, x = Area of trapezoidal

= x

f(x y) =

x

h

=(

)x h

4.

x

5.

=[

=*

( )

f(x ) = e f’(x ) = e

6.

(

]

)

(

)

+

0x cos x 0x sinx

20x

0sinx ] 0cosx

0 0

is

0 + 0

[Ans. D] x x x =0 (x )(x )=0 x =0 x =0 x= x= j

=x –(

(

The matrix at x = 0 x =

( )

=

.

[Ans. B] u(x x ) = 0x sin x 0. = 0 v(x x ) = 0x 0x cosx 0. = 0 The Jacobian matrix is u u x x v v [ x x ]

[Ans. A] Here f(x) = e f (x) = e The Newton Raphson iterative equation is

=

=x

= *x

|

Δ 2 o maximum permissible value of Δ is 2 .

i.e. x

e

[Ans. A]

since h = Δ here Δ

x

[

= x

h

=x

.

= 0.0099

h

or stability |

x

f(0.0 )] =

= [f(0)

Euler’s method equation is x = x h. f(x y ) x x = x h( )

2.

i=0

(

x =

2

as e = EE 1.

Now put

3.

=

Mathematics

)

)

th

th

th

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GATE QUESTION BANK

7.

[Ans. C] x =x =

.2

Hence, it will have atleast 5 (0+1)= 4 complex roots.

f(x ) f (x ) ( .2) 2( .2) ( .2) 2

4.

[Ans. C] Approach- 1 Given, x3 – 6x2 + 11x – 6 = 0 Or (x – 1)(x – 3)(x – 2) = 0 x= 1, 2, 3.

= 0.705 8.

[Ans. *] Range 0.05 to 0.07 Clearly, x = 0 is root of the equation f(x) = e =0 f (x) = e and x = .0 Using ewton raphson method (e . ) f(x ) x =x = = f (x ) e. e and x = x

f(x ) = f (x ) e =

(e

Approach- 2 For ax3 +bx2 + cx +d = 0 If the three roots are p,q,r then Sum of the roots= p+q+r= b/a Product of the roots= pqr= d/a pq+qr+rp=c/a

) e

5.

[Ans. D] dx x = dt f(x, y) =

e

e = 0. 7 0. = 0.0 Absolute error at 2nd itteration is |0 0.0 | = 0.0 IN 1.

x

=x h

=( [Ans. A] As k ∞ xk+1 ≈xk xk = x

h (x y ) = x )x

2.

3.

h

h(

x

)

)

h

|

h

x

/

(

or stability |

Δ

x = x x =5 x =5

Mathematics

Δ

= 1.70

[Ans. A] Assume x = √ f(x) = x =0 f(x ) x =x = [x f (x ) 2

6.

[Ans. A] dv =v t dt t v dv =v t dt 0 0 0 0+0 0. = 0 0.1 0 0+0.1 0. = 0.0

7.

[Ans. C] f(x) = x x f (x) = x = g(x) x = initial guess g (x) = x g (x ) x =x g (x )

2 ] x

[Ans. C] Given p(x) = x + x + 2 There is no sign change, hence at most 0 positive root ( rom escarte’s rule of signs) p( x) = x x+2 There is one sign change, hence at most 1 negative root ( rom escarte’s rule of signs)

2

th

th

th

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GATE QUESTION BANK

(

= x =x = .5 = .

)

= .5

g(x ) g (x ) 0.75 7

8.

[Ans. D] dy = 2xy x = 0 y = h = 0.2 dx y =y h. f(x y ) (0.2)f(0 ) = = and y = y [f(x y ) f(x y )] (0. )[f(0 ) f(0.2 )] = = 0. is the value of y after first step, using Euler’s predictor – corrector method

9.

[Ans. B] For convergence x

Mathematics

= x =x x=

x =

(2x

x

)

x= √

th

th

th

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GATE QUESTION BANK

Mathematics

Calculus ME – 2005 1.

ME – 2006

The line integral ∫ ⃗ ⃗⃗⃗⃗ of the vector function ⃗ ( ) 2xyz ̂+ x²z + ̂ k²y ̂ from the origin to the point P (1,1,1) (A) is 1 (B) is Zero (C) is 1 (D) cannot be determined without specifying the path

2.

be (A) (B) 8.

(A)

(C)

(B)

(D)

Changing the order of the integration in the double integral I = ∫ ∫

(

∫ (

What is q?

(A)

(C) X (D) 8 )

(A)

√

(C)

√

(B)

√

(D)

.

(

10.

/

)

(A) 0 (B) ⁄

is equal to 11.

∫

(C) (D) 1

The area of a triangle formed by the tips of vectors a , b and c is (A)

(

)(

(D) Zero

(B)

|(

)

Stoke’ theorem connects (A) A line integral and a surface integral (B) Surface integral and a volume integral (C) A line integral and a volume integral (D) Gradient of function and its surface integral

(C)

|

(D)

(

(C) 2∫ (

√

ME – 2007

(B) 2∫

6.

1 and t is a real number,

Let x denote a real number. Find out the INCORRECT statement + represents the set if all (A) S * real numbers greater then 3 + represents the empty (B) S * set + represents the (C) S * union of set A and set B + represents the set (D) S * of all real umbers between a and b, where and b are real number

)

leads to (A) 4y (B) 16y²

(C) 0 (D) ⁄

dt is:

∫

9.

4.

)

⁄ ⁄

√

By a change of variables x(u,v) = uv, y(u,v) = v/u is double integral, the integral f(x,y) changes to f(uv, u/v) ( ). Then, ( ) (A) 2 v/u (C) v² (B) 2 u v (D) 1

I =∫ ∫ (

2x2  7x  3 , then limf(x) will x 3 5x2  12x  9

Assuming i =

The right circular cone of largest volume that can be enclosed by a sphere of 1 m radius has a height of

3.

5.

If f( x ) =

7.

)

th

th

) (

)|

| )

th

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GATE QUESTION BANK

12.

√

If

√

y (2) = (A) 4 or 1 (B) 4 only 13.

, then

√

14.

20.

The divergence of the vector field )̂ ( ) ̂ is (x y) ̂ ( (A) 0 (C) 2 (B) 1 (D) 3

21.

Let

(C) 1 (D) 1/ln2

y2  4x and x2  4y is ⁄ (A) (B) 8 23.

⁄ (C) (D) 16

The distance between the origin and the point nearest to it on the surface

The directional derivative of the scalar

z2  1  xy is

function f(x, y, z) = x2  2y2  z at the

(A) 1

point P = (1, 1, 2) in the direction of the vector ⃗ ̂ ̂ is (A) 4 (C) 1 (B) 2 (D) 1

at x=2, y=1?

ME – 2009 22. The area enclosed between the curves

Which of the following integrals is unbounded? (C) ∫ (A) ∫ (D) ∫

What is

(A) 0 (B) ln2

The length of the curve

(B) ∫ 16.

In the Taylor series expansion of ex about x = 2, the coefficient of (x 2)4 is ⁄ (A) ⁄ (C) ⁄ (B) (D) ⁄

The minimum value of function y = x2 in the interval [1, 5] is (A) 0 (C) 25 (B) 1 (D) Undefined

between x = 0 and x = 1 is (A) 0.27 (C) 1 (B) 0.67 (D) 1.22 15.

19.

(C) 1 only (D) Undefined

ME – 2008

Mathematics

(C) √ (D) 2

(B) √ ⁄ 24.

A path AB in the form of one quarter of a circle of unit radius is shown in the figure. Integration of  x  y  on path AB 2

17.

Consider the shaded triangular region P shown in the figure. What is∬ xydxdy? y

P

18.

B

X

x

A

2

⁄ ⁄

The value of (A) (B)

Y

x+2y=2

1

0 (A) (B)

traversed in a counter-clockwise sense is

⁄ ⁄

(C) ⁄ (D) 1

(

)

(C) (D)

25.

is ⁄ ⁄

th

(A)

(C)

(B)

(D) 1

The divergence of the vector field ̂ at a point (1,1,1) is ̂ ̂ equal to (A) 7 (C) 3 (B) 4 (D) 0 th

th

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GATE QUESTION BANK

ME – 2010 26. Velocity vector of a flow field is given as ⃗ ̂ .̂ The vorticity vector at (1, 1, 1) is (A) 4 ̂ ̂ (B) 4 ̂ ̂ 27.

The function (A) o o

(C) ̂ (D) ̂

∀ ∀ (B) o o ∀ ∀ except at x = 3/2 (C) o o ∀ ∀ except at x = 2/3 (D) o o ∀

28.

29.

ME – 2012 33. Consider the function ( ) in the interval . At the point x = 0, f(x) is (A) Continuous and differentiable. (B) Non – continuous and differentiable. (C) Continuous and non – differentiable. (D) Neither continuous nor differentiable.

̂ ̂

R R R R

34.

R R

ME – 2011 30. If f(x) is an even function and is a positive real number, then ∫ ( )dx equals

31.

What is (A) (B)

32.

36.

For the spherical surface the unit outward normal vector at the point

is

(C) π (D) π

(C)

(B) (C)

is

√

(A) (B)

has

/

√

̂

√

̂

√

̂

√

̂

√

(C) ̂ (D) 37.

equal to?

A series expansion for the function (A)

.

∫ ( )

(C) 0 (D) 1

(C) 1 (D) 2

At x = 0, the function f(x) = (A) A maximum value (B) A minimum value (C) A singularity (D) A point of inflection

R except at x = 3 ∀ R

(D)

/ is

35.

The parabolic arc is √ revolved around the x-axis. The volume of the solid of revolution is (A) π (C) π (B) π (D) π

(A) 0 (B)

. (A) 1/4 (B) 1/2

The value of the integral ∫ (A) –π (B) –π

Mathematics

√

̂

√

̂

√

̂

The area enclosed between the straight line y = x and the parabola y = in the x – y plane is (A) 1/6 (C) 1/3 (B) 1/4 (D) 1/2

ME – 2013 38. The following surface integral is to be evaluated over a sphere for the given steady velocity vector field defined with respect to a cartesian coordinate system having i, j and k as unit base vectors. ∫∫ (

(D) th

th

) th

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GATE QUESTION BANK

Where S is the sphere, and n is the outward unit normal vector to the sphere. The value of the surface integral is (A) π (C) π⁄ (B) π (D) π 39.

45.

If a function is continuous at a point, (A) the limit of the function may not exist at the point (B) the function must be derivable at the point (C) the limit of the function at the point tends to infinity (D) the limit must exist at the point and the value of limit should be same as the value of the function at that point

The value of the definite integral ( )

∫ √

is

(A)

√

(C)

√

(B)

√

(D)

√

46.

Divergence of the vector field ̂ ( ̂ ̂ ) is (A) 0 (C) 5 (B) 3 (D) 6

(C) 3 (D)Not defined

47.

The value of the integral

ME – 2014 40.

is (A) 0 (B) 1

∫ 41.

Which one of the following describes the relationship among the three vectors ̂ ̂ ̂ ̂ + ̂ + ̂ ̂ ̂ ̂ (A) The vectors are mutually perpendicular (B) The vectors are linearly dependent (C) The vectors are linearly independent (D) The vectors are unit vectors

42.

.

(

)

/ is equal to

(A) 0 (B) 0.5 43.

̂

̂

)̂ )̂ ̂ ̂

̂ ̂ ̂ ̂

)

(

)

) (

)

(A) 3 (B) 0 48.

(C) 1 (D) 2

The value of the integral ∫ ∫ is (

)

(C) (

(B) (

)

(D) .

(A)

) /

).

Where, c is the square cut from the first quadrant by the lines x = 1 and y = 1 will ( G ’ h o o h h line integral into double integral) (A) ⁄ (C) ⁄ (B) 1 (D) ⁄

̂ ̂

( (

CE – 2005 1. Value of the integral ∮ (

(C) 1 (D) 2

Curl of vector ⃗ ̂ (A) ( (B) ( (C) (D)

44.

Mathematics

̂ ̂ 2.

A rail engine accelerates from its stationary position for 8 seconds and travels a distance of 280 m. According to the Mean Value theorem, the speedometer at a certain time during acceleration must read exactly. (A) 0 kmph (C) 75 kmph (B) 8 kmph (D) 126 kmph

The best approximation of the minimum value attained by (100x) for ≥ is _______

th

th

th

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GATE QUESTION BANK

CE – 2006 3. What is the area common to the circles o 2 (A) 0.524 a (C) 1.014 a2 (B) 0.614 a2 (D) 1.228 a2 4.

The directional derivative of f(x, y, z) = 2 + 3 + at the point P (2, 1, 3) in the direction of the vector a= k is (A) (C) (B) (D)

CE – 2007 5. Potential function  is given as = . When will be the stream function () with the condition  = 0 at x = y = 0? (A) 2xy (C) (B) + (D) 2 6.

Evaluate ∫ (C) π⁄ (D) π⁄

(A) π (B) π⁄ 7.

10.

12.

transformed to (A) (B)

9.

(C) √ (D) 18

parabola is y = 4h

(A) ∫ √ √

√

(D)

√

√

(C) ∫

= 0 by substituting (C)

where x is the

horizontal coordinate and y is the vertical coordinate with the origin at the centre of the cable. The expression for the total length of the cable is

= 0 can be

(D)

14.

√

∫

.

The



/

is

(A) 2/3 (B) 1

The inner (dot) product of two vectors ⃗ and ⃗ is zero. The angle (degrees) between in two vectors is (A) 0 (C) 90 (B) 30 (D) 120

(C) 40.5 (D) 54.0

√

(B) 2∫

+

is

CE – 2010 13. A parabolic cable is held between two supports at the same level. The horizontal span between the supports is L. The sag at the mid-span is h. The equation of the

CE – 2008 +

)

For a scalar function f(x, y, z) = the directional derivative at the point P(1, 2, 1) in the ⃗ is direction of a vector (A) (B)

A velocity is given as ̅ = 5xy + 2 y2 + 3yz2⃗ . The divergence

The equation

The value of ∫ ∫ ( (A) 13.5 (B) 27.0

CE – 2009 11. For a scalar function f(x, y, z) = + 3 + 2 the gradient at the point P (1, 2, 1) is (A) 2 + 6 + 4⃗ (C) 2 + 12 + 4⃗ (D) √ (B) 2 + 12 – 4⃗

of this velocity vector at (1 1 1) is (A) 9 (C) 14 (B) 10 (D) 15

8.

Mathematics

15.

th

(C) 3/2 (D)

Given a function ( ) The optimal value of f(x, y) th

th

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GATE QUESTION BANK

(A) Is a minimum equal to 10/3 (B) Is a maximum equal to 10/3 (C) Is a minimum equal to 8/3 (D) Is a maximum equal to 8/3

CE – 2013 21.

CE – 2011 16.

∫

√ √

√

?

22.

(C) a (D) 2a

/

o

magnitudes a and b respectively. |⃗ ⃗ | will be equal to (A) – (⃗ ⃗ ) (C) + (⃗ ⃗ ) (B) ab ⃗ ⃗ (D) ab + ⃗ ⃗ CE – 2012 19. For the parallelogram OPQR shown in the sketch, ̅̅̅̅ ̂ ̂ and ̅̅̅̅ R ̂ .̂ The area of the parallelogram is Q

(C) 1 (D)

24.

A particle moves along a curve whose parametric equation are : and z = 2 sin (5t), where x, y and z show variations of the distance covered by the particle (in cm) with time t (in s). The magnitude of the acceleration of the particle (in cm ) at t = 0 is ___________

25.

If {x} is a continuous, real valued random variable defined over the interval ( ) and its occurrence is defined by the density function given as:

(C) 1 (D) π

If ⃗ and ⃗ are two arbitrary vectors with

R P

.

( )

/

√

wh

‘ ’

‘ ’

the statistical attributes of the random variable {x}. The value of the integral

O

(A) ad –bc (B) ac+bd

.

With reference to the conventional Cartesian (x, y) coordinate system, the vertices of a triangle have the following coordinates: ( ) ( ) ( ) ( ) ( ) ( ). The area of the triangle is equal to (A) ⁄ (C) ⁄ (B) ⁄ (D) ⁄

π

(A) 0 (B) π

(C) ad + bc (D) ab – cd

∫

o

o

. √

/

dx is

(A) 1 (B) 0.5

The infinity series

(A) sec (B)

(C) 1 (D) 8/3

23.

π

20.

o

(A) (B)

Wh ho h o λ such that the function defined below is continuous π ? f(x)={

18.

The value of ∫ (A) 0 (B) 1/15

CE – 2014

(A) 0 (B) a/2 17.

Mathematics

o

26.

(C) o (D)

(C) π (D) π⁄

The expression

o

(A) log x (B) 0 th

th

(C) x log x (D) th

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GATE QUESTION BANK

CS – 2005 1.

Let G(x) 

CS – 2010

1   g(i)xi where |x| 0 is equal to (C) (D)

is equal to (A) 0 (B) 4

(B) .

∫ ∫

(C) (D)

is.

IN – 2008 12. Consider the function y = x2 6x + 9. The maximum value of y obtained when x varies over the interval 2 to 5 is (A) 1 (C) 4 (B) 3 (D) 9

The solution of the integral equation ()

(C)

For real x, the maximum value of (A) 1 (B) e 1

1

8.

Mathematics

√

√

√

/

(D) .

√

√

√

√

√

√

/

IN – 2010 17. The electric charge density in the region R: is given as σ( ) , where x and y are in meters. The total charge (in coulomb) contained in the region R is (A) π (C) π⁄ (B) π (D) 0 th

th

th

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GATE QUESTION BANK

18.

The integral ∫

.

evaluates to (A) 6 (B) 3 19.

/ sin(t) dt

23.

A scalar valued function is defined as ( ) , where A is a symmetric positive definite matrix with dimension n× n ; b and x are vectors of dimension n×1. The minimum value of ( ) will occur when x equals ) (A) ( (C) . / ) (B) – ( (D)

24.

Given ()

(

()

o .

(C) 1.5 (D) 0

The infinite series ( )

…………

converges to (A) cos (x) (B) sin(x)

(C) sinh(x) (D)

IN – 2011 20.

The series ∑ for (A) (B)

(

)

Mathematics

π) π

π /

The o w (A) A circle (B) A multi-loop closed curve (C) Hyperbola (D) An ellipse

converges

(C) (D)

IN – 2013 21. For a vector E, which one the following statement is NOT TRUE? (A) E E o o (B) If E E is called conservative (C) If E E is called irrotational (D) E E -rotational IN – 2014 22. A vector is defined as ̂ ̂ ̂ ̂ are unit vectors in Where ̂ ̂ Cartesian ( ) coordinate system. The surface integral ∯ f.ds over the closed surface S of a cube with vertices having the following coordinates: (0,0,0),(1,0,0),(1,1,0),(0,1,0),(0,0,1), (1,0,1),(1,1,1),(0,1,1) is________

th

th

th

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GATE QUESTION BANK

Mathematics

Answer Keys and Explanations ME 1.

= =

Since, potential function of ⃗ is x²yz ( ) ( ) ( ) 2.

0 o

1

[Ans. A]

[Ans. D] o

h

π h

π o

(

√

0

9.

[Ans. B]

10.

[Ans. B] (

)

For V to be max

)

This is of the form . /

Hence, h

Applying L hospital rule (

3.

√

1

)

[Ans. A]

. /

= (

)

|

|

|

|

= = 11.

4.

[Ans. A] (

After changing order ∫ ∫ 5.

[Ans. A] I= ∫ (

)

=2∫

[ ∫

[Ans. B] Let the vectors be

) ( )(⃗ )(⃗ )

]

= 2∫ 6.

[Ans. A] A Line integral and a surface integral is connected by stokes theorem

7.

[Ans. B]

Now Area vector will be perpendicular to plane of i.e. will be the required unit vector. And option (A), (D) cannot give a vector product )| |(⃗ ⃗ ) (⃗ 12.

[Ans. B]

Applying ’ Hospital rule, we get I= 8.

[Ans. A] ∫

√

Given:

I=

(

)

(

)

√

√

√

For 0 1

[

] th

th

th

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GATE QUESTION BANK

13.

14.

But y is always greater than x. Hence y= 4 only.

= ∫

[Ans. B] Since interval given is bounded, so minimum value of functions is 1.

= 0

(

)

)

)

Now by partial fractions, (a3 8) = (a 2)(a2+2a+4)

) |

⇒L=

[Ans. D] To see whether the integrals are bounded or unbounded, we need to see that the o ’ h h interval of integration. Let us write down the range of the integrands in the 4 options, Thus, (D) , i.e., ∫ [Ans. B] h

19.

o

o

o Φ (

Φ)

̂

̂ ). (

̂

( ̂

( )

(

( )

)

( )

( )

Coefficient of (x- )⁴ Now f(x)= ex ⇒ (x)= ex ⇒ (a)= ea ( )

Hence for a=2, 20.

⃗⃗

[Ans. D] div {( (

Hence directional derivative is (grad (x2+2y2+z)).

=

[Ans. C] Taylor series expansion of f(x) about a is given by ( )

dx is unbounded.

along a vector ⃗

(2x ̂

( (

A (0,1); B (0,1); C (0, ); D (0, )

16.

[Ans. B]

Let x= a3 ⇒ a=2

=1.22 15.

1

).dx

= (

)

L=

∫√

L =∫ (√

)

= 18.

h

(

= ∫ (

[Ans. D] h

Mathematics

)̂

(

)

)̂ (

)̂}

( )

(

)

=3

̂)

√ ̂)

21.

[Ans. C]

= Hence at (1,1,2), ⇒

Directional derivative = 17.

[Ans. A] I = ∬ .dx dy The limit of y is form 0 to

and limit

of x from 0 to 2 I =∫ ∫

⇒

( )

⇒

( (

∫

.

)(

) )

/ th

th

th

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GATE QUESTION BANK

22.

[Ans. A] Given:

23.

y2  4x x2  4y

Mathematics

[Ans. A] Short method: Take a point on the curve z = 1, x = 0, y=0 Length between origin and this point ) ( ) ( ) =1 √( This is minimum length because all options have length greater than 1.

(4,4) (0,0)

24.

x4   4x 16

[Ans. B] Y

or x4  64x

B

or x(x  64)  0 3

or x3  64 or x  4

x = cos  y=sin 

 y 4  Required area = ∫ .√

Path is x2  y2  1

/

R  e (x  y)2  1  2sin  cos 

4

 2 x3   2 x3 2    3 120  4 64  (4)3 2  3 12 32 16 16    3 3 3

2

2

 cos2    (1  sin2)d   2 0  0 =

Alternately For point where both parabolas cut each other

 1 1     1 2 2 2 2

Alternately Given: x2  y2  1 Put x=cos  , and y=sin 

y2  4x, x2  4y

 x  y 2  cos2  sin2  2sincos

 x  4 4x 2

= 1  sin2

or x2  8 x or x4  64x

∫ (

or x3  64

 x  4,0 ,(4,0)

2

4

x2 dx 0 4

     1 2 

  4x   0

)

 cos2    1 1        2 0  2 2 2  

 Required area 4

X

A

4

 2 x3  16  2 x3 2     3 120 3 

25.

[Ans. C]

F  3xzi  2xyj  yz2k ⃗ ⃗

th

th

th

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GATE QUESTION BANK

(

)

(

)

(

√

)

 3z  2x  2yz

π ∫

At point (1, 1, 1), divergence =3+2 2=3 26.

⃗

o

30.

⃗ ̂

̂

||

31. ( ̂

27.

)̂

(

∫ ( )

[Ans. D] Standard limit formulae

) ̂

32.

[Ans. B]

33.

[Ans. C] The function is continuous in [ 1, 1] It is also differentiable in [ 1, 1] except at x = 0. Since Left derivative = 1 and Right derivative = 1 at x = 0

34.

[Ans. B]

[Ans. C]

1

1

2

y is continuous for all x differentiable for all x since at

o

o

R, and R, except at

o

,

Using this standard limit, here a = 1 then = ( ) /2 =1/2

’ h

value towards the left and right side of

35.

[Ans. D] ( ) ( ) ( ) ( ) f(x) has a point of inflection at x =0.

36.

[Ans. A]

[Ans. D] ,

∫

29.

( )

̂

⇒

28.

π

)

[Ans. D] If f(x) even function ∫

||

o

π* +

π(

[Ans. D] ⃗ ̂

Mathematics

-

[Ans. D]

π

∫ π

(

) ̂

Volume from x = 1 to x = 2,

̂

∫π ( √ th

th

√

̂ ̂ ̂

̂

) th

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GATE QUESTION BANK

̂

√ √ ̂

√ ̂

̂ ̂

√ ̂

38.

(√ √

37.

̂

̂ √

[Ans. A] By Gauss Divergence theorem, ∬( ̅ ̅)

√ √ The unit outward normal vector at point P is (

Mathematics

)

∭(

(Surface Integral is transformed to volume Integral)

)

( )

( )

( )

)̂

√ ̂

)

∭(

∭

[Ans. A] The area enclosed is shown below as shaded

π π

(

∬( ̅ ̂)

)

)

∭(

( π) (

)

The coordinates of point P and Q is obtained by solving y = x and y = simultaneously, i.e. x = ) ⇒ ( ⇒ Now, x = 0 ⇒ which is point Q(0,0) and x = 1 ⇒ which is point P(1,1) So required area is

π 39.

[Ans. C] ∫(√ ) ( ) Using Integration by parts ∫

∫

Here, f=ln(x) and dg=√ and g=

∫

* +

∫

o ∫(√ ) ( )

* + [

]

∫

[

]

[

(

th

th

( ) ] )

th

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GATE QUESTION BANK

40.

π

[Ans. A] o

0

1

So the minimum value is

[Differentiating both o o Hospital method]

o w

.

=

’

. /

45.

o

41.

Mathematics

/

[Ans. D] o o o ( ) ( )

o

( )

o

o

otherwise it is said to be discontinuous. So the most appropriate option is D.

[Ans. B] G

o 46.

|

|

[Ans. C] ̂ ̂ Div

̂

(

)

Vectors are linearly dependent 42.

[Ans. B] (

) ) ( ) , o ( )( ) ( ) o ( )( ) 43.

(

47. -

ho

[Ans. B] Let ∫

(

)

(

( o (

)

[Ans. A] ̂

(

⃗ [ ̂[

(

] )

(

) ∫

̂ ̂

)

48.

)]

)

()

o

̂[ ,̂ ( 44.

(

) (

( ) )̂

∫

|

,

-

)] (

,̂ (

o

[Ans. B] ∫ ∫

̂[

o

∫

)] ̂,

)̂

(

-

∫ (

)

|

)̂ [

[Ans. *] Range 1.00 to 0.94 h o π

,

th

th

] -

,

th

-

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GATE QUESTION BANK

CE 1.

a =2a cos i.e, cos [Ans C] G ’ theorem is 

∮(

)

∬(

∮ ((



(

( )

)

= y and

=∫

∫

=∫

,

(

)-

=

×

( )

π . /

|

π . /

π (

π *

[Ans. D] Since the position of rail engine S(t) is continuous and differentiable function according to Lagrange’s mean value theorem more )

(

o ) o

∫(

w

∫

(t) = v(t) =

(

π * (

( )

π

π π

√

| π

√

√

)

√

)

)+

+

)

m/sec kmph

4.

[Ans. C] f = 2 +3

= 126 kmph Where v(t) is the velocity of the rail engine. 3.

∫

∫

(

-

= 2y

=∫

=

∫

)

’ h o I= ∫

)

, ∫

 = xy 

)

(

)

)

⇒

R

Here I = ∮ (

2.

Mathematics

[Ans. D] h ’ o h r=2acos (i) r = a represents a circle with centre ( ) ‘ ’ (ii) r = 2acos represents a circle symmetric about OX with centre at ( ) ‘ ’ The circles are shown in figure below. At h o o o ‘ ’ P y Q π 3

O

A

(

)⃗

= 4xi + 6yj + 2zk At P (2, 1, 3) Directional derivation ̂ ( ) ( ) ( ) √ ( ) ( ) ( ) √ √ 5.

[Ans. A] Potential function,

x

th

th

th

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GATE QUESTION BANK

8. Integrating ∫

[Ans. D] Put

∫

wh

Mathematics

o

⇒

√



=√



=√

…… ( )

Now given equation is ……….. (ii) 6.

[Ans. B] Let I(α) =∫ (

h ∫ )

.

/

h

(

dx …( ) =

(

)

(

h

)

√ ) [ from eqn(i)]

=

∫ Then Integrating by parts we get, =

0

=

.

( α

√

h

(

)

o )1

/

√

(

√

(

(

h

)

h

)

= dI = Integrating, I = ( )

α o

h

√

)

h

() ( )

+C=0 C= (α)  ( )

α

π

Now substitute in eqn (ii) we get h h

π

I(0) = But from equation (i), I(0) = ∫ ∫

h

⇒ dx

h

⇒

dx =

h h

Which is the desired form 7.

[Ans. D] ̅=5 +2

√

+ 3y ⃗

(⃗ )

9.

[Ans. C] ̅ ̅=0 ̅ ̅ If ̅ ̅ = 0

= 5y + 4y + 6yz At(1, 1, 1) div ( ) = 5.1 + 4.1 + 6.1.1 = 15

is the correct transformation.

o

 o Since P and Q are non-zero vectors  o 0  th

th

th

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GATE QUESTION BANK

10.

[Ans. A] Since the limit is a function of x. We first integrate w.r.t. y and then w.r.t. x )

∫ ∫(

∫

√ √

√

)

[Ans. D] Length of curve f(x) between x = a and x = b is given by ∫√

∫

*

√

√ 13.

∫(

Mathematics

(

)

+ Here,

∫ (

4h … … ( )

= 8h

)

Since ∫ (

and y = h at x =

)

* ( ) *

(As can be seen from equation (i), by substituting x = 0 and x = L/2)

( )+

(Length of cable)

+

√

=∫

.

/ ∫ √

ho 11.

[Ans. B] f = + 3 +2 f = grad f = i

+j

[Ans. A]

15.

[Ans. A] ( )

+k

= i(2x) + j(6y) + k(4z) The gradient at P(1, 2, 1) is = i(2×1) + j(6×2) + k(4 ( )) = 2i + 12j – 4k 12.

14.

[Ans. B] (

)

⃗

⃗ ⃗

̂

⃗

h

Putting,

√ o (

Given,

̂ )

.

/ is the only stationary point.

√ *

+ .

√

th

th

/

th

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GATE QUESTION BANK

*

Since the limit is in form of

+ .

*

’ ho and get λ

/

+ .

Since, We have either a maxima or minima at

o ()

⇒λ 18.

/

Also since, r=0 )

o

1

.

/

[Ans. A]

= 8 > 0, the point

(

o

)

,

)

19.

o -

[Ans. A] Area = |̅ ̅ | ̅̅̅̅ ̅̅̅̅ R (

So the optimal value of f(x, y) is a

|

) o

o

(

The minimum value is (

o

π

⇒λ

(

o

, we can use

/

Since,

.

Mathematics

)

(

)

|

minimum equal to 16.

̅̅̅̅ ̅̅̅̅ R ̅̅̅̅ ̅̅̅̅ R

[Ans. B] Let I = ∫

√ √

Since ∫ ( ) I=∫

√ √

…( )

√

√

∫ (

20.

[Ans. B]

21.

[Ans. B]

( )

( )

̅(

)

)

…( )

(i) + (ii)  2I = ∫

√

√

√

√

 2I = ∫

∫

2I = |

o ∫

o

o

 I = a/2 17.

⇒

[Ans. C] For a function f(x) to be continuous, at x=a ( ) ( )

o ∫ (

⇒

)

∫ (

)

If f(x) is continuous at x= π . /

*

λ o

+

[

th

th

]

th

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GATE QUESTION BANK

[

]

|

(

Mathematics

)

(

)

(

)

|

Substituting the values we get ( ) ( ) ( ) | | 24. π

π

∫

o

∫

o

∫

o

(

[Ans. 12]

o ) ( )

o

o ( ) ∫

o

( )

[ 22.

⇒ Magnitude of acceleration

]

=√

[Ans. C] ( ⇒

) (

25.

(

)

[Ans. B] We have

) ∫ ( )

⇒ , ow

-

∫ ( )

=1+0=1 Hence correct option is (C) 23.

∫ ( )

∫ ( )

[Ans. A] (4, 3) a (2, 2) b

c

x

( )μ

0.5

(1, 0)

0.5

o ∆ wh o –ordinate points are given is given by

th

th

th

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GATE QUESTION BANK

26.

[Ans. A]

=

α Use L – hospital Rule

= 4.

α

=1

[Ans. A] P=∑

o

…

‘ ’

= log x

 1 n 1 r  Cr xr  n 1  x  r0





r 0

r 0

‘ ’



r 0

i 0

5.

= 12

g(i) =i+1

– 24 48

)

+ 37

– 48 x = 0

√

x= =2

[Ans. A] f(x)= |x| Continuity: In other words, f(x) = x o ≥ x for x< 0 Since, = =0 , f (x) is continuous for all real values of x Differentiability:

=

√

96x

48

=

√

√ = 36

Now at x = 0 =

48

0

At 2 ± √ also

0 (using

calculator) There are 3 extrema in this function

( )

)

6.

( )

[Ans. D] Since ∫ ( )

R h So |x| is continuous but not differentiable at x=0 3.

(

x (12 – 48x 48 ) = 0 x = 0 or 12 – 48x – 48 = 0 4x – 4 = 0

∑ ()

(

k

)

[Ans. D] y = 3 – 16

(since r is a dummy variable, r can be replaced by i)

)

k

⇒

  r  1 xr   i  1 xi

(

–1)

)

w h (

  r1 Cr xr   r1 C1xr 

( …

 1 21 r  Cr xr  1  x 2 r0

Putting n=2,

2.

)

= Q=∑

[Ans. B]

(

w h a =1, l=2k 1

P= ( CS 1.

Mathematics

I =∫ =∫



=∫ (

)

 (

) (

)

Since tan (A B) =

[Ans. A] =

⁄ ⁄

th

th

th

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GATE QUESTION BANK 

[

]

[

]

Mathematics

∫ 

∫

0

1

0

1

9.

[Ans. B] f(x) = sin x ⇒ ( ) o ( ) ⇒ o π π π [



( (

∫

) )

( (

) )

π ]

( )





∫

∫

At

. /

gives maximum

value =,

)-

(

At

= ln ( sec ) – ln (sec 0) = ln (√ ) = ln (

. /

value

( ) 10.

)–0=

[Ans. A] For x =

7.

[Ans. B] (

8.

)

*

(

(

) [

*(

) +

) + .

/

]

11.

[Ans. C] By Mean value theorem

12.

[Ans. A] Define g(x) = f(x) – f(x + 1) in [0, 1]. g(0) is negative and g(1) is positive. By intermediate value theorem there is €( ) h h g(y) = 0 That is f(y) = f(y + 1) Thus Answer is (A)

13.

[Ans. 2] * w + * w + For min maximum non – common elements must be there ⇒ * + must be common to any 2 elements of V1 ( )minimum value = 2

o o ∫

∫

*

+ [

]

,

-(

o π

, f(x) =

For x = , f(x) = 3 – 1 = 2 For x = 3, f(x) = 2 ( ) ( ) = f(3)

[Ans. D] ∫

gives minimum

π

)

th

th

th

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GATE QUESTION BANK

14.

Mathematics

[Ans. 4] ∫ ∫

|

∫∫

π ( ) o π o (π) π Hence option (A) is correct

oπ π o π

∫

∫(

) ECE 1.

[Ans. C]

∫ ∫ (∫

∫ o

)

dy  0 for x< 0 dx dy  0 for x> 0 dx

∫ o

o Substituting the limits π o (π) o ( ) π

2.

[Ans. A] Given,

f x 

∫

 f ' x  

1  e .e  e 1  e 

|

∫∫

3.

= x cos

∫(

x 2

2x





ex

1  ex



2

0

o

)

Let cos = t ⇒ At o π o π o

o

∫ o

x

∫

[Ans. A] ∫

x

[Ans. C]

= π o ( π) π o π = π LHS = I + II = π π π⇒ 15.

ex 1  ex

∫

|

∫

∫ o

∫(

)

∫(

∫

∫

th

th

th

)

|

|

(

)

(

)

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GATE QUESTION BANK

8.

Mathematics

[Ans. A]

Given, f  x   x2  x  2

df  x  0 dx 4.

 2x 1=0

[Ans. A] o ’ h o )

∬(

 x

∮

1 2

d2f  x  = 2 ve dx2 So it shows only minima for interval [ 4, 4], it contains a maximum value that will be at x= 4 or x=4  f( 4)=18 and f(+4)=10 

5.

6.

[Ans. D] From vector triple product ( ) ( ) ( ) Here, ( ) ( ) ⇒ ( ) ( ) ( )

[Ans. D]

y  f  x ; x  0, 

[Ans. A] ( )

 f  x0 

For strictly bounded, 0  limy  

2 x  x0  f' x0   x  x0  f'' x0      ......

1

 e  (x  2)(e 2

x0

or 0  lim y  

2

 x  2 )

2

2

2

x 2

So, y  e x is strictly bounded

e ...... 2

  x  22  ......   e2  3  x    2   (Neglecting higher power of x)

7.

9.

10.

lim 0

=

ex  e x ex  ex

x x2 x3 e  1     .......... 1 2 3

11.

x

ex  1 

[Ans. B] Two points on line are ( 1, 0) and (0, 1) Hence line equation is,

 y y  y   2 1 x  c  x2  x1  y  x c y  x 1 … ( )

x x2 x3    .......... 1 2 3

x2 x4   .......... ex  ex 2 4  x x  e e x3 x5 x    .......... 3 5 1

or cot h (x)=

sin  /2 1  sin  /2  lim   0 2    /2 

1  sin  /2 1 = lim  2  0  /2  2

[Ans. C] coth (x)=

[Ans. A]

2 2 5 I   ydx    x  1dx   2.5 2 1 1

1 x

(Since at x=1,y=2)

(Neglecting x2 and higher order)

th

th

th

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GATE QUESTION BANK

12.

[Ans. B] Taking f(x, y)= xy, we can show that, xdx+ydy, is exact. So, the value of the integral is independent of path

15.

Mathematics

[Ans. A]

Given : g  x,y   4x3  10y 4 The straight line can be expressed as y=2x Then g(x,y)=4x3+ 10 (2x)4

(0, 1)

 1

1





4 I   4x3  10 2x   dx   4x3  160x4 dx   0 0 1

 4x4 160 5  =   x  33 5 0  4

(1, 0)

)

∫(

∫

[Ans. A] f(x)= + (x)= =0 x=0 (x)= + >0  x R. Hence minimum at x=0 f(0)=1+1=2 Alternatively: For any even function the maxima & minima can be found by A.M. >= GM => exp(x) + exp( x) ≥ 2 Hence minimum value = 2

17.

[Ans. B]

∫

[ |

13.

16.

| ]

[Ans. B] Let f(x)  ex  sinx o ’ 2 x  a  f  x   f a    x  a f'a   f''a  2!

Q

where, a= 2 x    f  x   f     x   f'   f''  2!

Coefficient of (x )2 is

f ''  2

P

f''   ex  sinx |at x   e

 Coefficient of (x )2=0.5 exp () 14.

∫(

)

[Ans. A]

o Thus, ( ( )w o ( )w o ( )w

)w h h h

h

∫

∫

[ |

| ]

o ow ow ow ow th

th

th

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GATE QUESTION BANK

18.

21.

[Ans. D] sinx = x = (x – π ) –

y=

(

or

19.

(

)

(

)

sin x = (x – π ) –

or

)

(

)

=1 –

(

) (

= 1

(

) (

)

o

.... ( )

)



...

o

....

(

)

( )



....

=

Therefore, at

22.

∬⃗ ⃗

̂

̂

̂

̂

∯

⃗

∭ ( )∭ and is the position vector)

(

23.

⃗⃗⃗ ⃗⃗⃗

has a maximum.

[Ans. D] Apply the divergence theorem

[Ans. C]

[Ans. A] ̂

Y

S

3

R

1

̅

Q

P

⇒

√

√

⇒

∮ ⃗ ⃗⃗⃗ ∫ ⃗ ⃗⃗⃗

∫ ⃗ ⃗⃗⃗ √

∫ ⃗ ⃗⃗⃗

∫

√

∫ ⃗ ⃗⃗⃗

∫ .√ /

∫ ⃗ ⃗⃗⃗

∫√ √

[ ∫ ⃗ ⃗⃗⃗

√

* +

[

) )

⇒

25.

[Ans. B]

, √ √

( )

)]

( (

)]

[Ans. C] ( ) , ( ) ( ) ( ) ⇒ are the stationary points ( ) ( ) ( ) and f(2) = 25 and f(4)=21 M o ( ) , f(6)=41

√

. /

(

(

24. ]

] ∫ .√ /

[

[

̂

⇒

∫ ⃗ ⃗⃗⃗

along PQ y =1 dy =0]

∫ ⃗ ⃗⃗⃗

⇒

⇒

X

= [

o

Since

[Ans. D] o ’ h o ⃗ ⃗ = ∮

⃗⃗⃗

 o

o

According Stokes Theorem ⃗⃗⃗ ⃗ ∮ ⃗ ⃗ =∮

20.

[Ans. A]

....

sin (x –π )

or

Mathematics

∮ ⃗ ⃗⃗⃗ ⇒ th

th

th

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GATE QUESTION BANK

o

30.

[Ans. C]

E o

(

E o

31.

o

[Ans. *] Range ( ) ( )

̂

̂

⇒ ⇒ ( )

=1+1+1 =3 [Ans. D] o ’ h o “ h integral of a ⃗ vector around a closed path L is equal to the integral of curl of ⃗ over the open ∮⃗

h

⃗

∬(

o

∫

∫

∫

∫

(

∫

*

32.

h ”

33.

)

) ) ) )

[Ans. C] Let x (opposite side), y (adjacent side) and z (hypotenuse side) of a right angled triangle

+

∫

29.

[Ans. *] Range 5.9 to 6.1 Maximum value is 6 ( ) ( ) ( ( ( (

⃗ )⃗

[Ans. *] Range 862 to 866 Volume under the surface ∫

( ) ( ) h

o

o

28.

to 0.01

( )

[Ans. D] ̅ ̂

=

27.

)

π

⇒

26.

Mathematics

Given

o

… )(

(

[Ans. A] o ( ) ̇( )

o

⇒ o

⇒ ( ) Since ( ) is negative, maximum value of f(x) will be where ( )

⇒

o 0(

⇒ ⇒

)

o ⇒ ( )

⇒

o

( )

( )

o

oh

(

)

)

1

( ( (

th

th

)(

))

) th

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GATE QUESTION BANK

By trial and error method using options π

34.

Now at x = 2 (2) = ( ) = ( ) = 2 0, minima at x = ( 2) =12( ) = 32 > 0; minima at x = There is only one maxima and only two minima for this function.

-

=√ 2.

,

4.

̂ )

So, directional derivative ⇒ ⃗ ̂ (̂ ̂ ) (̂

EE 1.

1

[Ans. C]

=

(

0

̂

At (1, 1, 1) ⃗ |⃗ | √

35.

Mathematics

[Ans. A] f(x) = (x) = ( ) = ( ) Putting ( (x) = 0 ( )=0 ( )=0 x = 0 or x = 2 are the stationary points. Now, ( ) ( (x) = )( ) = ( ( )) = ( ) ( )=2 At x = 0, (0) = Since (x) = 2 is > 0 at x = 0 we have a minima. th

th

th

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GATE QUESTION BANK

8.

Mathematics

= , =, ( = (

[Ans. A]

))

(

) =1

1

14.

[Ans. B] Dot product of two vectors =1+a+ =0 So orthogonal

15.

[Ans. C] f(x) = ( ) ( ) So the equation f(x) having only maxima at x = 1

16.

[Ans. B]

0

√

∫

9.

10.

(

∫

)

[Ans. C] ( ) ( ) ( ) ( ) ( )

̂

[Ans. D] ̅=( )̂ ( ( ) = (0, 2) ( ) = (2, 0) Equation of starting line

̂ ̂ ∫

11.

) y = 2 – x and dy = – dx

17.

C

)

o ( )

o

.(

)̂

̂

(

)̂

̂

̂ ||

( ̂ ( =0

⃗ ̂

̂

||

̂ ̂

)̂ /

̂

is undefined

[Ans. A] ̂ Div ( ) =.

‘ ’

(

Discontinuous

/(

̂

̂

̂)

18.

= 1+1+1= 3 13.

[Ans. D]

)

But at

12.

∫

[Ans. B] (

̂

Along x axis ,y=0,z=0 The integral reduces to zero.

=

∫ (

̂ ̂

̂

)̂

⇒ y = 2 – x , dy = – dx ̅ ̅ =( ( ) Putting ∫̅ ̅ ∫

̇̂

[Ans. B] P=∫ th

[Ans. A] ( ) o ⇒ M th

) ( ) ̂ (

( ( )

th

) ( ) ̂ (

)

(

) (

)

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) )

GATE QUESTION BANK

19.

Mathematics

[Ans. B]

.

0

π

/1

∫⃗ ∫

[

(

o

o π

)( )(

∫ ( ∫

⇒

o

23.

[Ans. A] ( ⃗) ⃗ ⃗ ( ) ( ) ⃗

24.

[Ans. B] ( ) ( ) ⇒ ⇒ )( ⇒( ⇒ ( ) ( ) ( )

) )

[

]

π 20.

]

[Ans. C] ( )

(

)

( )

( ) For number of values of ) o ( ) ( ( ) ( ( )

( ) ⁄

,

w

-

) ( ) ( )

M 21.

)

[Ans. B]

IN 1.

G o

h

[Ans. A]

o

o

⇒ o

⇒

(

o

)

Unit vector along y = x is G

∫ (∫ ∫ (∫

22.

[Ans. 2] ( ∫

)

̂

)

̂

π

√ o

√ ̂

.

) ∫

π

o

o .

π

/

.

π

/.

√

/

√

/

√

√

√ √

2.

[Ans. D] Using L Hospital Rule., numerator becomes =

From the graph, distance at

th

th

()

= ( )

th

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GATE QUESTION BANK

3.

[Ans. B]

Mathematics

When

Given integral is, I=∫

( )

Let f(x) = so curve of 1/

will be

(

)

(

)

And when

f(x)

( )

The possible expression for f(x) is 1

. 7.

-1

0

1

/

[Ans. B]

x

Error,

This curve will be discontinuous at x=0 o ’ w o

For error to be minimum (

4.

[Ans. A] ̅ (t) =x (t) ̂+y (t) + Let R ̂ z (t) ̂ ̅( ) =K (constant) |R i.e., (t) + (t) + (t) = constant. On analyzing the given (A) option, we find ̅(t) that R

̅( )

⇒ ⇒

[Ans. C] Given : f= + where,

(

)

⇒

√

will give constant magnitude, √

1

G …… + (i=0 to n) are constant.

=

+(n 1)

o

…… ⇒

+ and

)

o

so first differentiation of the integration will be zero. 5.

o

=0+

+

(n 1)

√

…… ⇒

+n

√

+ = , = nf 6.

+

+

⇒

-

(

)

⇒ ⇒

[Ans. B] ( )

(

)

8.

[Ans. B] ()

When ( )

(

)

(

)

( )

∫

…( )

Differentiating the above equation

When ( )

th

th

th

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GATE QUESTION BANK

()

,

14.

() ∫

Mathematics

[Ans. B] Given y = x2 + 2x + 10 = 2x + 2

( ) -

| From equation (i) () ()

⇒

()

()

This is Leibnitz linear equation Integrating factor I.F = ∫ the solution is ()

15.

[Ans. C] By definition

16.

[Ans. A]

()

Unit vector=

=xi+yj+zk

and 17.

∫

[Ans. C] R: Y

( ) 1 1

, [Ans. D]

10.

[Ans. A] This is a standard question of differentiability & continuity

Area =

[Ans. C] y= =(

X

- o

9.

11.

+1

( )

Total charge = σ = = 18.

).(cos x + sin x) = 0

⇒ tan x = 1 Or x =

coulomb.

[Ans. B] We know that ∫

() (

∫

.

) π

( )wh π . /

/

y will be maximum at x = y=

19.

= 12.

13.

[Ans. C] y(2) = y(5) =

=

√

[Ans. B] Expansion of sin x ........

( ) ( )

20.

[Ans. B] In a G.P

(

)

For a G.P to converge

[Ans. C] y= y=

(

⇒

)

⇒

(

)

⇒ th

th

th

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GATE QUESTION BANK

21.

[Ans. D] .E=0 is not irrational (it is solenoidal)

22.

[Ans. 1] From Gauss divergence theorem, we have ∫ ̅ ̅

̅

∫

/dxdydz

∫ [ ⇒

̅

∫

∫.

Mathematics

∫ ̅

[Ans. C]

24.

[Ans. D]

̂

) ̂

̂

]

o .

π

π /

23.

(

∫

(

π )

th

th

th

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GATE QUESTION BANK

Mathematics

Differential Equations ME – 2005 1.

If x

xy

n y

what is y (A) e (B) 1

then

⁄ ⁄

(C) (D)

(B) degree 1 order 1 (C) degree 2 order 1 (D) degree 2 order 2 ME – 2007 7.

2.

3.

Statement for Linked Answer Questions 2 and 3. The complete solution of the ordinary differential equation y y p qy s x x y Then, p and q are (A) p =3, q = 3 (C) p =4, q = 3 (B) p =3, q = 4 (D) p =4, q = 4 Which of the following is a solution of the differential equation y y p q y x x (A) (C) x (B) x (D) x

The solution of

For

ME – 2008 8. It is given that + 2y + y = 0, y (0) = 0, y(1) = 0. What is y (0.5)? (A) 0 (C) 0.62 (B) 0.37 (D) 1.13 9.

Given that ẍ + 3x = 0, and x(0) = 1, ẋ (0) = 0, what is x(1)? (A) 0.99 (C) 0.16 (B) 0.16 (D) 0.99

ME – 2009 10.

+ 3y =

, the particular

integral is: (A) (B) (C) (D) 5.

The solution of x

y

x

with the

s

(A) y

(C) y

(B) y

(D) y

ME – 2010 11. +

The solution of the differential equation

(A) (1+ x)

(C) (1 x)

(B) (1+ x)

(D) (1

The Blasius equation,

, is a

(A) second order non-linear ordinary differential equation (B) third order non-linear ordinary differential equation (C) third order linear ordinary differential equation (D) mixed order non-linear ordinary differential equation

2 dy  2xy  ex with y (0) = 1 is: dx

6.

(D) 2  x  2

(B)   x  1

condition y +4

with initial value

y (0) = 1 is bounded in the interval (C) x  1,x  1 (A)  x 

ME – 2006 4.

y

x)

The partial differential equation (

)

(

)= 0 has

(A) degree 1 order 2 th

th

th

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GATE QUESTION BANK

ME – 2011 12. Consider

the

differential

x ,yt x ,yt

equation

y x. The general solution with constant c is (A) y

t n

(B) y

t n (

(C) y

t n ( )

(D) y

t n(

17.

)

18.

differential y

equation

with the boundary

(B) s n ( )

(D)

partial u

15.

19.

s n( )

differential

equation

is a

linear equation of order 2 non – linear equation of order 1 linear equation of order 1 non – linear equation of order 2

subjected to the boundary conditions u(0) = 0 and u(L) = U, is (A) u (C) u ( ) (

)

(D) u

(

x

y and

x

x ,y-

*

x + ,y-

*

x + ,y-

y is

20.

with t __________

The general solution of the differential os x

constant, is (A) y s n x

y

(B) t n (

)

y

(C)

os (

)

x

(D) t n (

)

x

y with c as a x

Consider two solution x(t) = x t and x t x t of the differential equation x t x t t su t t t x t x | t x t | t

t s (A) 1 (B) 1

x t

x t

| t

(C) 0 (D)

The solution of the initial value problem xy y

)

ME – 2014 16. The matrix form of the linear system

x ,yt

at x

The wronskian W(t) =|

where k is a constant,

t

x + ,y-

If y = f(x) is the solution of

x

The solution to the differential equation

(B) u

*

x

conditions of y(0) =0 and y(1) = 1. The complete solution of the differential equation is (A) x (C) s n( )

(A) (B) (C) (D)

x + ,y-

the boundary conditions y

)

the

x

ME – 2013 14. The

*

equation

ME – 2012 13. Consider x

t n

Mathematics

is

(A)

(C)

(B)

(D)

CE – 2005 1. Transformation to substituting v = y

linear form by of the equation

+ p(t)y = q(t)y ; n > 0 will be (A) th

+ (1 n)pv = (1 n)q th

th

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GATE QUESTION BANK

2.

(B)

+ (1 n)pv = (1+n)q

(C)

+ (1+n)pv = (1 n)q

(D)

+ (1+n)pv = (1+n)q

CE – 2007 6. The degree of the differential equation + 2x = 0 is (A) 0 (B) 1

in the range (A) (B)

y x

( os x ( os x

(C)

( os x

(D)

( os x

,

( )

7.

The solution for the differential equation = x y with the condition that y = 1 at

is given by

x = 0 is

s n x)

(B) In(y) =

s n x) s n x)

8.

xy

x

+4

(D) y =

A body originally at 600C cools down to C in 15 minutes when kept in air at a temperature of 250C. What will be the temperature of the body at the end of 30 minutes? (A) 35.20C (C) 28.70C 0 (B) 31.5 C (D) 150C

CE – 2008 9.

The general solution of (A) (B) (C) (D)

The solution of the differential equation x

(C) In(y) =

(A) y =

s n x)

CE – 2006 3. A spherical naphthalene ball expanded to the atmosphere losses volume at a rate proportional to its instantaneous surface area due to evaporation. If the initial diameter of the ball is 2 cm and the diameter reduces to 1 cm after 3 months, the ball completely evaporates in (A) 6 months (C) 12 months (B) 9 months (D) Infinite time

+ y = 0 is

y = P cos x + Q sin x y = P cos x y = P sin x y=Psn x

given that at x = 1, 10.

y = 0 is

5.

(C) 2 (D) 3

The solution of y

4.

Mathematics

(A)

(C)

(B)

(D)

The differential equation

= 0.25 y is to be

solv us ng t b w r mpl t Eul r’s method with the boundary condition y = 1 at x = 0 and with a step size of 1. What would be the value of y at x = 1? (A) 1.33 (C) 2.00 (B) 1.67 (D) 2.33

Solution of (A) x (B) x

=

at x = 1 and y = √ is

y y

(C) x (D) x

y y

CE – 2009 11. Solution of the differential equation 3y

+ 2x = 0 represents a family of

(A) Ellipses (B) Parabolas

(C) circles (D) hyperbolas

CE – 2010 12. The order and degree of the differential equation

+ 4 √( )

respectively (A) 3 and 2 (B) 2 and 3 th

th

y

= 0 are

(C) 3 and 3 (D) 3 and 1 th

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GATE QUESTION BANK

13.

The solution to the ordinary differential equation (A) (B) (C) (D)

14.

+

y= y= y= y=

The following differential equation has 3

d2y  dy   4   y2  2  x dt2  dt  (A) degree=2, order=1 (B) degree=1, order=2 (C) degree=4, order=3 (D) degree=2, order=3

6y = 0 is

3

+ + + +

The partial differential equation that can be formed from z = ax + by + ab has the form (w t p (A) (B) (C) (D)

2.

n q

Mathematics

)

ECE – 2006

3.

For the differential equation

z = px + qy z = px + pq z = px + qy + pq z = qx + pq

the boundary conditions are (i) y=0 for x=0 and (ii) y=0 for x=a The form of non-zero solutions of y (where m varies over all integers) are m x y ∑ sn

CE – 2011 15. The solution of the differential equation + = x, with the condition that y = 1 at x = 1, is (A) y =

+

(D) y =

+

CE – 2012 16. The solution of the ordinary differential y=0 for the boundary

condition, y=5 at x = 1 is (A) y (C) y (B) y (D) y CE – 2014 17. The

y

∑

y

∑

y

∑

os

m x

(C) y = +

(B) y = +

equation

d2y  k2y  0 2 dx

integrating

for

the

equation (A) (B)

x

ECE – 2007 4. The solution of the differential equation

d2y  y  y 2 under the boundary dx2 conditions (i) y=y1 at x=0 and (ii) y=y2 at x=, where k, y1 and y2 are constants, is (A) y y y xp( x⁄ ) y (B) y y y xp x⁄ y (C) y y y s n x⁄ y (D) y y y xp x⁄ y k2

differential s

(C) (D)

ECE – 2005 1. A solution of the following differential equation is given by

d2y dy  5  6y  0 dx dx2

ECE – 2008 5. Which of the following is a solution to the differential equation

2x 3x (A) y  e  e

2x 3x (C) y  e  e

2x 3x (B) y  e  e

2x 3x (D) y  e  e

(A) t (B) x t

th

th

x t

x t

(C) x t (D) x t

th

t t

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GATE QUESTION BANK

ECE – 2009 6. The order of the differential equation

ECE– 2011 10. The solution of the differential equation

3

d2y  dy    y4  et is dt 2  dt  (A) 1 (C) 3 (B) 2 (D) 4

7.

Match each differential equation in Group I to its family of solution curves from Group II. Group I Group II 1. Circles dy y  P.

dx x dy y  Q. dx x

(A) (B) (C) (D)

3. Hyperbolas

dy x  dx y

y x

(C) y (D) y

ECE\EE\IN – 2012 11. With initial condition x(1) = 0.5, the solution of the differential equation, t

x

t is

(A) x

t

(C) xt

(B) x

t

(D) x

ECE\IN – 2012 12. Consider the differential equation y t y t y t t t t wt y t | n | num r

l v lu o

(A) (B)

x with the initial condition

y s ng Eul r’s rst or r m t o with a step size of 0.1, the value of y is (A) 0.01 (C) 0.0631 (B) 0.031 (D) 0.1 A function n x satisfies the differential equation

is

(A) x (B) x

P-2, Q-3, R-3, S-1 P-1, Q-3, R-2, S-1 P-2, Q-1, R-3, S-3 P-3, Q-2, R-1, S-2

ECE – 2010 8. Consider a differential equation

9.

y y

2. Straight Lines

dy x R.  dx y S.

Mathematics

where L is a

constant. The boundary conditions are: n and n . The solution to this equation is (A) n x xp x (B) n x xp x √ (C) n x xp x (D) n x xp x

y | t (C) (D) 1

s

ECE – 2013 13. A system described by a linear, constant coefficient, ordinary, first order differential equation has an exact solution given by y t or t when the forcing function is x(t) and the initial condition is y(0). If one wishes to modify the system so that the solution becomes – 2y(t) for t > 0, we need to (A) Change the initial condition to – y(0) and the forcing function to 2x(t) (B) Change the initial condition to 2y(0) and the forcing function to –x(t) (C) Change the initial condition to j√ y(0) and the forcing function to j√ x(t) (D) Change the initial condition to – 2y (0) and the forcing function to – 2x(t)

th

th

th

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GATE QUESTION BANK

ECE – 2014 14. If the characteristic equation of the differential equation y t

15.

has two equal roots,

n t v lu s o (A) ±1 (B) 0,0

r (C) ±j (D) ±1/2

xy

(C)

(B)

xy

(D)

(B) x t (C) x t (D) x t EE – 2011 3. With K as a constant, the possible solution for the first order differential equation is

Which ONE of the following is a linear non-homogeneous differential equation, where x and y are the independent and dependent variables respectively? (A)

Mathematics

xy

(C) (D)

(A) (B)

EE – 2013 4. A function y x x is defined over an open interval x = (1,2). At least at one point in this interval ,

16.

17.

18.

If z

xy ln xy then

(A) x

y

(C) x

y

(B) y

x

(D) y

x

If a and b are constants, the most general solution of the differential equation x x x s t t (A) (C) bt (B) bt (D)

(A) 20 (B) 25

tx

s

6.

x x (A) x t

n

with initial conditions | t

, the solution is

(B) s n t

os t

(C) s n t

os t

(D) os t

t

Consider

the

x

EE – 2005 1. The solution of the first order differential qu t on x’ t 3x(t), x (0) = x is (A) x (t) = x (C) x (t) = x (B) x (t) = x (D) x (t) = x EE – 2010 2. For the differential equation

(C) 30 (D) 35

EE – 2014 5. The solution for the differential equation x x w t n t l on t ons x t x n | s t (A) t t

With initial values y(0) = y (0) = 1, the solution of the differential equation y

is exactly

x

differential

equation

y

Which of the following is a solution to this differential equation for x > 0? (A) (C) x (D) ln x (B) x IN– 2005 1. The general solution of the differential equation (D2 4D +4)y = 0, is of the form (given D = d/dx), and C1 and C2 are constants (A) C1e2x (C) C1e2x + C2 e2x 2x (B) C1e + C2 (D) C1e2x + C2x th

th

th

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GATE QUESTION BANK

2.

urv s or w t urv tur ρ t 3 any point is equal to cos θ w r θ s t angle made by the tangent at that point with the positive direction of the x-axis, r

gv nρ

⁄

, where y and y

are the first and second derivatives of y with respect to x) (A) circles (C) ellipses (B) parabolas (D) hyperbolas IN– 2006 3. For an initial value problem ÿ ẏ y y n ẏ various solutions are written in the following groups. Match the type of solution with the correct expression. Group 1 Group 2 P. General solution 1. 0.1ex of homogeneous equations Q. Particular integral 2. (A cos 10 x + B sin 10 x) R. Total solution 3. cos 10 x + x satisfying boundary 0.1e conditions (A) P-2, Q-1, R-3 (C) P-1, Q-2, R-3 (B) P-1, Q-3, R-2 (D) P-3, Q-2, R-1 4.

A linear ordinary differential equation is given as

d2y dy  3  2y  δ(t) 2 dt dt Where  (t) is an impulse input. The solut on s oun by Eul r’s orw r difference method that uses an integration step h. What is a suitable value of h? (A) 2.0 (C) 1.0 (B) 1.5 (D) 0.2

Mathematics

IN– 2007 5. The boundary-value problem y λy y y w ll v non-zero solut on n only t v lu s o λ r (A) ± ± … (B) … (C) … (D) … IN– 2008 6. Consider the differential equation = 1 + y2. Which one of the following can be a particular solution of this differential equation? (A) y = tan (x + 3) (C) x = tan (y + 3) (B) y = tan x + 3 (D) x = tan y + 3 IN– 2010 7. Consider y

the

differential

equation

with y(0)=1. The value of

y(1) is (A)

(C)

(B)

(D)

IN – 2011 8. Consider the differential equation ÿ ẏ y with boundary conditions y(0) = 1, y(1) = 0. The value of y(2) is (A) 1 (C) – (B) (D) IN– 2013 9. The type of the partial differential equation

is

(A) Parabolic (B) Elliptic 10.

th

(C) Hyperbolic (D) Nonlinear

The maximum value of the solution y(t) of the differential equation y t ÿ t with initial conditions ẏ and y , for t is (A) 1 (C) (B) 2 (D) √

th

th

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GATE QUESTION BANK

Mathematics

IN– 2014 11. The figure shows the plot of y as a function of x

y

y

x

x

The function shown is the solution of the differential equation (assuming all initial conditions to be zero) is : (A) (B)

x

(C)

x

(D)

|x|

th

th

th

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GATE QUESTION BANK

Mathematics

Answer Keys and Explanations ME 1.

4. [Ans. D] x y y

lnx x

xy x

lnx x

y

omp r ng w g t ow y(I.F.) = ∫ x y

∫

olv ng bov utt ng x x 2.

3.

[Ans. B] The given differential equation may be written as y y y ux l ry qu t on s

w

lnx x

x ∫

∫

x

Substituting D=2, we get

x x

(

x

n t v lu o t n n t v lu o y t

5.

[Ans. B] First order equation,

sy

y

dy  Py  Q, dx

Where P = 2x and Q = Since P and Q are functions of x, then Integrating factor,

[Ans. C] Given equation is y p qy x x p q y p q ts solut on s y um o roots p p ro u t o roots q q [Ans. C] Given equation is y y p q x x p q ut p n q y

)

2

I.F. = e  Pdx  e x Solution is y

∫

y

x

∫

x

2

yex  x  c ,c=1

Since, y

 x2

(1 + x) e

y 6.

[Ans. A] Order: The order of a differential equation is the order of the highest derivative appears in the equation Degree: The degree of a differential equation is the degree of the highest order differential coefficient or derivative, when the differential coefficients are free from radicals and fraction. The general solution of differential qu t on o or r ‘n’ must nvolv ‘n’ arbitrary constant.

y

x

th

th

th

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GATE QUESTION BANK

7.

y ( )y x … x x Standard form y y … x Where P and Q function of x only and solution is given by

[Ans. C] y x

v n y

y x

y

nt gr t ng y

y

x

∫

x nx

y

n

x

x

x

olut on y x x

and x

Given condition y m ns t x

y

₂ r or yx

y [Ans. D] ẍ x Auxiliary equation is m2 + 3 = 0 i.e. m = ±√ x os√ t sn√ t ẋ os√ t s n√ t √ At t = 0 1=A 0=B x = os √ t x

11.

x

[Ans. B] is third order ( is linear, since the product

) and it is not

allowed in linear differential equation 12.

os √ t

[Ans. D] y x y ∫ y t n

10.

x

x

y 9.

∫x x x

x

yx [Ans. A] y y y A.E is, D2+2D+1 =0 2=0 m 1 The C.F. is (C1+C2x)e-x P.I. = 0 ow y ₁ n y ₂

∫

x

∫

x x

8.

x

Where, integrating factor (I.F) r

y

Mathematics

[Ans. A] Given differential equation is y x y x x

y

th

y t n.

th

y x ∫ x x x

/

th

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GATE QUESTION BANK

13.

[Ans. A] 17. y y x x y x x y n y Choice (A) satisfies the initial condition as well as equation as shown below y x y n y y lso x x y x 18. y y x x y x x x x x x x x x o y x is the solution to this equation with given boundary conditions

14.

[Ans. D]

15.

[Ans. B] m m u u At x=0, At x=L, (

[Ans. *] Range 34 to 36 y x y x y x tx y y tx x y x tx y [Ans. D] y os x y x Let x y z y z x x z os z x z os z x z s ( ) z x

z os ( )

Integrating z t n( ) x z t n( ) x x t n(

) n

19.

u x

Solving we get u = U( 16.

Mathematics

)

[Ans. A] x x y t y x y t So by observation it is understood that, x x ,y- * + ,yt

y )

[Ans. A] Since the determinant of wronskian matrix is constant values for, therefore it is same for both t=0 and t= t

20.

x

x t

x t t

[Ans. B] y ∫ ∫ x x y y ln ( ) x y

x t t

x t

ln y

x

ln

v ny n th

y th

th

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GATE QUESTION BANK

CE 1.

y tx x

[Ans. A] Given

+ p(t)y = q(t) y

n y

3.

Multiplying by (1 n) y we get v p t n y q t t Now since y = v we get v n pv n q t Where p is p (t) and q is q(t)

t Where, V =

n

4 r

=

sn x os x sn x os x sn x

os x

r )

 r

t utt ng r

n

y

r

r t  dr = kdt Integration we get r = kt + C At t = 0, r = 1 1= k×0+C C=1 r = kt + 1 Now at t = 3 months r = 0.5 cm 0.5 = k × 3 + 1

)

t

…

r r t t Substituting in (i) we get

±

os x

sn x)

r

y (

( os x

[Ans. A]

[Ans. A] y y y x x y y ( ) x This is a linear differential equation

n

s

+ p(t) y = q(t) y ; n > 0

Given, v = y v y n y t t y v t n y t Substitution in the differential equation we get

2.

Mathematics

n solv ng g v s t

sn x

 t = 6 months

y os

sn

4. sn x os x

[Ans. A] Given y x xy – x x y x xy x x Dividing by x

os x

sn x sn x os x

th

th

x

th

y

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GATE QUESTION BANK

y x ( )y ( ) x x x Which is a linear first order differential of the form y y x Integrating factor = I.F = ∫ = ∫ y × I.F = ∫ .(I.F)dx x yx ∫( )x x x Now at x = 1, y = 0

Hence, here the degree is 1, which is power of

7.

[Ans. D] y x y x This is variable separable form

x

= x dx

∫ x

∫

x

y y

∫x

tx log y

C x y

5.

–x x

[Ans. B] =

x

y

0C

= Now at t = 30 minutes Θ

±√

y =2

6.

t

 ln θ θ = kt + θ θ C. θ θ C. Given θ = 250C Now t t θ 60 = 25 + C.e0 C = 35 θ At t m nut s θ 40 = 25 + 35

y

y +1=0

t =∫

∫

y y y  0.25hy y +y =0 Putting k = 0 in above equation 0.25h y y +y =0 Since, y = 1 and h = 1 0.25 y

θ θ0) (Newton’s law of cooling)

θ θ θ

[Ans. C] y y y tx x h=1 Iterative equation for backward (implicit) Euler methods for above equation would be y

y x

y

8. y

x

x

log y

i.e. 0 ×

x

Mathematics

= 25 + 35 (

)

= 25 + 35 × ( ) (s n

[Ans. B] Degree of a differential equation is the power of its highest order derivative after the differential equation is made free of radicals and fractions if any, in derivative power.

)

= 31. ≈ C

th

th

th

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GATE QUESTION BANK

9.

12.

[Ans. A] +y=0

10.

y

.

13.

∫ x x

√ C=2 Solution is y x

[Ans. D] y y x n y x x This is a linear differential equation of the form y y wt n x x x IF = Integrations factor

x x ∫

y x

y

( )

( )

x

x (

x x

x

y

(

∫

∫

x

Solution is y (IF) = ∫ x  y. x = ∫ xx x  yx = ∫ x x

y )

+

15.

x x y

[Ans. C] y y x x Auxiliary equation is +D–6=0 (D 2) = 0 D = 3 or D = 2 Solution is y =

[Ans. C] Z = ax + by + ab … z p x z q b y Substituting a and b in (i) in terms of p and q we get z = px + qy + pq

[Ans. A]

∫ y y

y y / 0( ) y 1 x x The order is 3 since highest differential

14.

x +y =4

y x y y

y

is

x

3y

y ) x

Removing radicals we get

At x = 1, y = √

11.

√(

The degree is 2 since power of highest differential is 2

[Ans. D] y x x y y dy = x dx ∫y y

[Ans. A] y x

+1=0 E sm m ± General solution is y= [ cos (1 × x) + sin (1 × x)] = cosx + sinx = P cosx + Q sinx Where P and Q are some constants

Mathematics

)

Which is the equation of a family of ellipses

 yx =

th

th

+C

th

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GATE QUESTION BANK

y=

 sin ka=0 m x 

+

Now y(1) = 1  ot

 solut on s y

y

17.

m x

[Ans. D] k2D2y= y

y2

y2  2 1  D  k2  y  k2   1 m1 =  k x/k x/k C.F. = C1e  C2 e

[Ans. D]

x/k x/k  y2 y= C1e  C2 e

∫

At y=y1, x=0  y1 = C1+ C2+y2 … At y=y2 , x=  Hence C1 must be zero  y1 = C2+y2  C2 =y1 - y2

[Ans. B]

d2y dy  5  6y  0 dx dx2 A.E. is D2 5D  6  0  D=2,3 2x 3x Hence, solution is y  e  e

2.

sn

x

[Ans. D] y y x y y y

Particular integral (P.I) = = ECE 1.

∑

x 4.

16.

Mathematics

x  y=(y1 – y2) exp    + y2  k 5.

[Ans. B] x t x t t (D +3) x(t) = 0

[Ans. B] 3

d2y  dy   4   y2  2  x dt2  dt  Order of highest derivative=2 Hence, most appropriate answer is (B) 3

3.

[Ans. A] Given, Differential equation,

d2y  k2y  0 dx2 Auxilary equation is y ± Let y os x sn x At x=0, y = 0  A=0 y sn x At x=a, y=0  B sin ka=0 B0 otherwise y=0 always

So, x  t   ke3t , Hence x  t   2e3t is one solution (for some boundary / initial condition) 6.

[Ans. B] The order of a differential equation is the order of the highest derivative involving in equation, so answer is 2. The degree of a differential equation is the degree of the highest derivative involving in equation, so answer is 1.

7.

[Ans. A] P.

∫

∫

log y log x log y xw s qu t on o str g t l n th

th

th

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GATE QUESTION BANK

Q.

∫ log y y

x

log x

∫y y y

x

8.

yp rbol

y

∫x x

9.

t ∫t t

xt

t

xt 12.

x

y

[Ans. D] Approach 1: y t y t y t t t t Converting to s-domain s y s sy y sy s y s s s y s s s y s s s n nv rs pl tr ns orm y t t u t y t t t y t | t

y

[Ans. D] Approach 2: y t y t y t t t t Applying Laplace Transform on both sides y s y s sy | t (sy s y ) y s s y s s sy s y s s s y s s s s s

n x m

Auxiliary equation m olut on n x Since, n

±



Since, n  must be zero) Therefore

(hence

The solution is, n x 10.

∫

r l

y old y +0.1 ( ) new x y x y 0 0 0+0 0+0.1×0=0 =0 0. 0 0.1+0 0+0.1×0.1=0.01 1 =0.1 0. 0.0 0.2+0.01 0.01+0.21×0.1 2 =0.21 1 =0.031 The value of y at x= 0.3 is 0.031. x

x=1

Using initial condition, at t = 1, x = 0.5

ypr bol

… Equ t on o

[Ans. B] y x y x x  x y

t

ol s xt

qu t on o

∫y y x

x

∫x x

…

S.

[Ans. C] t

log

… qu t on o

R.

11.

∫

Mathematics

[Ans. C] Given y ln y When y y

and x

y

y t

x

t

t

t

y

y t y t t

y

t

t t t t

th

th

th

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GATE QUESTION BANK

z y

x ln xy

ox

z x

y t | t 13.

[Ans. D] Let the differential equation be y t y t x t t Apply Laplace transform on both sides y t {x t } 2 y t 3 t sy s y y s x s s y s x s y x s y y s s s Taking inverse Laplace on both sides x s {y s } 2 3 y { } s s y t x t y So if we want y t as a solution both x(t) and y(0) has to be multiplied by . Hence change x(t) by x t and y(0) by y

14.

[Ans. A] y y y x x The auxiliary equation is m m ± then either m or m i.e., roots of the equation are equal to or

y

z y

18.

EE 1.

xy ln xy

xy

[Ans. C] z xy ln xy z y ln xy x

y

y ln xy

…

x (t) x

x ∫ t x lnx = t x Putting x Now putting initial condition x(0) = x x x Solution is x = x i.e. x(t) = x

0 is a first order linear

xy

.

∫

omog n ous

xy

xy

t

equation (homogeneous) r non l n r qu t ons 16.

x

xy

[Ans. *] Range 0.53 to 0.55 E m m m olut ons s y bx y bx b … s ng y y n gv s n b y x tx y

[Ans. A] v n x’ t

is a first order linear

equation non

x ln xy

[Ans. B] x x x t t Pre auxiliary equation is m m Pre roots of AE are m Repeated roots are present. So, most general solution in n t bt

[Ans. A] xy

x

xy

z y

y

i.e. 15.

xy

xy ln xy

z x

ox 17.

Mathematics

y

th

th

th

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GATE QUESTION BANK

2.

[Ans. B] x x x t t Auxiliary equation m m m m (m+4)(m+2)=0 m= 2, 4 x(t) =

6.

[Ans. C] x y xy y y x

m

t

nx

 … On solving (1) & (2), we get

IN 1.

y

x

x( s ts

x

)

x

s

[Ans. C] y

and

m

x(t)= 2 3.

x

x

… (1)



|

y y x

w subst tut y

n x(0) = 1 1=

Mathematics

Since there is double root at 2, so general solution of the given differential equation would be +

[Ans. A] y

m

x

Integrate on both sides 2.

y 4.

5.

[Ans. B] v n ρ

os θ … y n ρ … y now y’ t nθ … Equating equations (i) and (ii) and using equation (iii) in equation (ii), we get y os θ= os θ

[Ans. B] y x x p n nt rv l x y x x y x x Value is in between 20 and 30 So it is 25 [Ans. C] x x gv n t x os t sn t x n x sn t os t t x | t x

os t

y= .x Which is equation of a parabola. 3.

[Ans. A] A.E. D= 1+ 10i C.F = (A cos10 x + B sin 10 x) x

4.

[Ans. C]

5.

[Ans. C]

x

sn t

th

th

th

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GATE QUESTION BANK

6.

[Ans. A] Given

Hyperbolic if El ps Compare the given differential equation with standard from A = 1, B = 0, C = 0

= 1 + y2

Integrating ∫

=∫ x

Or t n y = x + c Or y = t n x 7.

Parabolic

[Ans. C] y y x Auxiliary equation, m + 1 = 0 m= 1 C.F =

10.

y

[Ans. C]

The solution for the differential equation is y x Now, y and y , placing these values We get, and y

s nx

s nx

os x

So, y os x s n x or m x m y s nx os x s nx os x x y os x s n x y or x m xm y m x os sn

y

√ 11.

[Ans. A] Given partial differential equation is x

± os x

ẏ ẏ

y

9.

[Ans. D] y t ÿ t

y y

y=

8.

Mathematics

√

√

√

[Ans. D] By back tracking, from option (D) y |x| x or x x = x or x Integrating y ∫ ∫ x x or x x

t

∫ x x or x

x t We know that

x

y (x y

or x

)

or x

is said to be Parabolic if

th

th

th

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GATE QUESTION BANK

Mathematics

Complex Variables ME – 2007 1. If x y and (x, y) are functions with continuous second derivatives, then x y + i (x, y) can be expressed as an analytic function of x + i (i = √ ), when (A)

ME – 2014 6.

The argument of the complex number where i = √ π π 2

7. +

(D)

+

(A) 2πi (B) 4πi

An analytic function of a complex variable z x + i y is expressed as z u x y +iv x y where i √ f u(x,y)= 2xy, then v(x,y) must be (A) x + y + onst nt (B) x y + onst nt (C) x + y + onst nt (D) x y + onst nt

9.

An analytic function of a complex variable z = x + i y is expressed as f(z) = u(x, y) + i v(x, y), where i = √ . If u (x, y) = x – y , then expression for v(x, y) in terms of x, y and a general constant c would be (A) xy + (C) 2xy +

ME – 2009 3. An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x, y) +iv(x, y) where i = 1 . If u = xy, the expression of v should be

x

2

2

y 2

2

 k

y (C) (D)

2

 x2 2

 k

 x  y 2  k 2

ME – 2010 4. The modulus of the complex number

(B)

) is

(A) 5 (B) √

(C) 1/√ (D) 1/5

traversed in

8.

(C) 2πi (D) 0

 x  y 2  k

is evaluate

counter clock wise direction. The integral is equal to π (A) 0 – 2 π π – 4 4

+

ME – 2008 2. The integral ∮ z z evaluated around the unit circle on the complex plane for z is

(

x y

along the circle x + y

(C)

(B)

π 2 π

The integral ∮ y x

(B)

(A)

, is

10.

+

(D)

+

If z is a complex variable, the value of is

∫

(A) i (B) 0.511+1.57i (C) i (D) 0.511+1.57i

ME – 2011 5. The product of two complex numbers 1 + i and 2 – 5i is (A) 7 – 3i (C) 3 – 4i (B) 3 – 4i (D) 7 + 3i th

th

th

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GATE QUESTION BANK

CE – 2005 1. Which one of the following is NOT true for complex number and ? (A) (B) | (C) | (D) |

2.

̅̅̅̅

=|

|

|≤| |+| | |≤| | | | | +| | 2| | + 2| |

+ +

CE – 2011 6. For an analytic function, f(x + iy) = u(x, y)+iv(x, y), u is given by u = 3x 3y . The expression for v considering K to be a constant is (C) 6x 6y+k (A) 3y 3x + k (D) 6xy +k (B) 6y – 6x + k CE – 2014

Consider likely applicable of u hy’s integral theorem to evaluate the following integral counter clockwise around the unit circle c. ∮s

z z

7.

z

πn

2

+

i i

ECE – 2006 1. The value

of

∮|

2.

-

π/2: singul riti s s t { nπ n 2 } (D) None of the above

∮

dz is

(A)

4πi

(C)

(B)

πi

(D) 1

πi

(C) (D)

the

+

contour

i i

integral

z in positive sense is

|

(A)

(C)

(B)

(D)

For the function of a complex variable W = In Z (where, W = u + jv and Z = x + jy), the u = constant lines get mapped in Z-plane as (A) set of radial straight line (B) set of concentric circles (C) set of confocal hyperbolas (D) set of confocal ellipses

(C) I

CE – 2006 3. Using Cauchy’s is integral theorem, the value of the integral (integration being taken in counter clockwise direction)

can be expressed as

(A) (B)

z being a complex variable. The value of I will be (A) I = 0: singularities set = ϕ (B) I = 0: singularities set =,

Mathematics

ECE – 2007 3. If the semi-circular contour D of radius 2 is as shown in the figure, then the value of the integral ∮

is

j

CE – 2009 4.

The analytic function f(z) = singularities at (A) 1 and 1 (B) 1 and i

5.

has

j2

(C) 1 and i (D) i and i

The value of the integral ∫

j2

2

dz (A) jπ (B) jπ

(where C is a closed curve given by |z| = 1) is (A) –πi (C) (B) (D) πi th

th

(C) π (D) π

th

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GATE QUESTION BANK

ECE – 2008 4. The residue of the function

f z 

5.

1

z  2 z  22 2

(A) (B)

at z=2 is

(A)

(C)

(B)

(D)

(C) (D) 2

∮(

) z is

(A) 4π (B) 4π

The equation sin(z)=10 has (A) no real or complex solution (B) exactly two distinct complex solutions (C) a unique solution (D) an infinite number of complex solutions

12.

+ j2 j2

(C) 4π (D) 4π

+ j2 j2

The real part of an analytic function z where z x + jy is given by cos(𝑥). The imaginary part of z is (A) os x (C) sin x (B) sin x (D) sin x

EE – 2007

If f(z) =

+

is given by (A) 2π (B) 2π +

z

, then ∮

z

(C) 2πj (D) 2πj

z and 1 and

(C) (D)

The value of

+

at its poles are

(A)

1.

where C is the

∮

contour |z-i/2| = 1 is (A) 2πi (C) t n z (B) π (D) πi t n z

ECE – 2010 7. The residues of a complex function

(B)

2

ECE – 2014 11. C is a closed path in the z-plane given by |z|=3. The value of the integral

ECE – 2009 6.

Mathematics

EE – 2011 2. A point z has been plotted in the complex plane, as shown in figure below. nit ir l

and z

and

ECE – 2011 8.

The value of the integral ∮

z

ECE\EE\IN – 2012 9. If x = √ then the value of x is ⁄ (C) x (A) ⁄ (D) 1 (B) 10.

Given f (z)

nit ir l

lm

where is the circle |z| is given by (A) 0 (C) 4/5 (B) 1/10 (D) 1

lm

nit ir l

lm

nit ir l

y y lmlm

. If C is a

nit ir l

y

counterclockwise path in the z – plane such that |z+1| =1, the value of ∮

y

z z is th

th

th

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GATE QUESTION BANK

EE – 2013 3.

z evaluated anticlockwise around

∮

the circle |z (A) 4π (B) 4.

i|

2 where i √ (C) 2 + π (D) 2 +2i

Square roots of – i, where i = √ (A) i, i (B)

os (

) + i sin (

, is

, are

)

IN – 2005 1. Consider the circle | | 2 in the complex plane (x, y) with z = x + iy. The minimum distant form the origin to the circle is (C) √ 4 (A) √2 2 (B) √ 4 (D) √2 2.

Let ̅, where z is a complex number not equal to zero. The z is a solution of (C) z (A) z (D) z (B) z

os ( ) + i sin ( ) (C)

os ( ) + i sin ( ) os ( ) + i sin ( )

(D) os ( ) + i sin ( os (

)

) + i sin ( )

EE – 2014 5. Let S be the set of points in the complex plane corresponding to the unit circle. {z: |z| } . Consider the (That is, function f(z)=zz* where z* denotes the complex conjugate of z. The f(z) maps S to which one of the following in the complex plane (A) unit circle (B) horizontal axis line segment from origin to (1, 0) (C) the point (1, 0) (D) the entire horizontal axis

IN – 2006 3. The value of the integral of the complex function 3s  4 f(s) (s  1)(s  2) Along the path s  3is (A) 2j (B) 4j

7.

All the values of the multi-valued complex function , where i √ are (A) purely imaginary. (B) real and non-negative. (C) on the unit circle. (D) equal in real and imaginary parts. Integration of the complex function z

, in the counter clockwise

(C) 6j (D) 8j

IN – 2007 4.

For the function

of a complex variable

z, the point z=0 is (A) a pole of order 3 (B) a pole of order 2 (C) a pole of order 1 (D) not a singularity 5.

6.

Mathematics

Let j = √ (A) √j (B) 1

.Then one value of (C)

is

(D)

IN – 2008 6. A complex variable x+j has its real part x varying in the range to + . Which one of the following is the locus (shown in thick lines) of 1/Z in the complex plane?

direction, around |z 1| = 1, is (A) πi (C) πi (B) (D) 2πi

th

th

th

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l xis

(Note:

j m gin ry xis

m gin ry xis

j

l xis

IN – 2009 The value of ∮

where the contour

of integration is a simple closed curve around the origin, is (A) 0 (C) (D) (B) 2πj 8.

If z = x+jy, where x and y are real. The value of | | is (A) 1 (C) (D) (B) √

9.

One of the roots of the equation 𝑥 =j, where j is positive square root of 1, is √ (A) j (C) j +j

(D)

√

)

x

j

√

√ y

pl n

l xis

(B)

z is.

∮

l xis

j

7.

Mathematics

IN – 2010 10. The contour C in the adjoining figure is described by x + y . The value of

m gin ry xis

m gin ry xis

GATE QUESTION BANK

(A) 2πj (B) 2πj

(C) 4πj (D) 4πj

IN – 2011 11. The contour integral ∮ / with C as the counter-clockwise unit circle in the zplane is equal to (A) 0 (C) 2π√ (B) 2π (D)

j

th

th

th

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GATE QUESTION BANK

Mathematics

Answer Keys and Explanations ME 1.

2.

4. [Ans. B] By definition C-R equation holds [Ans. A] f(z)=

has simple pole at z = 0

Residue of f(z) at z = 0 lim z z lim os z ∫ z z 2πi (residue at z = 0) 2πi 2πi 3.

[Ans. B] + 4i + 2i 2i + 2i + i + 4i +4 Modulus = √

[Ans. C] Given u=xy For analytic function u v  x y and

[Ans. A] +i 2 2 i + 2i

6.

dw u v  i dz x x 

7.

∫y x

∮

Replacing x by z and y by 0, we get

∮ 8.

+ 2i + i i

r os x

r sin

r sin

r os

r

r

2π

π 2

[Ans. C] u v x y v 2y y 2y + x v 2 y + x v u v y x 2x x 2x + x 2 x x

z2 C 2 Where C is a constant, z v m0 i + 1 2 Integrating, w  i

 (x2  y2  2ixy)   mi  2   or v 

i

x y

y = r sin x y r os

dw  y  ix dz

Where, z = x + iy dw = izdz

i

[Ans. C]

u u i x y

dw  0  iz dz

i

[Ans. C] +i +i i i +i 2i i + i 2 +i rg ( ) t n ( ) i π⁄ 2

u v  y x

By Milne Thomson method Let w = u + iv

or

5.

+ 2i

y 2  x2 2

th

th

th

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GATE QUESTION BANK

v

y y

9.

10.

[Ans. B] z ∫ ln z| z

[Ans. B] ∫s

z os z The poles are at z = n + /2 π = π/2 π/2 + π/2 None of these poles lie inside the unit circle |z| =1 Hence, sum of residues at poles = 0 Singularities set = ϕ and 2πi [sum o r si u s o t z t th poles] 2 πi

[Ans. C] iv n u x y v v v x+ y x y v u v u y x x y u u v x+ y y x 2y x + 2x y rm ont ing y t rms only llow v 2 xy +

3.

z z

ln i

ln

ln + ln i ln ln z os z i z i ln i ln z π i ( 2

ln + ln i + i sin i sin π/2

=

=| ̅̅̅̅

z

z= ∮

pplying z z

(

)

u hy’s int gr l th or m, using i .2πi ( )/

/

i

2πi

Now, ∮

/

ln

z

∮

i.e. ∮

)

[Ans. C] (A) is true since ̅̅̅̅

∫

[Ans. A] u hy’s int gr l th or m is f(a) =

+

CE 1.

2.

x x + onst nt

Mathematics

i .2πi 0( )

1/

i .2πi 0( )

1/

2π

̅̅̅̅

4πi

2π

|

(B) is true by triangle inequality of complex number (C) is not true since | |≥| | | | (D) is true since | + |2 = ( + ) ̅̅̅̅̅̅̅̅̅̅̅̅ + = ( + ) (z̅ + z̅ ) = z̅ + z̅ + z̅ + z̅ i ̅̅̅̅̅̅̅̅̅̅̅̅ And | |2 = ( + )

i

4.

4πi

[Ans. D] z z z z + z z i z+i The singularities are at z = i and –i

z

5.

[Ans. C]

= ( + ) (z̅ z̅ ) = z̅ + z̅ z̅ + z̅ ii Adding (i) and (ii) we get | + |2 + | |2 = 2 z̅ + 2 z̅ = 2| | + 2| |

r

∫

os 2πz 2z z *

2

th

th

∫

+ *z

+

th

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GATE QUESTION BANK

in

z

is point with in |z|=1(the

los urv w n us integral theorem and say that

7.

[Ans. B] 2 i z +i Multiplying by conjugates 2 i i +i i 2i + i 2 + + i 2 + i

u hy’s

os 2πz [2πi ( )] wh r z 2 2 z [Notice that f(z) is analytic on all points inside |z| ] 2

[2πi

os 2

π

(

/2 )

Mathematics

]

2πi

6.

[Ans. D] f = u + iv u = 3x2 – 3y2 For f to be analysis, we have CauchyRiemann conditions, u v i x y u v ii y x From (i) we have u v x x y ∫ v

ECE 1.

Given ,

∮

I

2.

x +

x

z

z +4 j| 2

|z

2j   2j  2j 2

[Ans. B] iv n

log

1 y  u  iv  loge  x  iy   log x2  y2  i tan1   2 x Since, u is constant, therefore



x v + x 2 i.e. v = 3x2 + f(x) iii Now applying equation (iii) we get u v y x [ x+

1 1  z  4 z  2jz  2j 2

Pole (0, 2) lies inside the circle |z j|=2 y u hy’s nt gr l ormul

∫ x y

y

[Ans. D]



1 log x2  y2  c 2





x +y Which is represented set of concentric circles.

x

y

3.

[Ans. A] s

∮

y x x By integrating, f(x) = 6yx – 3x2 + K Substitute in equation (iii) v= 3x2 + 6yx – 3x2 + K v yx + K

2πj sum o r si u

Singular points are s = Only s= +1 lies inside the given contour lims  1 f s Residues at s= +1 = S1

lims  1 S1

n

th

∮

th



1 1  S 1 2

s

2



s

th

2πj ( ) 2

πj

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GATE QUESTION BANK

4.

[Ans. A] Residue of z=2 is

Mathematics

z

d 2 z  2 f z   z2 dz  d 1  2 1   lim lim   2 3 z2 dz  z  2  z  2 32   z  2 

z+

)

z

+

lim

5.

+

(z

+ F(z 2πj 7.

+

)

[Ans. C] X(z) =

[Ans. D] sin z

Poles are Z= 0, Z =1, Z=2 Residue at Z=0 is lim

2i

Residue at Z =1 is lim

2 i

Residue z =2 is lim

2 i (

)

2 i

ut m m

8.

[Ans. A] z+4 ∮ z + 4z +

2 im 2 i

m 2 i

m

i

iz

log

4

2 2 i + i√ 2

√ 2 √

i

2 i

√

z

i

9.

√

i

i

i

√

log i + log( √ π iz log + i ( 2nπ) 2 +log √ π iz i ( 2nπ) + log 2 π z ( 2nπ) ilog( 2

i

)

[Ans. D] f(z) = + + z z z

∮ ∮(

+

log y i log i π i i 2

x log x i log π 2

⟹y √ √

)

10.

z

z+

z F z 2 π j r si u o Residue at z = 0 ( 2- order )

[Ans. C] z z

∫ 2πj

∫

z

*∫

∫

z+

z

z+

where f (z) =1

11. + z

x

log y

z +

x

√ ty

⟹ log y

( 2 infinite number of complex solutions sin z has infinite no. of complex solutions 6.

z + 4z + z+2 + 2 j will be outside the unit circle o th t int gr tion v lu is ‘z ro’

[Ans. A]

i√

i

iz

z

z

[Ans. C] s z lim

2j z + 2j

4+ j

2πj[ 4 + j

)

th

th

4π

th

+ 2j

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GATE QUESTION BANK

12.

EE 1.

[Ans. B] Suppose that z u x y + iv x y is analytic then, u and v satisfy the Cauchy Riemann equation u v u v n x y y x r u xy os x u v sin x x y u v os x y x v sin x

z

z +

∮

z

∮ 2 πi 2.

√ + o / is outside the unit circle is IV quadrant 3.

[Ans. A] z 4 ∮ z +4 |z i| 2 z +4 z 4 z 2i For z 2i Residue at z +2i 4 4 +2i z + 2i +4i t z 2i li insi tz 2i li outsi z 4 o∮ 2πi sum o r si u z +4 2πi 2i 4π

4.

[Ans. B] Let + i √ i Squaring both sides we get +2 i i Equating real and imaginary parts

[Ans. B] Pole (z=i) lies inside the circle. |z-i/2|=1. Hence ∮

z+i z

i

2 πi i , wh r

z z

-

π

2i

[Ans. D] Let + i Since Z is shown inside the unit circle in I quadrant, a and B are both +ve and +

√ ow

2

+ i i

+ Since

Mathematics

+

+

wh n

i

2

2

i √2

+

√

wh n

+ o

+i in

qu

r nt wh n

| | √ in

√(

) +(

+

+

√ √

+

i

i

√2 √2 i i +i ( ) √2 √2 i i

√2

+

i √2

√2 √2 i i i +i + i( ) + √2 √2 √2 √2 i +( ) √2 √2 π π os ( ) + i sin ( ) 4 4

)

+

+

th

th

th

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GATE QUESTION BANK

or

x π π os ( ) + i sin ( ) 4 4

5.

[Ans. C] z zz

} n s {z: |z| z All point of s will be mapped on the point (1, 0) 6.

[Ans. B] z log z i log z r l n Non-negative

7.

[Ans. C] ∫ x x

)

lim z

IN 1.

2.

int gr tion

2πi

z+

2

2 √2

o |z | king z |z| |z| z

2

uis o th ir l y 4

2

√2

[Ans. C] z z̅ Multiply both the side by z, we get z z̅ z |z| z |z| |z| wh r is ngl o z |z| since is a real quantity so in order to satisfy above equation has to be real quantity = 1 and , (where n = +2 )

z z

r +

√2 x

|

z lim z+ quir

x

2πi r s (f(a)) where a is a

singularity in contour c |z | r n pol s o z z nly z li s insi |z s(

y x x

|z|

Mathematics

π/2 ⁄

z 3.

[Ans. C]

πi

X X -2 -1 Cx y y (Cx ( -3

[Ans. A] | + i | 2 Radius of the circle is 2 and centre is at + i

3 Cx

y(n) n n y(n) )y(n)) 3s  4 1 2 C3 = F(s)  C3  . CC3 (sC 1)(s  2) s  1 s  2 y(n) 3 3 y(n) dz y(n)

2 + i

By Formula, y y ( ( Since, both n n contour, ) )

xy

 z  a  2j

the poles are enclosed by

therefore Value of integral=2πj + 2 2πj For distance to be min. The point P will be on the line passes through origin and centre of the circle. Slope of line OP = Slope of line OC

4.

πj

[Ans. B] Expand by Laurent series

𝑥 th

th

th

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GATE QUESTION BANK

5.

[Ans. D]

10. ⁄

tx

j

⁄

(

⁄

log

log

)

2πj 11.

2πj

2

+

[sum o r si

2 o pol

4πj

j

x +

x +

j

∮

⁄

z

∮( + + + + ) z z 2z z The only pole of z is at z , which lies within |z| ∫ z z 2πi (residue) Note: Residue of z at z is coefficient ⁄ of z i.e. 1, here.

x j x + x

j j x +

lim { x +

j ption

⟹z j j ⟹ 2[ j

[Ans. C] z

x+j

|

z

⁄

[Ans. B] x+j

|

z

∮

)

/

x

7.

⁄

log (

π j 2 π j j 2

log

6.

⁄

log (

z=∮

Pole z j Residue at z

⁄

log x

[Ans. D] ∮

)

Mathematics

s tis y th

ov

}

on itions

[Ans. A] u hy’s int gr l ormul is ∫ Here a = 0, then f(0) = sin 0 = 0

8.

[Ans. D] z x + iy p | |= | = |

9.

| |=

|

|=

[Ans. B] Given x3 = j = e+jπ/2 x

⁄

x

os

π

+ j sin

π

√ +j 2 2

th

th

th

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GATE QUESTION BANK

Mathematics

Laplace Transform ME – 2007 1. If F (s) is the Laplace transform of function f (t), then Laplace transform of t

 f (t) dt is 0

(A)

F (s)

(C) sF (s) – f (0)

(B)

F (s) f (0)

(D) CE – 2009 1. Laplace transfrm of the function f(x) = cosh(ax) is (A) (C)

ME – 2009 2.

The inverse Laplace transform of is (A) (B)



1 s s 2



(B) (C) 1 – (D)

ME – 2010 3. The Laplace transform of a function . The function

is

is

(A) (B)

(C) (D)

ME – 2012 4. The inverse Laplace transform of the function F(s)

is given by

(A) (B)

(C) (D)

(D)

ECE - 2005 1. In what range should Re(s) remain so that the Laplace transform of the function exists. (A) (C) (B) (D) ECE – 2006 2. A solution for the differential equation x’(t)+2x(t)= (t) with initial condition x( )=0 is (C) (A) (D) (B) ECE – 2008

ME – 2013 5. The function equation

3.

Consider the matrix P = *

satisfies the differential

value of eP is

and the auxiliary

conditions,

+ . The

(A) *

+

(B) [

]

. The

Laplace transform of

is given by

(A)

(C)

(B)

(D)

ME – 2014 6. Laplace transform of The Laplace transform of

(C) [

]

(D) [ is

.

]

ECE - 2010 4. The trigonometric Fourier series for the waveform f(t) shown below contains th

th

th

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GATE QUESTION BANK

Mathematics

ECE – 2013 9. A system is described by the differential equation

=x(t). Let x(t)

be a rectangular pulse given by , Assuming that y(0) = 0 and (A) Only cosine terms and zero value for the dc component (B) Only cosine terms and a positive value for the dc component (C) Only cosine terms and a negative value for the dc component (D) Only sine terms and a negative value for the dc component. 5.

the Laplace transform of y(t) is

Given [

10.

]

then the value of K is (A) 1 (C) 3 (B) 2 (D) 4 ECE– 2011 6.

[

If

]

then the initial

and final values of f(t) are respectively (A) 0, 2 (C) 0, 2/7 (B) 2, 0 (D) 2/7, 0

The maximum value of the solution y(t) of the differential equation y(t) + ̈ with initial condition ̇ and ≥ (A) 1 (C) (B) 2 (D) √

ECE – 2014 11. The unilateral Laplace transform of . Which one of the following is the unilateral Laplace transform of ?

ECE/EE/IN – 2012 7. The unilateral Laplace transform of f(t) is . The unilateral Laplace transform

8.

of t f(t) is (A) –

(C)

(B)

(D)

Consider the differential equation

|

|

The numerical value of (A) (B)

EE – 2005 12. For the equation (t) + 3 (t) + 2x(t) = 5, the solution x(t) approaches which of the following values as t ? (A) 0 (C) 5 (D) 10 (B) EE – 2014

|

is

13.

(C) (D)

Let

be

the

transform of signal x(t). Then, (A) 0 (C) 5 (B) 3 (D) 21 th

th

th

Laplace is

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GATE QUESTION BANK

14.

Mathematics

[ Let g: [ be a function [ ] where [x] defined by g(x) represents the integer part of x. (That is, it is the largest integer which is less than or equal to x). The value of the constant term in the Fourier series expansion of g(x) is_______

Answer Keys and Explanations ME 1.

5.

[Ans. C]

[Ans. A] From definition, We know ∫

2.

Taking Laplace transformation on both sides [ ] [ ] ( ) ( )

[Ans. C]

1 1 1 1    (s  s) s(s  1) s (s  1) 2

(

3.

)

( )

(

[

)

(

) (

[

[Ans. A] [

]

6.

4.

)

]

s and constant

]

[Ans. B] It is the standard result that L (cosh at) =

ECE 1.

[Ans. A] [

[Ans. D] {

]

[Ans. D]

CE 1.

[

)

]

[

Matching coefficient of in numerator we get,

(

]

} 2. {

[Ans. A] ̇ (t) + 2x (t) = (t) Taking Laplace transform of both sides , we get

}

th

th

th

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GATE QUESTION BANK

sX(s) X(0) + 2X(s) = 1 1 X(s) = s2 From Inverse Laplace transform gives, we get [ ] 3.

[Ans. D ] eP= [

Mathematics

cosine terms and a negative value of the dc component. 5.

[Ans. D]

[

]

] [

0 1  and P=   2  3 s 1  Where =   2 s+3 s  3 1  1  s  1s  2 2 s s 3    s  1 s  2 =   2  s  1 s  2  

]

[

1

1  s  1 s  2      s s  1s  2 

6.

]

[Ans. B]

Using initial value theorem:

 eP

 2 1  s  1  s  2   2  2  s  1 s  2

=

1 1   s  1 s  2    2 1   s  2 s  1  

=[

⁄

] =2

4.

[Ans. C] Since f(t) is an even function, its trigonometric Fourier series contains only cosine terms

∫

7.

∫

*∫

∫

[Ans. D]

+ t

[

(

)] 8.

[

[Ans. D]

]

Therefore, the trigonometric Fourier series for the waveform f(t) contains only

Taking Laplace transform on both the sides. We have, th

th

th

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GATE QUESTION BANK

11.

Mathematics

[Ans. D]

By Laplace transform property, [

]

[

]

(

[

] [

|

] 12.

9.

[Ans. B] =5 By taking Laplace transform

[Ans. B] Writing in terms of Laplace transform

( ⁄

X(s) = (

) 13.

[Ans. B]

(

)

( ( 10.

)

[

)

]

(

)

)

[Ans. D] 14.

[Ans. 0.5]

∫ For t =

∫

|

Value of constant term = 0.5 + sin

√

th

th

th

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GATE QUESTION BANK

Network Theory

Network Solution Methodology ECE - 2006 1. A negative resistance R is connected to a passive network N having driving point impedance (s) as shown below. For (s) to be positive real, Rneg

4.

In the interconnection of ideal source shown in the figure, it is known that the 60V source is absorbing power. 20 V + I

60 V 12 A

N

Z2(s)

Which of the following can be the value of the current source I? (A) 10 A (C) 15A (B) 13A (D) 18A

Z1(s)

(j )

(A) |R

|

Re

(B) |R

|

| (j )|

(C) |R

|

(D) |R

|

m

ECE - 2010 5. In the circuit shown, the power supplied by the voltage source is

(j ) (j )

ECE - 2007 2. For the circuit shown in the figure, the Thevenin voltage and resistance looking into X –Y are

1Ω 1Ω

1Ω

10V 1

1Ω

2

1Ω

1Ω

i 2i 1Ω

2

(A) 4/3 V 2Ω (B) 4V 2/3 Ω

2Ω

(C) 4/3 V 2/3Ω (D) 4V 2Ω

ECE - 2009 3. In the circuit shown, what value of RL maximizes the power delivered to RL? V 4Ω

4Ω V

(A) 0 W (B) 5 W

(C) 10 W (D) 100 W

ECE - 2011 6. In the circuit shown below, the value of R such that the power transferred to R is maximum is 10 Ω

10 Ω

R

10 Ω

4Ω

V 100V

(A) 2.4 Ω (B) 8⁄3 Ω

1Ω

R

5V

(C) 4 Ω (D) 6 Ω

1

2V

(A) 5Ω (B) 10Ω th

th

(C) 15Ω (D) 20Ω th

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GATE QUESTION BANK

7.

In the circuit shown below, the current I is equal to

11.

VB =6 V, then VC R

j4Ω

j4Ω

If VA

Network Theory

V

R

R

2Ω

VD is R

V

R

R

R

~

1Ω

6Ω

14 0 V

R V

5V

6Ω

(C) 2.8 0 A (D) 3.2 0 A

In the circuit shown below, the Norton equivalent current in amperes with respect to the terminals P and Q is

25Ω

(A) 6.4 – j4.8 (B) 6.56 – j7.87

Q

ECE/IN - 2012 9. The average power delivered to an impedance (4 j3) by a current 5cos (100t+100) A is (A) 44.2 W (C) 62.5 W (B) 50 W (D) 125 W

RC

RB RA

(C) 1/k

(A) k (B) k 13.

ECE/EE/IN - 2012 10. The impedance looking into nodes 1 and 2 in the given circuit is i

9kΩ

Rc

Rb

(C) 10 + j0 (D) 16 + j0

99i

1kΩ

(C) 3 V (D) 6 V

Ra

j50Ω

15Ω

V

ECE/EE/IN - 2013 12. Consider a delta connection of resistors and its equivalent star connection as shown. If all elements of the delta connection are scaled by a factor k, k>0, the elements of the corresponding star equivalent will be scaled by a factor of

P

j30Ω 16 0

2

(A) 5 V (B) 2 V

6Ω

(A) 1.4 0 A (B) 2.0 0 A 8.

10V

(D) √k

Three capacitors C1, C2 and C3 whose values are 10μF 5μF and 2μF +respectively, have breakdown voltages of 10V, 5V and 2V respectively. For the interconnection shown below , the maximum safe voltage in Volts that can be applied across the combination, and the corresponding total charge in μC stored in the effective capacitance across the terminals are respectively, C2 C3

1 100Ω 2

(A) 50  (B) 100 

C1

(C) 5 k (D) 10.1 k

(A) 2.8 and 36 (B) 7 and 119 th

th

(C) 2.8 and 32 (D) 7 and 80 th

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GATE QUESTION BANK

ECE - 2014 14. For maximum power transfer between two cascaded sections of an electrical network, the relationship between the output impedance of the first section to the input impedance of the second section is (A) (C) (B) (D)

18.

15.

19.

Consider the configuration shown in the figure which is a portion of a larger electrical network

Network Theory

In the figure shown, the value of the current I (in Amperes) is______. 5Ω 5Ω

5V

10Ω

1

In the circuit shown in the figure, the value of node voltage V is 10 0

i i

V

V 4Ω

R R

i

i

i

For R 1Ω and currents i =2 i 1 i 4 which one of the following is TRUE? (A) i 5 (B) i 4 (C) Data is sufficient to conclude that the supposed currents are impossible (D) Data is insufficient to identify the currents i i and i

17.

j6Ω

6Ω

j3Ω

R

i

16.

4 0

A Y-network has resistances of 10Ω each in two of its arms, while the third arm has a resistance of 11 Ω. In the equivalent -network, the lowest value (in Ω.) among the three resistances is ________

(A) 22 + j 2 V (B) 2 + j 22 V 20.

(C) 22 – j 2 V (D) 2 – j 22 V

The circuit shown in the figure, the angular frequency (in rad/s) at which the Norton equivalent impedance as seen from terminals b-b′ is purely resistive is_____________. 1Ω

1F b

10 cos t (volts)

~

0.5

b′

21.

Norton’s theorem states that a complex network connected to a load can be replaced with an equivalent impedance (A) in series with a current source (B) in parallel with a voltage source (C) in series with a voltage source (D) in parallel with a current source

For the Y-network shown in the figure, the value of R (in Ω) in the equivalent ∆-network is ____. R

5Ω

3Ω 7.5Ω

th

th

th

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GATE QUESTION BANK

22.

The magnitude of current (in mA) through the resistor R in the figure shown is__________ R 1kΩ R

10 m

R

2kΩ

R

23.

2m

4kΩ

3kΩ

The equivalent resistance in the infinite ladder network shown in the figure, is R . 2R R

R

R

R

R

R

R

(A) Very low (B) L/3

(C) 3L (D) Very high

EE - 2007 2. A 3V dc supply with an internal resistance of 2Ω supplies a passive non-linear resistance characterized by the relation VNL = . The power dissipated in the non-linear resistance is (A) 1.0 W (C) 2.5 W (B) 1.5 W (D) 3.0 W EE - 2008 3. In the circuit shown in the figure, the value of the current i will be given by 1Ω 1Ω

R

The value of R /R is________ 24.

Network Theory

5V

a + V

1Ω 1Ω

3Ω 1Ω

b

4V

1Ω 1Ω

i

The circuit shown in the figure represents a (A) 0.31 A (B) 1.25 A

R

4.

Assuming ideal elements in the circuit shown below, the voltage Vab will be 2Ω

a

(A) voltage controlled voltage source (B) voltage controlled current source (C) current controlled current source (D) current controlled voltage source EE - 2006 1. The three limbed non ideal core shown in the figure has three winding with nominal inductances L each when measured individually with a signal phase AC source. The inductance of the winding as connected will be R

(C) 1.75 A (D) 2.5 A

+ 1A

Vab

+

i

5V

b

(A) – 3 V (B) 0 V

(C) 3 V (D) 5 V

EE - 2009 Statements for Linked Answer Questions 5 & 6: 2kΩ 5V

+

+

3VAB A

2kΩ

1kΩ B

th

th

th

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GATE QUESTION BANK

5.

For the circuit given above, Thevenin’s resistance across terminals A and B is (A) 0.5kΩ (C) 1kΩ (B) 0.2kΩ (D) 0.11kΩ

the the

Network Theory

EE - 2010 10. If the 12 Ω resistor draws a current of 1A as shown in the figure, the value of resistance R is 1Ω

6.

7.

8.

For the circuit given above, the Thevenin’s voltage across the terminals and B is (A) 1.25V (C) 1V (B) 0.25V (D) 0.5V How many 200W/220V incandescent lamps connected in series would consume the same total power as a single 100W/220V incandescent lamp? (A) not possible (C) 3 (B) 4 (D) 2

(A) (B) (C) (D)

2Ω

Is = 5A

4V

(5A; Put Vs=20V) (2A; Put Vs =8V) (5A; Put Is = 10A) (7A; Put Is= 12A)

11.

1kΩ 6V

(A) 0mA (B) 1mA

(C) 2mA (D) 6mA

6V

(C) 8Ω (D) 18Ω

As shown in the figure, a 1Ω resistance is connected across a source that has a load line v + i = 100. The current through the resistance is i Source

1Ω

V

(A) 25A (B) 50A

(C) 100A (D) 200A

EE - 2011 12. In the circuit given below, the value of R required for the transfer of maximum power to the load having a resistance of 3Ω is R

10 V

The current through the 2 kΩ resistance in the circuit shown is C 1kΩ 1kΩ 2kΩ 1kΩ

12Ω

1

(A) 4Ω (B) 6Ω

For the circuit shown, find out the current flowing through the 2Ω resistance. Also identify the changes to be made to double the current through the 2Ω resistance.

V

9.

2

R

6Ω

(A) Zero (B) 6Ω

3Ω

oad

(C) 3Ω (D) Inifnity

EC/EE/IN - 2012 13. Assuming both the voltage sources are in phase the value of R for which maximum power is transferred from circuit A to circuit B is

th

th

th

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GATE QUESTION BANK 2Ω

~

16.

R

j1Ω

10V

Circuit

~

3V

In the figure, the value of resistor R is (25 + I/2) ohms, where I is the current in amperes. The current I is______

Circuit

(A) 0.8 Ω (B) 1.4 Ω

Network Theory

300V

R

(C) 2 Ω (D) 2.8 Ω

EC/IN/EE - 2013 14. The following arrangement consists of an ideal transformer and an attenuator which attenuates by a factor of 0.8. An ac voltage V 100V is applied across WX to get an open circuit voltage V across YZ. Next , an ac voltage V = 100V is applied across YZ to get an open circuit voltage V across WX. Then,V /V V /V are respectively. W

17.

Two identical coupled inductors are connected in series. The measured inductances for the two possible series connections are 380 μH and 240 μH. Their mutual inductance in μH is_______

18.

The total power dissipated in the circuit, shown in the figure, is 1 kW. 10

~

2

1Ω

R

V

ac source

oad 200V

1:1.25

The voltmeter, across the load, reads 200 V. The value of is__________

Y

19.

The line A to neutral voltage is 10 15° V for a balanced three phase star-connected load with phase sequence ABC. The voltage of line B with respect to line C is given by (A) 10√3 105 V (C) 10√3 75 V (B) 10 105 V (D) 10√3 90 V

20.

The Norton’s equivalent source in amperes as seen into the terminals X and Y is _______

Z

X

(A) 125/100 and 80/100 (B) 100/100 and 80/100 (C) 100/100 and 100/100 (D) 80/100 and 80/100 EE - 2014 15. The three circuit elements shown in the figure are part of an electric circuit. The total power absorbed by the three circuit elements in watts is ___________ 10

2.5V

8

2.5Ω 100V

5Ω

80V

5Ω 15V

5Ω

2

5V

th

th

th

THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30 Cross, 10 Main, Jayanagar 4 Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 154

GATE QUESTION BANK

21.

The power delivered by the current source, in the figure, is_____________

3.

Which one of the following equations is valid for the circuit shown below? 1Ω 1Ω I3 I1

1V

1Ω

1Ω

I2

1Ω

1V

Network Theory

+

1Ω

2

1Ω

5V

I5

I6

1Ω

1Ω

1Ω

22.

An incandescent lamp is marked 40 W, 240V. If resistance at room temperature (26°C) is 120 Ω, and temperature coefficient of resistance is 4.5 10 /°C, then its ‘ON’ state filament temperature in °C is approximately___________

IN - 2006 1. The root – mean – square value of a voltage waveform consisting of a superimposition of 2V dc and a 4V peak – to – peak square wave is (A) 2 V (C) √8 V (B) √6 V (D) √12 V

(A) (B) (C) (D) 4.

I

1V

5.

1A

(C) 1A (D) 2A

In the circuit shown below the maximum power that can be transferred to the load is 10√2 sin(1000t)

1Ω

(C) 2.5W (D) 3.0W

10Ω

10 m

i(t)

(A) 0W (B) 1.0W

1Ω

(A) 0A (B) 0.5A

3Ω 3Ω

6Ω

0 0 0 0

The current I supplied by the dc voltage source in the circuit shown below is

IN - 2008 2. The power supplied by the dc voltage source in the circuit shown below is

3V

I4 I7

(A) 250 W (B) 500 W

th

th

(C) 1000 W (D) 2000 W

th

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GATE QUESTION BANK

IN - 2009 6. The source network S is connected to the load network L as shown by dashed lines. RL

2Ω

+

+ 10 V

3V

Source Network S

Network Theory

IN - 2014 9. The circuit shown in the figure contains a dependent current source between A and terminals. The Thevenin’s equivalent resistance in kΩ between the terminals C and D is ___________. 5kΩ 5kΩ C

10 V

10 V

Load Network L

The power transferred from S to L would be maximum when RL is (A) 0Ω (C) 0.8 Ω (B) 0.6 Ω (D) 2Ω

V

IN - 2010 7. A 100 Ω , 1W resistor and a 800 Ω , 2W resistor are connected in series. The maximum dc voltage that can be applied continuously to the series circuit without exceeding the power limit of any of the resistors is (A) 90V (C) 45 V (B) 50 V (D) 40V IN - 2011 8. The current I shown in the circuit given below is equal to 10Ω

10 V

(A) 3 A (B) 3.67 A

10Ω

10

10Ω

(C) 6 A (D) 9 A

th

th

th

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GATE QUESTION BANK

Network Theory

Answer Keys and Explanations ECE 1.

4. [Ans. A] (s) = (s)

I

(S)

R ( (s))

+

60V

|

I

V x

2i

i 1Ω

2

2Ω

y

For V Apply nodal analysis, V V 2i 2 V 0 V i 2 1 V V 2 2V 2i 0 2 2 2 V V 4V Similarly, 2 V V 4V R 2Ω 3.

[Ans. C] For maximum power transfer, R V 4Ω

4Ω

5.

12 A

[Ans. A] 3 1Ω 1Ω

1Ω (

3)

3 10V

(

2)

1

1Ω

2 (

2)

1Ω Fig .1

R

The current through all the branches are marked as shown in Fig. 1. Apply KVL to outer loop 2( 3) 2( 2) 10 4 10 10 0 Power supplied by 10 V 10 0 0

4Ω

V 100V 100V 100 (100 V ) 8 4 Also V 50V 12.5 12.5 25 V R 4Ω R ⁄

12 A

In the given circuit, the current through the branch of 60 V source is (12 –I) as shown in Fig. The source of 60 V absorbs power, only if P =(12 – I)60 is +ve. i.e., I0, the capacitor current iC (t), where t is in seconds, is given by 20k

Common Data for Questions 4 and 5: The following series RLC circuit with zero initial condition is excited by a unit impulse function δ(t)

i

1H

±

20k 10V

(A) (B) (C) (D)

0.50 exp ( 0.25 exp ( 0.50 exp ( 0.25 exp (

+ 4 F -

VC

25t)mA 25t)mA 12.5 t)mA 6.25 t)mA

ECE - 2008 3. The circuit shown in the figure is used to charge the capacitor C alternately from two current sources as indicated. The switches S1 and S2 are mechanically coupled and connected as follows For 2nT ≤ t< (2n+ )T, (n = 0, , 2…) S to P and S2 to P2 For (2n+ )T ≤ t< (2n+2)T, (n = 0, , 2…) S to Q and S2 to Q2

δ(t)

4.

Ω

+ 1F

For t > 0, the output voltage (A) (B) (C) (D)

5.

√ √ √ √

(e

√

e

VC(t)

(t) is

)

te e e

cos ( ⁄

√

t)

√

t)

sin (

For t > 0, the voltage across the resistor is (A)

√

(e

(B) e (C) (D) th

√ √ th

√

e

*cos (

√

) )

√

√

)

√

)

e

sin (

e

cos (

th

sin (

√

)+

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GATE QUESTION BANK

6.

Network Theory 0Ω

In the following circuit, the switch S is closed at t=0. The rate of change of current

1.5A

(0 ) is given by

15mH

R

i(t)

(A) 0

(C)

(B)

(D)

(A) (B) (C) (D)

L

Rs

(

)

ECE - 2009 7. The time domain behavior of an RL circuit is represented by L + Ri =

0Ω

t=0 0Ω

S

IS

S i(t)

⁄

( + e

i(t) = 0 i(t) = i(t) = 0 i(t) = 0

ECE - 2011 10. In the circuit shown below, the initial charge on the capacitor is 2.5 mC, with the voltage polarity as indicated. The switch is closed at time t = 0. The current i(t) at a time t after the switch is closed is i(t)

sint) u(t)

For an initial current of i(0) =

0 2 e 0 2 e 0 e e

, the

0

steady state value of the current is given by

00 0 F

(A) i(t) →

+

(B) i(t) →

8.

(C) i(t) →

(1+B)

(D) i(t) →

(1+B)

The switch in the circuit shown was on position ‘a’ for a long time and is moved to position ‘b’ at time t = 0. The current i(t) for t > 0 is given by 0 kΩ a

b i(t)

100 V

0.2 F

+

0.2e 20e 0.2e 20e

i(t) = exp( 2 0 t) i(t) = exp( 2 0 t) i(t) = 0 exp( 2 0 t) i(t) = exp( 2 0 t)

ECE/EE/IN - 2012 11. In the following figure, C1 and C2 are ideal capacitors. has been charged to 12 V before the ideal switch S is closed at t = 0. The current i(t) for all t is. S

kΩ

t=0

0.3 F

0.5 F

(A) (B) (C) (D)

(A) (B) (C) (D)

u(t)m u(t)m u(t)m u(t)m

i(t)

(A) (B) (C) (D)

ECE - 2010 9. In the circuit shown, the switch S is open for a long time and is closed at t = 0. The current i(t) for t 0 is th

Zero a step function an exponentially decaying function an impulse function

th

th

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GATE QUESTION BANK

Network Theory

ECE - 2013 Common Data for Questions 12 and 13: Consider the following figure

R k +

1 10V

2

13.

e

⁄

), =

(

e

⁄

), =

( ) (t) = (

e

⁄

), =

msec

( ) (t) = ( 2

e

⁄

), =

msec

( ) (t) =

The current Is in Amps in the voltage source, and voltage Vs is Volts across the current source respectively , are (A) , 20 (C) , 20 (B) , 0 (D) , 20 The current in the 1 resistor in Amps is (A) 2 (C) 10 (B) 3.33 (D) 12

17.

ECE - 2014 14. In the figure shown, the ideal switch has been open for a long time. If it is closed at t = 0, then the magnitude of the current (in m ) through the 4 k resistor at t = 0 is _______. k

4k

2

(

( ) (t) =

2A

12.

F

2k

Is

5

Vs

R

2

2

msec msec

A series RC circuit is connected to a DC voltage source at time t = 0. The relation between the source voltage , the resistance R, the capacitance C, and the current i(t) is given below: = R i(t) + ∫ i(u)du Which one of the following represents the current i(t)? ( )

k

() 0

15.

16.

+ 0 F

mH

A series LCR circuit is operated at a frequency different from its resonant frequency. The operating frequency is such that the current leads the supply voltage. The magnitude of current is half the value at resonance. If the values of L, C and R are H, F and , respectively, the operating angular frequency (in rad/s) is ________.

0

t

0

t

( ) i(t)

In the figure shown, the capacitor is initially uncharged. Which one of the following expressions describes the current I(t) (in mA) for t >0?

th

th

th

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GATE QUESTION BANK

Network Theory

( ) R

R L1

i(t)

+ I

0

R

(A) 1,4 (B) 5,1

t

V

(C) 5,2 (D) 5,4

( )

2.

An ideal capacitor is charged to a voltage Vo and connected at t = 0 across an ideal inductor L. (The circuit now consists of a capacitor and inductor alone). If we let

i(t)

=

0

18.

In the circuit shown in the figure, the (t) (in volts )for t → value of is ________ i

0u(t)

2H 2i

, the voltage across the capacitor

at time t > 0 is given by (A) Vo (B) Vocos ( t) (C) Vosin ( t) (D) Voe cos ( t)

t

+

√

+ (t)

EE - 2007 3. In the circuit shown in figure switch is initially CLOSED and Sw is OPEN. The inductor L carries a current of 10 A and the capacitor is charged to 10 V with polarities as indicated. Sw is initially caps at t = 0 and Sw is OPENED at t = 0. The current through C and the voltage across L at t = 0 + is SW2 R2 0Ω

EE - 2006 1. In the circuit shown in the figure, the current source I = 1A, voltage source V = 5V, R = R = R = 1Ω, = = = 1H, = = 1F. The currents (in A) through R3 and the voltage source V respectively will be

R1= 0Ω

SW1

L 10A

(A) 55 A, 4.5 V (B) 5.5 A, 45 V 4.

C

+ _10V

(C) 45 A, 5.5 V (D) 4.5 A, 55 V

The state equation for the current I1 shown in the network shown below in terms of the voltage Vx and the

th

th

th

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GATE QUESTION BANK

Network Theory

independent source V, is given by 0 2H

1F +

+

1F 0 H

+

1F

3A

02

⁄ s ⁄4 s

(A) (B) (A)

= 1.4 Vx – 3.75I1 + V

(B)

= +1.4 Vx – 3.75I1

(C)

= 1.4 Vx + 3.75I1 + V

(D)

= 1.4 Vx + 3.75I1

V

V

EE - 2008 Statement for Linked Answer Questions 5 and 6 The current i(t) sketched in the figure flows through an initially uncharged 0.3 nF capacitor.

(C) 4 s (D) 9s

EE - 2009 8. In the figure shown, all elements used are ideal. For time t 0, the switch, K is closed and the relevant circuit is shown in Fig. 3 1

V2

R2

20 F

Fig. 1 2k

C1

V1

V2

R1

I2

I

2

1k

9V

I1 5V

0 F

V1

1k

9V

Fig. 2(at t = 0 )

K

A

R1

10 V

C

(0 )

= (0 ) =

[Ans. B] The given circuit is shown in Fig. 1 t steady state i e , as t → , capacitor behaves as open circuit. The circuit at steady state is shown in Fig.2

I2(0 ) I1(0 )

)

5.

10 V

I=0

10 V

[Ans. B] From Fig. 3: Write the Outer loop equation: d (t) (t) ]+ + 0= dt At t = 0 d (0 ) = (0 ) = + dt = /sec

Fig. 1(for t 0)

10 V

Apply voltage division across R1 = 2k and R2 = 1k

th

th

th

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GATE QUESTION BANK

7.

Node voltage, V1= =

[Ans. C] (

=

0 0

From the given circuit, = j = ( )

10 V ±

=0

= 8.

i

+

j

= 10

0 + 0

[Ans. D] For t < 0, the status of the circuit is shown in Fig. 1 inductor behaves as short circuit after a long time. i(0 ) =

) j

V1 is divided between C1 = 0 F and C2 = 20 F Apply, again voltage division across C1 and C2 0 = = = + 0 6.

Network Theory

[Ans. tt=0

]

1H Fig. 1

for t 0, the status of the circuit is shown in Fig. 2 Current through the inductor cannot change instantaneously. i(0 ) = i(0 ) = initial value (I.V) After a long time inductor behaves as short circuit. i( ) = 2 = final value (F. V) Time constant =

=

6

6 = 2 At t= 0 T =

, = 20

+ 0

+

i

10

10

(0 )

=

2

10 4A

(inductor in steady state)

+ 0

6

1H

=

0

=

0 ]

Fig. 2

i(t) =

+ (F

for t 0 = + (2 =2 e

)( )(

e

e

( )

9.

[Ans. 32] Load impedance = = (4 Ideal current source i(t) = 4 sin ( t + 20 ) Average real power = i

)

)

=( ) √

Power =

th

th

6 2

j2)

4

4= 2

th

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GATE QUESTION BANK

10.

[Ans. *]Range 1.5 to 1.6 at t = 0 , switch is open so (0 ) = at (t = )so capacitor is in steady state

11.

[Ans. C] Time constant of RC Circuit is = R In case I Time constant T = R In case II Time constant (T ) = R Given T = k T So R =k R R =k R ]

12.

[Ans. *] Range 186 to 188

2k

2k

( ) = 2k

4k

Network Theory

( )

=2

We know that (t) = (0 ) v ( )]e = 0 2 ]e +2

4

+

( ) 60

= (2k

)

4

2k = k 4 = 0 4 0 =4 0 s 000 = =2 0 4 (t) = 2 e +2 (t) = 2 2 e we know that dv (t) i (t) = dt d i (t) = c 2 2 e ] dt = 2 c( 2 0)e =4 F i (t) = +2 4 0 2 0e i (t) = 2 0 e (t) ] i = m

0 Hz

0 Hz H 0

j

0 Hz

This is a tank circuit configuration of a Parallel R-L-C circuit. Resonant frequency of tank circuit √

f =

=4

f =

=

(

+(

)

)

=

R=4 , f= 0H +

( )

=

=

= =

2

0

=

th

F

th

th

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GATE QUESTION BANK

Network Theory

Sinusoidal Steady State Analysis ECE - 2007 1. The RC circuit shown in the figure is R

C +

+

ECE - 2008 4. The Thevenin equivalent impedance ZTh between the nodes P and Q in the following circuit is 1

1

C

R

1

(A) (B) (C) (D)

1

1

1

a low-pass filter a high-pass filter a band-pass filter a band-reject filter

(A) 1

(C) 2 + s +

(B) 1 + s+ 2.

3.

An independent voltage source in series with an independence ZS =RS + jXS delivers a maximum average power to a load impedance ZL when (A) j (C) j (B) (D) j In the AC network shown in the figure, the phasor voltage VAB (in Volts) is

(D)

ECE - 2009 5. An AC source of RMS voltage 20V with internal impedance Zs (1 2j) Ω feeds a load of impedance ZL (7 4j) Ω in the figure below. The reactive power consumed by the load is (1

2j)Ω

A

2 ∠

5∠3 j3

B

(A) 0 (B) 5∠3

(7

5

5

(C) 12.5∠3 (D) 17∠3

(A) 8VAR (B) 16 VAR

j3

4j)Ω

(C) 28 VAR (D) 32 VAR

ECE - 2010 6. The current I in the circuit shown is 2 m

~

2 ∠

1 1

(C) 0 A (D) 20 A

(A) –j1 A (B) j1 A

th

th

5

rad s

th

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GATE QUESTION BANK

7.

For the parallel RLC circuit, which one of the following statements is NOT correct? (A) The bandwidth of the circuit decreases if R is increased (B) The bandwidth of the circuit remains same if L is increased (C) At resonance, input impedance is a real quantity (D) At resonance, the magnitude of input impedance attains its minimum value

ECE/EE - 2013 8. A source v (t) cos 1 t has an internal impedance of (4 + j3) . If a purely resistive load connected to this source has to extract the maximum power out of the source , its value in  should be (A) 3 (C) 5 (B) 4 (D) 7 ECE/EE/IN - 2013 9. In the circuit shown below, if the source voltage Vs= 10053.130 V then the Thevenin’s equivalent voltage in volts as seen by the load resistance RL is 3

j4

(A) 100900 (B) 80000 10.

j6

j4 

Network Theory

ECE - 2014 11. A 230 V rms source supplies power to two loads connected in parallel. The first load draws 10 kW at 0.8 leading power factor and the second one draws 10 kVA at 0.8 lagging power factor. The complex power delivered by the source is (A) (18 j 1.5) k (B) (18 j 1.5) k (C) (2 j 1.5) k (D) (2 j 1.5) k 12.

A periodic variable x is shown in the figure as a function of time. The rootmean-square (rms) value of x is ______. x 1

t T 2

13.

T 2

The circuit shown in the figure, the value of capacitor C (in mF) needed to have critically damped response i(t) is ________. 4

4

5 i(t)

1

RL= 10

(C) 800900 (D) 100600

Two magnetically uncoupled inductive coils have Q factors q1 and q2 at the chosen operating frequency. Their respective resistances are R1 and R2. When connected in series, their effective Q factor at the same operating frequency is (A) q q (B) (1⁄q ) (1 q ) ) ⁄( (C) (q q ) ) ⁄( (D) (q q )

14.

In the magnetically coupled circuit shown in the figure, 56 % of the total flux emanating from one coil links the other coil. The value of the mutual inductance (in H) is ______ . 1 6 cos (4t 3 )

15.

th

(1⁄16)

~

4

5

The steady state output of the circuit shown in the figure is given by

th

th

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GATE QUESTION BANK

Network Theory

V1 at a frequency which causes resonance with a current of I.

( )). If the y(t) = A( )sin( t amplitude | ( )| 0.25, then the frequency is

I

y(t) sin t

~ ( ) ( )

1

( )

√3 2

( )

1

The phasor diagram which is applicable to this circuit is

2

(A)

√3

I

EE - 2006 1. In the figure the current source is 1 ∠0 A, R = 1Ω, the impedances are Zc = j Ω, and ZL = 2jΩ. The Thevenin equivalent looking into the circuit across X-Y is

(B)

I

x

(C)

y

(A) (B) (C) (D)

√2∠0 V, (1 + 2j) Ω 2 ∠450 V, (1 – 2j) Ω 2 ∠450 V, (1 + j) Ω √2∠450 V, (1 + j) Ω

I

(D)

2.

3.

An energy meter connected to an immersion heater (resistive) operating on an AC 230 V, 50 Hz, AC single phase source reads 2.3 units (kWh) in 1 hour. The heater is removed from the supply and now connected to a 400 V peak to peak square wave source of 150 Hz. The power in kW dissipated by the heater will be (A) 3.478 (C) 1.540 (B) 1.739 (D) 0.870 The circuit shown in the figure is energized by a sinusoidal voltage source

I

EE - 2007 4. In the figure transformer T1 has two secondaries, all three windings having the same number of turns and with polarities as indicated. One secondary is shorted by a 1 Ω resistor , and the other by a 15m th

th

th

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GATE QUESTION BANK

capacitor. The switch SW is opened (t =0) when the capacitor is charged to 5 V with left plate as positive At (t =0+) the voltage VP and Current IR are

A

T1

L

R B

+ +

Vp

Im

(A) 25 V, 0.0A (B) Very large voltage, very large current (C) 5.0 V, 0.5 A (D) 5.0 V, 0.5 A 5.

C

(A)

C

25V

R

~

IR SW

Network Theory

Re

(B)

In the figure given below all phasors are with reference to the potential at point “O”. The locus of voltage phasor YX as R is varied from zero to infinity is shown by

Im

Re

~

∠

R

(C) C

XY

~

∠

Im

O 2V

(A) 0

(C)

2V 0

VYX

VYX Locus of VYX

Re

Locus of VYX

(D) Im

(B)

Locus of VYX

(D)

Locus of VYX

VYX 0

VYX 2V

0

2V Re

6.

The R-L-C series circuit shown is supplied from a variable frequency voltage source. The admittance locus of the network at terminals AB for increasing frequency is

EE - 2008 7. The Thevenin's equivalent of a circuit operating at = 5rad/s, has = 3.71∠ 15.90 V and ZO =2.38 – j 0.667Ω. At this frequency, the minimal realization of the Thevenin's impedance will have a th

th

th

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GATE QUESTION BANK

(A) resistor and a capacitor and an inductor (B) resistor and a capacitor (C) resistor and an inductor (D) capacitor and an inductor

Network Theory

EE - 2011 10. The r.m.s value of the current i(t) in the circuit shown below is 1

1 1

8.

The resonant frequency for the given circuit will be

i(t)

1

0.1H

~ (1. sin t)

(C) 1 (D) √2

(A) (B) 1

1

√

Common Data Q. 11 and Q. 12 The input voltage given to a converter is (A) 1 rad /s (B) 2 rad /s

(C) 3 rad /s (D) 4 rad /s

EE - 2010 9. If the electrical circuit of figure (b) is an equivalent of the coupled tank system of figure (a), then

v 1 √2 sin(1 t) The input drawn by the converter is i

11.

h

(a) oupled tank B A

3

) t

2√2 sin(5 t The input power of the converter is (A) 0.31 (C) 0.5 (B) 0.44 (D) 0.71

4

) 6)

The active power drawn by the convert is (A) 181 W (C) 707 W (B) 500 W (D) 887 W

and C, D

EE - 2012 13. A two–phase load draws the following phase currents: ), i (t) sin( t i (t) cos( t ). These currents are balanced if 1 is equal to ) (A) (C) ( 2 ) (B) (D) ( 2

and B, D

14.

D C

(b) Electrical equivalent

(A) A, B are resistances capacitances (B) A, C are resistances capacitances (C) A, B are capacitances resistances (D) A, C are capacitances resistances

t

5√2 sin(3

12.

h

(1 √2 sin (1

and C, D and B, D

th

The average power delivered to an impedance (4 j3)  by a current 5 cos (100t+100) A is (A) 44.2 W (C) 62.5 W (B) 50 W (D) 125 W

th

th

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GATE QUESTION BANK

EE - 2014 15. A combination of 1 µF capacitor with an initial voltage ( ) 2 in series with a 100 resistor is connected to a 20 mA ideal dc current source by operating both switches at t = 0s as shown. Which of the following graphs shown in the options approximates the voltage across the current source over the next few seconds?

Network Theory

16.

A non-ideal voltage source has an internal impedance of . If a purely resistive load is to be chosen that maximizes the power transferred to the load, its value must be (A) 0 (B) real part of (C) magnitude of (D) complex conjugate of

17.

A series RLC circuit is observed at two frequencies. At = 1 krad/s, we note that source voltage 1 ∠ results in a current . 3∠31 A. At = 2krad/s, the source voltage 1 ∠ results in a current 2∠ A. The closest values for R,L,C out of the following options are (A) 5 25 m 1 (B) 5 1 m 25 (C) 5 5 m 5 (D) 5 5m 5

t t

( )

t 2

IN - 2006 1. Consider the AC bridge shown below. If

( )

RC = 1 and

V ΔC 0 For

j 0.1H

1F

z

1

j .1 j .1

Therefore, [ ] On the basis of above analysis, the admittance locus is

1 j ( .1 ) 1 1 At resonance, imaginary part must be zero. .1

th

1 th

th

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GATE QUESTION BANK

.1

1

11.

1 1

1 9 3 rad/sec

9.

l

C=

10.

[Ans. B] rms value of input voltage 1 √2 1 √2 rms value of current 1 √2 5√2 2√2 √( ) ( ) ( ) √2 √2 √2 11.358 Let input power factor cos l cos =active power drawn by the converter 1 11.358 cos 5 cos .44

[Ans. D] In such system, volumetric flow rate C is analogous to current and pressure is analogous to voltage. The hydraulic capacitance due to storage in gravity field is defined as Where A = Area of the tank ρ ensity of the fluid g = Acceleration due to gravity The hydraulic capacitance is represented by A & C. Liquid trying to flow out of a container, can meet with resistance in several ways. If the outlet is a pipe, the friction between the liquid and the pipe walls produces resistance to flow. Such resistance is represented by B & D.

Network Theory

12.

[Ans. B] 1 i

√2 sin(1

t)

1 √2 sin (1

t

3

)

5√2 sin (3

) 4 2√2sin(5 t 4) Fundamental component of input voltage ( ) 1 √2 sin(1 t) 1

( )

,

(i )

,

t

√2

1 √2 Fundamental component of current (i ) 1 √2 sin(1 t 3)

[Ans. B] 1 sint sin t 1 and 1 rad sec Impedance of the branch containing inductor & capacitor j( ) 1 j( ) 1 1 j (1 1 ) 1 1 So, this branch is short – circuit and the whole current flow through it 1. sin t i(t) 1. sint 1 rms values of the current 1

1 √2

√2 Phase difference components

1 between

these

two

, cos cos .5 3 3 Active power due to fundamental components ( ), (i ) , cos 1 1 .5 5 Since, 3rd & 5th harmonics are absent in input voltage, there is no active power due to the these components. Hence, active power drawn by the converter =Active power due to fundamental components = 500 W

√2

th

th

th

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GATE QUESTION BANK

13.

[Ans. D] i sin( t i cos( t

17.

[Ans. B]

) )

j(

sin (

t ) 2 As these two currents are balanced. i i sin

sin

2. sin ( )

sin( t 2 sin ( t

14.

4

2

1k rad s 1 ∠ . 3∠31 1 ∠ . 3∠31 j( )

)

j (1

) t

)

1

2

1

1

4

1 1

2 (2

α)

5

IN 1.

4

1 and 25

(

)

(

) )

1

d dt

1 ∫ dt (

k

At t = 0, k 2 t t

2

16.

)

1

[

1 1

1 2

1 1 2 1

1

t

1 2

)

(

2

2

.

[Ans. D]

(

t

5 ∠

∠

c

)

1

[Ans. C] tt

∫d

∠

)

On solving equation 5 , 1 m ,

[Ans. B] Z = 4 j3 = RL jXc; RL=4; 5cos(1 t 1 ) mcos(cot

1

.

5 2

5

1 ∠ . 3∠31

) 1 c 2k rad s

j (2

) )

)

t

) . cos ( t

2

P= 15.

2

sin (

1 cos ( 2 2 1 ( 2 2

Network Theory

1

2

1

2

1

t

[Ans. C] Magnitude of | | (using transfer theorem)

1 1 2 maximum

1 2

power

th

th

1

]

1 1 1 2 1 2 1

1

1

1

1 2

1 2

2 1 2

( ) 1

2

th

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GATE QUESTION BANK

2

2

2 (2

)

i 3.

i

Network Theory

(1

5 cos 2t)

[Ans. D] The A.C bridge is shown in Fig

. 1, it can neglected in comparison to 2 [Ans. A] The circuit is shown in the figure 1

Let

1

.25

i

(5 cos 2t)

1

1

Fig. 1

Superposition theorem can be used to find i Due to 1 V d.c source alone, the circuit is shown in Fig. 2, where the current source is open circuited (I = 0). Also, in steady state, L behave as short circuit and C behave as open circuit

1

,

1

,

,

j√

1

With R, L and C having positive finite value, becomes imaginary which is not possible. cannot be made zero 4.

1

,

j , of the vridge is balanced if product of the impedances of opposite arms is same i.e., j or j

[Ans. A] The coupled circuit is shown in Fig. 1

1 i Fig. 2

From figure 2, i 1 Due to AC source (5 cos 2t) alone, the circuit is shown in figure 3, where the voltage source is short circuited (V = 0)

ig 1

Write the mesh equations: j j (1) j j (2) The equivalent T network is shown in Fig .2

1

1 .25

i

(5 cos 2t)

2.

Fig. 3

For As

2, 2 1

2

.25

.5,

2 ig 2

, the parallel RLC circuit is

Write the mesh equations: ( )

under resonance i 5 cos 2t th

th

th

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GATE QUESTION BANK

(

or

)

. (3)

For C = 1000 1 1 l

( ) ( ) or . (4) omparig (1)with (3)and (2)with (4, ) j , j , j ( ) j , j ( ) 5.

[Ans. D] Let v (t) v (t)

1 √2 in (1 1 √2 cos(7

RMS value of v (t)

j(

t) )

√

√ √

1

MSV of v (t) 1 1 Similarly M.S.V of v (t) 1 Let v(t) = v (t) v (t) . . of v(t) 1 1 1 1 1 1 ,1 Power is consumed or dissipated only in 1 . if it is , . . of v(t) 1 ,1

8.

2

=

∠

1∠

)

= 10

circuit it under

1

( )

Q= √

A

1

5Ω

[Ans. B]

[Ans. D] Circuit is under resonance I=

l

[Ans. D] = 1000 1 resonance 1 1 1 1

9. 6.

1

The circuit is open for the excitation frequency 1 r s steady state current, It may also be noted that in steady state, as t v(t) = 0 because of the exponential,

t) and

(

Network Theory

10.

100JV

√

2

[Ans. A] The given circuit is shown in fig. 1 2

cos(1

1

5√2e

)

( ) 2 sin(5t)

[Ans. A] The circuit is shown in the figure

1m

Given:

For the given input, v(t) 5 √2e cos(1 1 r s Z=1+ For L = l mH 1 1 1

.1 i(t)

i( ) cos(5t)

Fig. 1 1

R=2 ,

5 rad sec .1 ,

2

Taking cos (5t) = Re[1e e ] 1 cos (5t) e[1 e e ] ⃗⃗⃗ hasor, 1e for (t) 2 sin(5t) 2 cos(5t hasor, ⃗⃗⃗⃗ 2 e

v(t)

7.

t)

9

)

The phasor equivalent circuit is shown in fig – 2

th

th

th

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GATE QUESTION BANK

Network Theory

9 j2

2

⃗⃗⃗

⃗⃗⃗⃗ 2∠

j2 ⃗ Fig. 2

The circuit is further simplified to the circuit in Fig. 3 and Fig. 4 j1

2

1

j2 ⃗

Fig. 3

(1

j)

2

j2 ⃗

Fig. 4 From Fig 4 (1 j)2 (1 j) √2∠45 ⃗ 2 j2 1 j √2∠ 45 1∠9 1∠ 9 j ⃗ i(t) e[ e ] e[1e e ]

e[e ( sin(5t) 11.

)

]

cos(5t

[Ans. C] Phasor current through 2

1 1 1

9

)

j j

j

i (t) 1 cos(5 t) Power is dissipated in the resistance, 2 . Assuming that the source values are given in RMS value Power delivered by the 2 sources = Power dissipated in the equivalent 2 of Fig. 4 (1) 2 2 Note: Assuming Peak value for the sources, Power delivered by the 2 sources =( ) √

2

1

th

th

th

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GATE QUESTION BANK

Network Theory

Laplace Transform ECE - 2006 1. The first and the last critical frequencies (singularities) of a driving point impedance function of a passive network having two kinds of elements, are a pole and a zero respectively. The above property will be satisfied by (A) RL network only (B) RC network only (C) LC network only (D) RC as well as RL networks 2.

A 2 mH inductor with some initial current can be represented as shown below, where s is the Laplace Transform variable. The value of initial current is I(s)

ECE - 2009 4. If the transfer function of the following network is

= R

+

+ RL

Vi

V0

C

_

_

The value of the load resistance RL is (A) R/4 (C) R (B) R/2 (D) 2R ECE - 2011 5. The circuit shown below is driven by a sinusoidal input . The steady state output is R

C

0.002

R

C

1mV +

(A) 0.5 A (B) 2.0A

(C) 1.0 A (D) 0.0A

ECE - 2007 3. Two series resonant filters are as shown in the figure. Let the 3-dB bandwidth of Filter 1 be B1 and that of Filter 2 be B2. The Value of C1

is

⁄ ⁄ ⁄ ⁄

(A) (B) (C) (D)

⁄ ⁄ ⁄ ⁄

ECE/EE/IN - 2013 6.

The transfer function

of the circuit

shown below is

L1

00μF

+ Vi

+ R

Vo

10k V1(s)

Filter 1

V2(s) 00μF

+ Vi

+ R

Vo

Filter 2

(A) 4 (B) 1

(C) (D)

(A)

(C)

(B)

(D)

⁄ ⁄ th

th

th

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GATE QUESTION BANK

Network Theory

EE – 2014 1. The driving point impedance Z(s) for the circuit shown below is

F

F

(A)

(C)

(B)

(D)

Answer Keys and Explanations ECE 1.

[Ans. B]

2.

[Ans. A]

5.

[Ans. A] Redrawing the circuit s – domain

I(S) +

~

( ‖ )

LS V(S)

(

)

Li (0 )

+

…………… ⁄ 0

3.

0

So here, Now,

[Ans. D] ⁄

4.

0

⁄

⁄

Put ,

[Ans. C] (

(

(

⁄

(

⁄

So,

)

⁄ (

⁄

)

) ⁄ (

⁄

)

)) ⁄ (

+ ………………………

I(s) =

⁄ (

⁄

*

⁄

Now ,

)

( ‖ )

)

Satisfies above equation th

th

th

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GATE QUESTION BANK

Network Theory

In time domain,

( )

6.

[Ans. D] 100 μF 0 100 μF

⁄ ⁄ Substituting the values we get

EE 1.

Ans. A]

F

peda e f

F

du a e

peda e f apa

‘ ’d

a

f F

(

)

th

th

th

THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30 Cross, 10 Main, Jayanagar 4 Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 206

GATE QUESTION BANK

Network Theory

Two-Port Networks ECE - 2006 1. In the two port network shown in the figure below, Z12 and Z21are, respectively

r

(A) r n (B) n

(C) n r (D) r n

r

The z-parameter matrix for this network is

4.

r

r

3.

*

+[

(C) *

+

(B) *

+

(D) *

+

(A) *

+

(C) *

(B) *

+

(D) *

Z(s)

. The component values are

]

(A)

(C)

(B)

(D)

ECE - 2008 Statement for linked Answer Questions 3 and 4 A two port network shown below is excited by external dc sources. The voltages and currents are measured with voltmeters and ammeters (All assumed to be ideal) as indicated. Under following switch conditions, the readings obtained are: (i) –open , - closed –closed ,

+

The driving point impedance of the following network is given by

If port-2 is terminated by RL, then input impedance seen at port-1 is given by

(ii)

+

r

A two-port network is represented by ABCD parameters given by [ ]

+

The h-parameter matrix for this network is

5. 2.

(A) *

s

(A) (B) (C) (D)

R

L

L H R Ω L H R Ω L = 0.1H, R = 2Ω L H R 2Ω

F F 0.1F F

ECE - 2010 6. For the two-port network shown below, the short circuit admittance parameter matrix is Ω

1

2 Ω

Ω

Ω 2

(A) *

– open

(B) *

2 2

+S

(C) * +S

(D) *

+S 2 2

+S

2 wo port n twork

ECE - 2011 7. In the circuit shown below, the network N is described by the following Y matrix:

2

* th

th

+. th

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GATE QUESTION BANK

The voltage gain

s

is

Network Theory

s s

s s s

2

s

s s

s

s s

s

s s

s s s

N

s s s s s s

(A) 1/90 (B) 1/90

(C) (D)

1/99 1/11

11.

ECE/EE/IN - 2012 Common Data for questions 8 and 9 With 10 V dc connected at port A in the linear nonreciprocal two-port network shown below, the following were observed: (i) 1 connected at port B draws a current of 3 A (ii) 2.5 connected at port B draws a current of 2 A

2

2 2

12.

8.

9.

In the h-parameter model of the 2-port network given in the figure shown, the value of h (in S) is ______ .

Consider the building block ‘N twork N’ shown in th figur L t F n R k N twork N

With 10 V dc connected at port A, the current drawn by 7Ω connected at port B is (A) 3/7 A (C) 1 A (B) 5/7 A (D) 9/7 A

called

R

Two such blocks are connected in cascade, as shown in the figure.

For the same network, with 6 V dc connected at port A, 1Ω connected at port B draws 7/3 A. If 8 V dc is connected to port A, the open circuit voltage at port B is (A) 6 V (C) 8 V (B) 7 V (D) 9 V

s

N twork N

N twork N

h tr nsf r fun tion ECE - 2014 10. A two-port

2

2

network

has scattering s s parameters given by [ ] *s s + If the port-2 of the two- port is short circuited, the s parameter for the resultant one-port network is th

of the cascaded

network is (A)

(C) (

(B)

(D)

th

th

s

)

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GATE QUESTION BANK

13.

For the two-port network shown in the figure, the impedance (Z) matrix (in ) is

I1

Network Theory k

k

k

2

49i1

Input loop F F

2

(A) * (B) *

2 2

(C) *

+

2

(D) *

+

2 2

(A) 2 F (B) F

+ +

EE - 2006 1. The parameters of the circuit shown in the figure are R MΩ R = 10 Ω, A = V/V.If = 1 V, then output voltage, input impedance and output impedance respectively are R

R +

(C) 2 F (D) Fs

EE - 2010 4. The two-port network P shown in the figure has ports 1 and 2, denoted by terminals (a, b) and (c, d), respectively. It has an impedance matrix Z with parameters denoted by . A 1Ω resistor is connected in series with the network at port 1 as shown in the figure. The impedance matrix of the modified twoport network (shown as a dashed box) is

+ _

e

a

c

_

(A) 1V, , 10 Ω (B) 1 V,0, 10Ω 2.

P

(C) 1 V, 0, (D) 10 V, ,10Ω

f

The parameter type and the matrix representation of the relevant two port parameters that describe the circuit shown are

d

b

(A) [

]

(B) [

]

(C) [

]

(D) [

]

IN - 2007 (A) z parameters,*

+

(B) h parameters,*

+

(C) h parameters,*

+

1.

The DC voltage gain circuit is given by. R

(D) z parameters,*

in the following R

R

+

EE - 2009 3. The equivalent capacitance of the input loop of the circuit shown is

th

(A) AV

(C) AV

(B) AV

(D) AV

th

th

+R

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GATE QUESTION BANK

IN - 2008 2. For the circuit shown below the input resistance R11 =

|

3I2

2V3

Ω

+

I2 V3

2Ω

V1

+

2

2

1

2Ω

+I 1 V1

IN - 2013 3. Considering the transformer to be ideal, th tr nsmission p r m t r ‘ ’ of th 2 port network shown in the figure below is

is +

Network Theory

1:2

I1

5

5

V2

(C) (D)

I2

V2 2

(A) 1.3 (B) 1.4 (A) Ω (B) 2 Ω

2

(C) 0.5 (D) 2.0

Ω Ω

Answer Keys and Explanations ECE 1.

3.

[Ans. C] = =

[Ans. B] |

current source will be

s (1)

open)

|

=0

|

| |

2.

=0

=

r

s (2) |

[Ans .D] The ABCD parameter equations are given by,

=* 4.

wh n th n twork is t rmin l R R R

R

= 1.5 +

[Ans. A] =h h =h h From giv n p r m t rs 2

fig

R R

= 1.5 H=*

RL

= 3 +

Fig. 1

th

th

th

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GATE QUESTION BANK

5.

Network Theory

[Ans. D] R

L R L ⁄

(R L

Note:

R ⁄ R L⁄

R L

is Independent of

L⁄ ) 8.

R

[Ans. C] As per the given conditions, we can draw the following two figures.

L⁄

RL R L

L

R

(

⁄R ⁄L ) 2 s 2 Comparing (1) and (2) 2

⁄ ⁄R ⁄L

A

B

± 10 V

2

F R

2

2

L

2A

H

B A

6.

3A

N

⁄

±

N

10V

2.5

[Ans. A] | | | [ ]

7.

*

Let Vth and Rth be the Thevenin voltage & resistance as seen from part B.

2 2 2

+

R

[Ans. D] iv n

*

+ 2

2 From the circuit shown in Fig. 1

R 2

2

N

Vth = 3Rth + 3 (1) & Vth = 2Rth + 5 (2) Solving (1) & (2) Rth 2 So, Vth = 3 x 2 + 3 = 9V Now,

Fig

From eqn (2)

th

th

th

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GATE QUESTION BANK

Network Theory

i

N

2

N

i= 9.

The y parameter of the parallel network is equal to the sum of the individual network y parameter For network A

[Ans. B] R

2

So, Vth = 7/3 x 2 + =

= 7V.

The open circuit voltage at port B is 7V. 10.

2

[Ans. B]

p r m t rs

ort

ort 2

Lo

2 For network B

s s s Port 2 is short circuited h n ws

2 2

2

From s

s [

s ] s

s s

2

From s

s [

s s 11.

s

2

s ] s s [ s s s s s s

s s

[Ans. *] Range 1.24 to 1.26 In the figure two port networks in parallel

th

th

th

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GATE QUESTION BANK

EE 1.

h

Network Theory

[Ans. A] = 1V =

2

2

= 12.

s

2.

s

[Ans. C] h h

s

V1

Apply mesh analysis to determine the current s )

s

(2R

s

s )

R s

s

Since port – 1 is open – circuit , I1=0 Port – 2 is short – circuit, V2 =0

s

)

s

[from

)

R]

s

)

R +

s

]

n

) (2R s R ) (R *(2R

s

R

s R s s

(2R

R

s

s

) (R

)

R s Rs

2Rs

k F s s

h

|

h

|

h

|

h

|

s

R s

o h – p r m t rs

R 3.

}R

h h

h ] h

*

+

k

k

i

s s

[

[Ans. A] Assume a 1A current source at input terminals, = 1A

R s sr s R R

V2

s

(2R

[(R

s

s

R

om ining

s

I2

s s

(R

h h

I1

R

R s

13.

R = 10

[Ans. B]

i

k

s

~

F F

[Ans. C]

i

[

]

*

Applying KVL i i [2

+ *

2

+

i

j ]

j

Input impedance

2

j

As imaginary part is negative, input impedance has equivalent capacitive reactance . th

th

th

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GATE QUESTION BANK

Network Theory

2 h r

2

2

2

put it in qu tion

2 2 F 4.

o[

[Ans. C] The impedance matrix of the modified network is calculated from fig. given below:

3.

]

[Ans. A] 2

1 I1

[ ][ ]

[ ]

|

V1

2

5

2

5

2

V2

2

2 [

[ ]

n

2

|

][ ]

[

[ ]

]

*

t

+

2 2 From qu tion

1

2

2

From qu tion

Fig.

2 IN 1.

From qu tion

2 n

[Ans. A] V= =

2.

[Ans. D] 2 2V3

Ω + I1

+

f

2Ω

d

+ +

b

V1

3I2 I2

c 2

+

V3

2Ω

e

Apply KVL on abcdef

th

th

th

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GATE QUESTION BANK

Network Theory

Network Topology ECE - 2008 1. In the following graph, the number of trees (P) and the number of cut-sets (Q) are

Which of the following statements is true? (A) The equations v1 v2+v3 = 0, v3 +v4 –v5 =0 are KVL equations for the network for some loops (B) The equations v1-v3-v6 = 0, v4 +v5 –v6 =0 are KVL equations for the network for some loops (C) E = AV (D) AV = 0 are KVL equations for the network

(1)

(2)

(3) (4)

(A) P = 2, Q = 2 (B) P = 2, Q = 6

(C) P = 4, Q = 6 (D) P = 4, Q = 10

EE - 2007 1. The matrix A given below is the node incidence matrix of a network. The columns correspond to branches of the network while the rows correspond to nodes. Let V = [v1 v2… v6]T denote the vector of branch voltages. While I = [i1 i2… i6]T that of branch currents. The vector E=[e1 e2 e3 e4]T denotes the vector of node voltages relative to a common ground. [

EE - 2008 2. The number of chords in the graph of the given circuit will be

±

(A) 3 (B) 4

(C) 5 (D) 6

]

th

th

th

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GATE QUESTION BANK

Network Theory

Answer Keys and Explanations ECE 1.

The graph of the network is shown in fig. Where From the graph it can be observed that (i) are not KVL equations as set of branches (1, 2, 3) and (3, 4, 5) do not form closed paths. (ii) and are KVL equations for the loops (1, 3, 6) and (4,5, 6) From the matrix, A it can be concluded that (i) E ≠ (ii) AV = 0 are not KVL equations Statement in option (b) is true.

[Ans. C] Different trees (P) are shown below.

Different cut sets (Q are shown below

(1)

(2)

(3)

(5)

(6) (4)

So P = 4, Q = 6 2. EE 1.

[Ans. A] The graph of the given circuit is shown in Fig. Number of nodes = N = 4 Number of branches = B = 6 Number of tree branches = (N – 1) = 3 Number of links = L = B – (N – 1) = 3

[Ans. B] For the given node – to – branch incidence matrix [

] 1 5

2

I

III

II

Fig. 4

3

IV

6

Fig.

th

th

th

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GATE QUESTION BANK

Signals and Systems

Introduction to Signals & Systems ECE - 2006 1. The Dirac delta function t is defined as t (A) t , t rw s t (B) t , t rw s t (C) t , t rw s ∫ (D)

t

, ∫

t t t t

rw s

t t

ECE - 2007 2. The 3-dB bandwidth of the low-pass signal u t , where u(t) is the unit step function, is given by (C) (A) z (D) z √√ (B) z ECE - 2009 3. A function is given by f(t) = sin2t +cos2t . Which of the following is true? (A) f has frequency components at 0 and / π z (B) f has frequency components at 0 and /π z (C) f s fr qu cy c mp ts t / π /π z (D) f has frequency components at 0, / π /π z ECE - 2011 4. If the unit step response of a network is , then its unit impulse response is (A) (C) (B) (D)

ECE /EE/IN- 2013 5. Two systems with impulse responses (t) and (t) are connected in cascade. Then the overall impulse response of the cascaded system is given by (A) Product of h1(t) and h2(t) (B) Sum of h1(t) and h2(t) (C) Convolution of h1(t) and h2(t) (D) Subtraction of h2(t) from h1(t) ECE - 2014 6. A discrete-time signal x[n] s π t (A) Periodic with period π (B) Periodic with period π (C) Periodic with period π/ (D) t p r c

r s

7.

A system is described by the following differential equation, where u(t) is the input to the system and y(t) is the output of the system. ẏ t y t u t When y (0) = 1 and u(t) is a unit step function, y(t) is (A) (C) (B) (D)

8.

The sequence x[n] = u[n] , where u[n] is the unit step sequence, is convolved with itself to obtain y[n]. Then ∑ y is___________

EE - 2006 1. Which of the following is true: (A) A finite signal is always bounded (B) A bounded signal always possesses finite energy (C) A bounded signal is always zero outside the interval [ t , t ] for some t (D) A bounded signal is always finite

th

th

th

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GATE QUESTION BANK

EE - 2007 2. If u(t), r(t) denote unit the unit step and unit ramp functions respectively and u(t) * r(t) their convolution, then the function u (t+1) * r(t 2) is given by (A) (1/2) (t 1) (t+2) (B) (1/2) (t 1) (t 2) (C) (1/2) (t 1)2 u(t 1) (D) none of the above

Signals and Systems

EE - 2014 5. The function shown in the figure can be represented as

t

EE - 2008 3. Given a sequence x[n], to generate the sequence y[n] = x[3 4n], which one of the following procedures would be correct?. (A) First delay x[n] by 3 samples to generate z [n], then pick every 4th sample of z1[n] to generate z2[n], and then finally time reverse z2[n] to obtain y[n] (B) First advance x[n] by 3 samples to generate Z1[n], and then pick every 4th samples of z1[n] to generate z2 [n] and then finally time reverse Z2[n] to obtain y[n] (C) First pick every fourth sample of x[n] to generate v time-reverse v to obtain v & finally advance v by 3 sample to obtain y[n] (D) First pick every fourth sample of x[n] to generate V1[n], time-reverse V1[n] to obtain V2[n], and finally delay V2[n] by 3 samples to obtain Y[n] EE - 2011 4. A zero mean random signal is uniformly distributed between limits and its mean square value is equal to its variance. Then the r. m. s. value of the signal is (A) (C) √ √ (D) √ (B) √

(A) u t

u t

u t

u t (B) u t (C) u t

u t

u t

u t

u t

u t (D) u t

u t

u t

6.

An input signal x t s πt is sampled with a sampling frequency of 400 Hz and applied to the system whose transfer function is represented by z z ( ) z Where, N represents the number of samples per cycle. The output y(n) of the system under steady state is (A) 0 (C) 2 (B) 1 (D) 5

7.

For the signal f t s πt s πt s πt the minimum sampling frequency (in Hz) satisfying the Nyquist criterion is ________

IN - 2007 1. Consider the periodic signal x t c s πt c s πt where t is in seconds. Its fundamental frequency, in Hz, is (A) 20 (C) 100 (B) 40 (D) 200 th

th

th

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GATE QUESTION BANK

IN - 2008 2. The fundamental period of the discretetime signal x[n] = (A) ⁄ π ⁄ (B) 3.

(

The integral

is (C) 6 (D) 12

The step response of a linear time invariant system is y t u t , where u(t) is the unit step function. If the output of system corresponding to an impulse input t is h(t), then h(t) is (A) u t (B) t (C) u t u t (D) t u t

π ) s

∫ (t

)

IN - 2009 4. The fundamental period of x(t) = 2sin2πt +3sin3πt , with t expressed in seconds, is (A) 1s (C) 2s (B) 0.67s (D) 3s 5.

7.

Signals and Systems

t

t v u t t

(A) 6 (B) 3

(C) 1.5 (D) 0

IN - 2011 8. Consider a system with input x(t) and output y(t) related as follows y(t) =

{

x t }

Which one of the following statements is TRUE? (A) The system is nonlinear (B) The system is time-invariant (C) The system is stable (D) The system has memory 9.

The continuous-time signal x(t) = s t is a periodic signal. However, for its discrete-time counterpart x[n] = s to be periodic, the necessary condition is (A) ≤ π

For input x(t), an ideal impulse sampling system produces the output

(B)

to be an integer

(C)

to be a ratio of integers

y t

(D) none

∑ x

t

Where t is the dirac delta function. The system is (A) Non-linear and time invariant (B) Non-linear and time varying (C) Linear and time invariant (D) Linear and time varying IN - 2010 6. The input x(t) and the corresponding output y(t) of a system are related by . y(t) = ∫ (A) (B) (C) (D)

x

. The system is

EC/IN - 2013 10. If the A-matrix of the state space model of a SISO linear time invariant system is rank deficient, the transfer function of the system must have (A) A pole with a positive real part (B) A pole with a negative real part (C) A pole with a positive imaginary part (D) A pole at the origin 11.

time invariant and causal time invariant and non-causal time variant and non-causal time variant and causal

th

The impulse response of a continuous time system is given by h(t) = t t . The value of the step response at t = 2 is (A) 0 (C) 2 (B) 1 (D) 3

th

th

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GATE QUESTION BANK

IN - 2014 12. Time domain expressions for the voltage t t r v s t s t t = c s t Which one of the following statement is TRUE?

(A) (B) (C) (D)

Signals and Systems

t t t t

s

t y t y t y t y

s s s

Answer Keys and Explanations ECE 1.

s Every after s

[Ans. D]

t

π s π trigonometric function repeats π t rv π π s π π

π

) π S c ‘ ’ s y teger, there is no p ss v u f‘ ’f rw c ‘ ’c an integer, thus non-periodic

t 2.

[Ans. A] f m

tu

√

s

w

7.

s t

fr qu

cy

[Ans. A] ẏ t y t I.F. =

√

y t

w

√

r f

3.

u t

8. c s t

π

c s t

π

π

∑ y

∑ x

Here x

z

∑ x

t ∑ u

∑

[Ans. A] t

6.

∫

[Ans. *] Range 3.9 to 4.1 We know, if y x

Frequency components are f

5.

u t , P = 5, Q = u t ,

y t

π

f t

4.

(

y

[Ans. B]

f

π

t

s

y t

[Ans. C ] Cascade means convolution [Ans. D] Assume x N) x x

EE 1.

∑ y

[Ans. D] If the amplitude of a signal have some finite boundaries for all values of time then it is called as bounded signal.

to be periodic, (with period

th

th

th

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GATE QUESTION BANK

|f t | }f r t r |f t | f tt v v u So a bounded signal may possess finite energy or infinite energy. For example u (t) is bounded signal but it possess infinite energy because it is a power signal. It can be zero or nonzero outside a finite interval ( t t But it is always true that it will be always finite for any value of time t.

Signals and Systems

i.e, z z x Now reverse (time reverse) z give y[n] = z x 4.

will

[Ans. A] Variance

Mean square value 2.

[Ans. C] u t

RMS value

r t ∫r

z t

5.

u t

√

[Ans. A] Result graph

r p r p

r p

∫ r

r p u

z t

u t

∫

For (t

r p

≥ t

z t

(

u t

)

r p

f r t≥ t z t

t

For

t

t

t

t≥

z t

u t

t

For

t≥

z t 3.

t

t

u t

6.

[Ans. C] x t t πft

[Ans. B] Y[n] = x[3 – 4n] = x[ 4n+ 3] So to obtain y[n] we first advance x[n] by 3 unit. i.e, z x Now we will take every fourth sample of z

m p r

s πt

πt f

fx t ms

s mp um th

th

p r

r f s mp s th

ms c cyc

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GATE QUESTION BANK

Signals and Systems

Now, fundamental period of x(t) is the LCM of T1, T2 and T3 , which is x z

So, fundamental frequency = 20 Hz

s mp p r r st y st t my m m [

z

z

2. z

y z

IN 1.

[Ans. D]

z x z

x Fundamental period

z πt πt

3.

]

[Ans. D] Step response s(t)

u t

Impulse response h(t)

z

zs zc s

z + z (Using L Hospital rule) m*

m

z z zs zc s

z z z

m*

7.

s

πt πt z m z

{

t +

[Ans. 14] r qu cy f s πt z r qu cy f s πt z r qu cy f s πt z By Nyquist theorem, sampling frequency f z x mum fr qu cy f z

u t u t 4.

u t / t t t

[Ans. C] x t S πt S P r f S πt s P r

f

S

πt

πt s ⁄

P r of final original is LCM of = LMC / = 2 sec. 5.

[Ans. D] Given signal is y t

[Ans. A] X(t) = (1+0.5 cos πt c s πt c s πt c s πt c s πt c s πt c s πt c s πt x t x t x t w r x c s πt x c s πt x c s πt Now, tm p r fx t π s π

}

∑ x

t

This is the representation of a signal x(t) in weighted and sum form and it obeys the principle of superposition and homogeneity. So, this is linear but it is time varying as for x(t t ), y t ∑x t t And y t t ∑x t t Therefore y t t y t s t m v ry 6.

[Ans. C] x t

y t

∫ x

s Value of y t ‘t’ depends on values of x for times from t t As output is depending on future values of input, the system is Non casual

s

th

th

th

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GATE QUESTION BANK

For the input x t y t

∫ x

y t

∫ x

x t

t

As y(t) depends only on x(t), the system has no memory. 9.

t

[Ans. C] If x(n) is periodic with period N, then the condition to be satisfied is x(n) = x(n+N). If x(n) = s then the necessary condition is s s Or N= m π m 1, 2,

t

From (1), y(t-t

x

∫

Let From (3),

t

Or

y t

x ∫ As (4) & (5) are not same, the system is Time variant. syst m s m v r t causal. 7.

∫

r t

Or

[Ans. D]

11.

[Ans. B] Y(s)

π

(t

t

t

rs

(step response)

u t ) s

f

should be expressible as π ( )

10.

[Ans. B] Given signal is x t

Signals and Systems

t

u t

u t

u t

By shifting property of unit impulse function ∫x t ∫

t (t

t π

x t {

t

) s

t

t

t

t t s w r s

u t

u t

π At t = 2 value is 1

Hence, correct option is (B) 8.

12.

[Ans. C]

[Ans. A] t

Given the system, y(t) = y t

t

x t

{

t

x t }

x t

[

x t x t ] t As y(t) is obtained from linear operation on x(t), the system is linear . As the input x(t) is multiplied by a time varying function , the system is Time – varying. For a bounded input x(t) , x(t) is bounded, y(t) is also bounded. Therefore the system is stable. th

v t t t

t

t

s

c s c s c s

th

s t t t

t

t y

th

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GATE QUESTION BANK

Signals and Systems

Linear Time Invariant (LTI) systems ECE - 2006 1. A system with input x given

as

y

(s

5. and output y π )x

.

Let x(t) be the input and y(t) be the output of a continuous time system. Match the system properties P1, P2 and P3 with system relations R1, R2, R3, R4. Properties Relations P1: Linear but NOT : y(t) = t x (t) time-invariant P2: Time-invariant : y(t) = t |x (t)| but NOT linear P3: Linear and : y(t) = |x (t)| time-invariant :y t x t 5) (A) (P1, R1), (P2, R3), (P3, R4) (B) (P1, R2), (P2, R3), (P3, R4) (C) (P1, R3), (P2, R1), (P3, R2) (D) (P1, R1), (P2, R2), (P3, R3)

is The

system is (A) linear, stable and invertible (B) non-linear, stable and non-invertible (C) linear, stable and non-invertible (D) linear, unstable and invertible ECE - 2008 2. A discrete time linear shift-invariant system has an impulse response h[n] with , and zero otherwise. The system is given an input sequence x[n] with x[0] = x[2] =1, and zero otherwise. The number of nonzero samples in the output sequence y[n] and the value of y[2] are, respectively (A) 5, 2 (C) 6, 1 (B) 6, 2 (D) 5, 3 3.

4.

ECE - 2009 6. Consider a system whose input x and output y are related by the equation :

The impulse response h(t) of a linear time-invariant continuous time system is described by h(t) = exp( t)u(t) + exp( t)u( t), where u(t) denotes the unit step function, and and are real constants. This system is stable if (A) is positive and is positive (B) is negative and is negative (C) is positive and is negative (D) is negative and is positive

y t

∫ x t H(t)

0

(t)

Where h(t) is shown in the graph. Which of the following four properties are possessed by the system? BIBO: Bounded input gives a bounded output Causal: The system is casual LP : The system is low pass LTI : The system is linear and timeinvariant (A) Casual ,LP (B) BIBO ,LTI (C) BIBO, Casual, LTI (D) LP , LTI

The input and output of a continuous time system are respectively denoted by x(t) and y(t). Which of the following description corresponds to a causal system? (A) y(t) = x(t 2) + x (t + 4) (B) y(t) = (t 4) x (t + 1) (C) y(t) = (t + 4) x (t 1) (D) y(t) = (t + 5) x (t + 5)

th

th

th

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GATE QUESTION BANK

Signals and Systems

ECE - 2010 7. Two discrete time systems with impulse responses and are connected in cascade. The overall impulse response of the cascaded system is (A) (B) (C) (D)

ECE - 2013 12. The DFT of a vector c is the vector . Consider the product c c pqrs c [ ]. c c The DFT of the vector p q r s is a scaled version of (A)

ECE - 2011 8. A system is defined by its impulse response u . The system is (A) Stable and causal (B) Causal but not stable (C) Stable but not causal (D) Unstable and noncausal

ECE - 2014 13. The value of the ∫ s c t t is ________.

9.

An input is applied response (A) (B) (C) (D)

x t xp t u t t to an LTI system with impulse t u t The output is xp t u t u t xp t u t u t xp t u t u t xp t u t u t

(B) [√ √ √ √ ] (C) (D)

14.

A continuous, linear time-invariant filter has an impulse response h(t) described f r ≤t≤ by t , t rw s When a constant input of value 5 is applied to this filter, the steady state output is_____.

15.

Consider a discrete-time signal f r ≤ ≤ x , t rw s If y[n] is the convolution of x[n] with itself, the value of y[4] is _________.

16.

The input-output relationship of a causal stable LTI system is given as y y x If the impulse response of this system satisfies the condition ∑ , the relationship between and is (A) / (C) (B) / (D)

17.

Let h(t) denote the impulse response of a

ECE /EE/IN- 2012 10. The input x(t) and output y(t) of a system are related as y(t) = ∫ x c s The system is (A) time-invariant and stable (B) stable and not time-invariant (C) time-invariant and not stable (D) not time-invariant and not stable 11.

Let y[n] denote the convolution of h[n] and g[n], where h[n]= (1/2)n u[n] and g[n] is causal sequence. If y[0]=1 and y[1]=1/2, then g[1] equals (A) 0 (C) 1 (B) 1/2 (D) 3/2

integral

causal system with transfer function

.

Consider the following three statements. S1: The system is stable. th

th

th

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GATE QUESTION BANK

S2:

y [ 0 ] = 3, y[ 2 ] = 2 is the output of a discrete – time LTI system. The system impulse response h [ n ] will be (A) h [ n ] = 0; n < 0, n > 2, h[ 0 ] = 1, h[1]=h[2]= 1 (B) h [ n ] = 0; n < 1, n > 1, h[ 1 ] = 1, h[0]=h[1]=2 (C) h [ n ] = 0; n < 0, n > 3, h[ 0 ] = 1, h [ 1 ] =2, h [ 2 ] = 1 (D) h [ n ] = 0; n < 2, n > 1, h[ 2 ] = h [ 1 ] =h [ 1] = h[0]=3

is independent of t for t > 0.

S3: A non-causal system with the same transfer function is stable. For the above system, (A) only S1 and S2 are true (B) only S2 and S3 are true (C) only S1 and S3 are true (D) S1, S2 and S3 are true 18.

A real-valued signal x(t) limited to the frequency band |f| ≤ is passed through a linear time invariant system whose frequency response is |f| ≤ f

{ |f|

The output of the system is (A) x t (C) x t (B) x t (D) x t 19.

A stable linear time invariant (LTI) system has a transfer function H(s)

. To make this system causal

it needs to be cascaded with another LTI system having a transfer function s . A correct choice for s among the following options is (A) s (C) s (B) s (D) s EE - 2006 1. A continuous – time system is described by y(t) = | | , where y(t) is the output and x(t) is the input. y(t) is bounded (A) only when x(t) is bounded (B) only when x(t) is non – negative (C) only for t ≤ 0 if x(t) is bounded for t≥0 (D) even when x(t) is not bounded 2.

x[n] = 0; n < 1, n > 0, x [ 1] = 1, x[ 0 ] = 2 is the input and y[ n ] = 0; n < 1, n > 2, y [ 1 ] = 1 = y [1],

Signals and Systems

EE - 2007 3. X(z) = 1 z , Y(z)= 1+ z are Z – transforms of two signals x[n], y[n] respectively. A linear time invariant system has the impulse response h[n] defined by these two signals as h[n] = x[n 1] y[n] where denotes discrete time convolution. Then the output of the syst m f r t put -1] (A) Has Z-transform X(z) Y(z) (B) qu s 2] 4] 5] (C) s tr sf rm z z z (D) Does not satisfy any of the above three. EE - 2008 4. A signal sin ( t) is the input to a Linear Time Invariant system. Given K and are constants, the output of the system will be of the form sin (vt ) where (A) t qu t ; but equal to (B) v need not be equal to ut qu to (C) qu t and equal to (D) t qual to and need not be equal to

th

th

th

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GATE QUESTION BANK

5.

6.

The impulse response of a causal linear time-invariant system is given as h(t). Now consider the following two statements Statement (i): Principle of superposition holds Statement (ii) : h(t) = 0 for t < 0 Which one of the following statements is correct? (A) Statement (i) is correct and Statement (ii) is wrong (B) Statement (ii) is correct and Statement (i) is wrong (C) Both Statement (i) and Statement (ii) are wrong (D) Both Statement (i) and Statement (ii) are correct A system with input x(t) and output y(t) is defined by the input-output relation, y (t) = ∫

The system will be

(A) causal, time-invariant and unstable (B) causal, time-invariant and stable (C) non – causal, time-invariant and unstable (D) non - causal, time-variant and unstable EE - 2009 7. A Linear Time Invariant system with an impulse response h(t) produces output y(t) when input x(t) is applied. When the input x (t s pp t system with impulse response h( ), the output will be (A) y(t) (C) y(t (B) y(2(t (D) y(t 8.

The z transform of a signal x[n] is given by z + z + 2 – 6z2 + 2z3. It is applied to a system, with a transfer function H(z) = z 2. Let the output be y[n]. Which of the following is true? (A) y[n] is non causal with finite support (B) y[n] is causal with infinite support

Signals and Systems

(C) y[n] = 0;|n|>3 (D) Re[Y[z] = Re[Y[z] Im[Y[z] = Im[Y[z] π≤ θ π 9.

;

A cascade of 3 Linear Time Invariant systems is causal and unstable. From this, we conclude that (A) Each system in the cascade is individually causal and unstable (B) At least one system is unstable and at least one system is causal (C) At least one system is causal and all systems are unstable (D) The majority are unstable and the majority are causal

EE - 2010 10. Given the finite length input x[n] and the corresponding finite length output y[n] of an LTI system as shown below, the impulse response h[n] of the system is h[n] y[n] = {1, 0, 0, 0, 1}

x[n] = {1, 1}

{

(A)

{

(B)

}

{

(C)

}

{

(D) 11.

}

}

The system represented by the inputoutput relationship y t (A) (B) (C) (D)

t is ∫ x Linear and causal Linear but not causal Causal but not linear Neither linear nor causal

EE - 2011 12. The response h(t) of a linear time v r t syst m t mpu s t under initially relaxed condition is th

th

th

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GATE QUESTION BANK

13.

h(t) = . The response of this system for a unit step input u(t) is (A) u(t) + (B) u t (C) u t (D) t u t Given two continuous time signals x(t) = and y(t) = which exist for t > 0 the convolution z(t) = x(t) * y(t) is (A) u t (B) u t (C) (D)

EE - 2013 14. Which one of the following statement is NOT TRUE for a continuous time causal and stable LTI system? (A) All the poles of the system must lie on the left side of the j axis. (B) Zeros of the system can lie anywhere in the s-plane (C) All the poles must lie within |s| (D) All the root of the characteristic equation must be located on the left side of the j axis.

If the input to the system is cos(3t) and the steady state output is A sin(3t + ), then the value of A is (A) 1/30 (C) 3/4 (B) 1/15 (D) 4/3 17.

Consider an LTI system with impulse response h(t) = u t . If the output of the system is y t u t u t then the input, x(t), is given by (A) u t (B) u t (C) u t (D) u t

18.

A 10 kHz even-symmetric square wave is passed through a bandpass filter with centre frequency at 30 kHz and 3 dB passband of 6 kHz. The filter output is (A) a highly attenuated square wave at 10 kHz. (B) nearly zero. (C) a nearly perfect cosine wave at 30 kHz. (D) a nearly perfect sine wave at 30 kHz.

19.

A continuous-time LTI system with system function H( ) has the following pole-zero plot. For this system, which of the alternatives is TRUE?

EE - 2014 15. x(t) is nonzero only for t , and similarly, y(t) is nonzero only for t . Let z(t) be convolution of x(t) and y(t). Which one of the following statements is TRUE? (A) z(t) can be nonzero over an unbounded interval. (B) z(t) is nonzero for t (C) z(t) is zero outside of t (D) z(t) is nonzero for t . 16.

(A) | (B) | (C) | (D) |

Consider an LTI system with transfer function s

Signals and Systems

| | | |

| | | | s mu t p m x m | | | | c st t

t

s s

th

th

th

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GATE QUESTION BANK

IN - 2007 1. The signals x(t) and h(t) shown in the figures are convolved to yield y(t) .

3.

Consider a discrete-time LTI system with input x [n] = [n] + [n 1] and impulse response h[n] = [n] – [n 1]. The output of the system will be given by (A) [n] – [n 2] (B) [n] – [n 1] (C) [n 1]+ [n 2] (D) [n]+ [n 1]+ [n 2]

x(t) 1

1 1

t

1 h(t)

1 0

4

2

t

Which one of the following figures presents the output y(t)? (A) y(t)

IN - 2009 4. A linear time-invariant casual system has frequency response given in polar form as:

1

t

√

1

3

1

1

For input x(t) = sint, the output is

t

7

5

(A) (B)

(B) y(t)

(C)

1 4

(C)

6

2

1

(D)

t

y(t)

1 1

Signals and Systems

3

1

t

5

√ √ √ √

c st c s (t s t s

(t

)

IN - 2011 5. Consider the signal t≥ x(t) = { t Let X( ) denote the Fourier transform of this signal. The integral is

∫ (D)

)

(A) 0 (B) ⁄

y t) 1 3 1

2

5

t

(C) 1 (D)

IN - 2013 6. The impulse response of a system is h(t) = t u(t). for an input u(t 1), the output is (A)

IN - 2008 2. Which one of the following discrete-time systems is time invariant? (A) y [n] = nx [n] (C) y[n] = x[ n] (B) y [n] = x [3n] (D) y [n] = x[n 3]

u t

(B)

u t

(C)

u t

(D)

th

u t

th

th

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GATE QUESTION BANK

Signals and Systems

IN - 2014 7. The impulse response of an LTI system is given as : {s

π

π It represents an ideal (A) non-causal, low-pass filter (B) causal, low-pass filter (C) non-causal, high-pass filter (D) causal, high-pass filter

Answer Keys and Explanations ECE 1.

[Ans. C] x

→

x[k]

y

[s

( π )] x

The system is linear as y(n) is proportional to the input x(n) Times varying as x(n) is multiplied not by a constant but by a function of n. Stable as bounded input, x(n) gives bounded output , y(n) : |s ( π )| ≤

and |y

| ≤ |x

2 1

0 1

h[k]

|

+1 2

0

Not invertible as several inputs: k (n) , where k is any arbitrary integer, give the same output equal to zero. The system is Linear, stable and non – invertible (It is also Time – varying). 2.

1

1

1

[Ans. D] y

∑ x y y y y y y y

3.

[Ans. D] h(t) = u t u t For the system to be stable , ∫

t t

For the above condition, h(t) should be as shown below.

th

th

th

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GATE QUESTION BANK

Signals and Systems

t

∑ u t

u t

9.

s

v

syst m s

[Ans. D]

t

s

s

s y(s) = X(s).H(s)

Therefore Graph for

s

s s

y t [Ans. C] A system is casual if the output at any time depends only on values of the input at the present time and in the past.

5.

[Ans. A]

6.

[Ans. B] y t

10.

x t

y t

∫ x

c s

check for time invariant ∫x

y t f r

∫x t

c s

t y

utput

∫ x

c s

For delayed input

t

y t

It is not low pass filter. But the system is LTI and BIBO.

8.

u t

[Ans. D]

y t

∫x t

t

c s

P p

[Ans. C]  H(z) = 

s

u t

h(t)

7.

s

s

s s taking inverse w t

4.

t st

z

where t P t t When P Also P t

z =

[Ans. B] u For causal system h(n) = 0 for n < 0. Hence given system is casual. For stability:

y t

∫ x P c s

P

t

p

s c y t y t t It is not time invariant for a bounded input, x (t) = cos (3t) u(t) th

th

th

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GATE QUESTION BANK

Signals and Systems

x y t

∫c s

{x

x

} { c um v ct r }

x

x ∫ su

11.

{x x Designing the sequence x x x x

∫ c s u syst m s

st t stable.

[Ans. A]

g(1) g(0)

∑ 1[- k] 1/2

1

1/4 k

0 1 0 1 c c y

x

x x x x x [ x ] Now circular convolutions given by x x x c c [ ] c c Now circular convolution of c with it self c c c c [ ]0 1 c c c Taking transpose both side c c 0 1 0 1 c [ ] c c c c Let

( ) u

y

} { c um v ct r } x matrix using same

∑

P

S

Take DFT both side

y P

y

0 1 0 1 c c

c

c

∑

y h[1 – k] will be zero for k > 1 and g[k] will be zero for k 2.Which of the following is true? (A) x(t) has finite energy because only finitely many coefficients are nonzero (B) x(t) has zero average value because it is periodic (C) The imaginary part of x(t) is constant (D) The real part of x(t) is even

1 KHz |U

and

/

(A)

(A)

x

zero otherwise). Then if sinc(x) =

x t { Which among the following gives the fundamental Fourier term of x(t)?

3.

)

(where rect (x) = 1) for

EE- 2007 2. A single x(t) is given by t≤

Signals and Systems

|

(B)

EE- 2010 7. x t is a positive rectangular pulse from t t t with unit height as shown in the figure. The value of | { where is the Fourier ∫ |

(C)

(D)

transform of x t } is 𝜔

th

th

th

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GATE QUESTION BANK x(t)

1

1

t

(A) 2 (B) π 8.

(C) 4 (D) π

The second harmonic component of the periodic waveform given in the figure has an amplitude of

EE- 2014 11. For a periodic square wave, which one of the following statements is TRUE? (A) The Fourier series coefficients do not exist. (B) The Fourier series coefficients exist but the reconstruction converges at no point. (C) The Fourier series coefficients exist and the reconstruction converges at most points. (D) The Fourier series coefficients exist and the reconstruction converges at every point. 12.

+1

Let f(t) be a continuous time signal and let F( ) be its Fourier Transform defined by

0 T

T/2

Signals and Systems

∫ f t

t

t

Define g(t) by -1

t (A) 0 (B) 1 9.

(C) 2/π

The period of the signal πt

) is

(A) 0.4πs (B) 0.8πs

(C) 1.25s (D) 2.5s

EE- 2011 10. The Fourier series expansion ∑ f(t) = c s t s t of the periodic signal shown below will contain the following nonzero terms

13.

f(t)

0

(A) (B) (C) (D)

and , and , and and ,

u

u

What is the relationship between f(t) and g(t) ? (A) g(t) would always be proportional to f(t). (B) g(t) would be proportional to f(t) if f(t) is an even function. (C) g(t) would be proportional to f(t) only if f(t) is a sinusoidal function. (D) g(t) would never be proportional to f(t).

(D) √

x(t) = 8 s (

∫

A discrete system is represented by the difference equation [

]

+[

*

]

It has initial conditions . The pole locations of the system for a = 1, are (A) ± (C) ± (B) ± (D) ±

t

,

th

th

th

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GATE QUESTION BANK

14.

A function f(t) is shown in the figure.

2.

If the waveform, shown in the following figure, corresponds to the second derivative of a given function f(t), then the Fourier transform of f(t) is

f t /

f t t

/ /

Signals and Systems

t

1

/ 1

The Fourier transform f f t is (A) real and even function of . (B) real and odd function of . (C) imaginary and odd function of . (D) imaginary and even function of . 15.

A signal is represented by |t| x t { |t| The Fourier transform of the convolved signal y t x t x t⁄ is

4.

The magnitude of fourier transform of a function x(t) is shown below in figure (a). The magnitude of fourier transform of another function y(t) is shown below in figure(b). The phases of and are zero for all . The magnitude frequency units are identical in both the figures. The function y(t) can be expressed in terms of x(t) as | | | |

A differentiable non constant even function x(t) has a derivative y(t), and their respective Fourier Transforms are X( ) and Y( ). Which of the following statements is TRUE? (A) X( ) and Y( ) are both real. (B) X( ) is real and Y( ) is imaginary. (C) X( ) and Y( ) are both imaginary. (D) X( ) is imaginary and Y( ) is real.

IN- 2006 1. The Fourier transform of a function g(t) is s

Then the function

g(t) is given as, (A) t xp( 3|t|) (B) c s t xp( 3t) (C) s t c s t (D) s t xp( 3t)

(D)

The Fourier series for a periodic signal is v s x t c s πt c s πt cos πt fu m t fr qu cy of the signal is (A) 0.2 Hz (C) 1.0 Hz (B) 0.6 Hz (D) 1.4 Hz

s

v

(C)

3.

s

16.

t

2

(A) 1 + sin (B) 1 + cos

s ( )s s ( )

+1

(A)

x( )

(C)

x t

(B)

x t

(D)

x( )

IN- 2007 5. Consider the discrete-time signal x[n] =( )

u

, where u

=,

≥

.

Define the signal y[n] as y[n] = x[ n],

th

th

th

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GATE QUESTION BANK

∑ y (A) (B)

⁄

IN- 2011 9. Consider a periodic signal x(t) as shown below

qu s

(C) ⁄ (D) 3

⁄

Signals and Systems

x(t) 1

6.

Let the signal x(t) have the Fourier tr sf rm s rt s y(t) =

x t

t

where t

0 1

2 3

4 5 6

is an It has a Fourier series representation

arbitrary delay. The magnitude of the Fourier transform of y(t) is given by the expression | | | (A) | | (B) | | (C) .| |. (D) | |.| IN- 2008 7. The fourier transform x(t) = u t when u(t) is unit step function, (A) Exist for any real value of a (B) Does not exist for any real value of a (C) Exists if any real value of a is strictly negative (D) Exists if the real value of a is strictly positive

x t

/

∑

Which one of the following statements is TRUE? (A) for k odd integer and T = 3 (B) , for k even integer and T = 3 (C) for k even integer and T = 6 (D) , for k odd integer and T = 6 IN- 2014 10. x(k)is the Discrete Fourier Transform of a 6-point real sequence 𝑥(𝑛). If X(0)= 9 + j0, X(2)= 2 + j2, X(3)= 3 – j0, X(5)= 1 – j1,x(0) is (A) 3 (C) 15 (B) 9 (D) 18

IN- 2010 8. f(x), shown in the adjoining figure is represented by f x

∑{

The value of

c s x

s

x }

is f x

π

(A) 0 (B) π⁄

π

π

π

π

x

(C) π (D) π

th

th

th

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t

GATE QUESTION BANK

Signals and Systems

Answer Keys and Explanations ECE 1.

6.

[Ans. C] c s

[Ans. A] x t x [ (t

)]

(

c s s c s c

[Ans. B] ( )

s

∫

f

| π

y

t

[Ans. A]

∫

⃡

t

(

t r ct ( )

{ t { t t {r c (

]

t } }

t )

r ct (

t

)}

1

|

½

-1

+1

3

s

) 7.

[Ans. D] The concept of matched filter assumes that the input signal is of the same form g(t) as the transmitted signal(except difference in amplitude ). This requires that the shape of the transmitted signal not change on reflection. h(t) = g( t ⇔ f (f) G f f f sr

± π π [Ans. A ] P & S are correct [Ans. C] S xp s f have sine terms.

)

t r ct ( )

)⃡

t

-3

5.

s c(

c

[

π

∫x t

4.

s

| π

+

c

) ys c

3.

*

Now r ct ( ) ⃡ s c(

∫ (

s

c s

)

Using scaling and shifting property. 2.

s

g(t) =

(fourier transform)

y t h(t)

g(t)

v

fu ct

g(t)

s ’t y(t)= h(t) * g(t) [convolution] th

th

th

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GATE QUESTION BANK

8.

[Ans. B] Given

9.

x(t T) = x( t+T) but signal is periodic with period T. therefore x(t T) = x(t) therefore x(t) = x(T t Now since signal contains only odd harmonics i.e no terms of frequency π

t

S

y t

S

∫ y t t

∫

y i.e. no even harmonics. This means signal contains half wave symmetry this implies that x(t) = x(t / From (i) and (ii) x(t) = x(T t) = x(t T/2)

[Ans. *] Range 9.99 to 10.01 π x s * + /

P r S

fx π

Signals and Systems

/

π

2.

[Ans. A] The given signal x(t) is a periodic waveform with period T0 =2 T and satisfies half – wave symmetry:

So is non-zero for k = 1 and . Fourier co-efficient are periodic with N N= 10 So the value of k for which are

x(t) =

x (t ±

)

x t±

as shown

in Fig.1 Comparing We see m x 10.

±

x(t) 1

[Ans. *] Range 3.36 to 3.39 ( ) u

( )

( ) u ( ) ( )

s

u

( )

0 -1

↔ u u

Fig . 1 Fundamental frequency =

↔

x t

( ) ↔

EE 1.

m is any integer

x (t

)

shown

π/ = π/ in

Fig.

2

possesses Even symmetry and also half wave symmetry. Fundamental Fourier term of

( )

x t

( )

c st ( t)

Where M = peak to peak value of x t

[Ans. C] Since trigonometric fourier series has no sine terms and has only cosine terms therefore this will be an even signal i.e. it will satisfy x(t) =x( t) or we can write th

th

th

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GATE QUESTION BANK x1(t)

4.

[Ans. C] ⁄ x t r ct t x t ≤t≤ = 0 otherwise

1 t

0

Signals and Systems

x t

1

∫x t

t

Fig. 2

∫

⁄ ) As x(t)= x (t Fundamental component of π x t c s [ (t )] π π π c s* t + π 3.

t

(

⁄

⁄

*

x t

≤t≤

∫x

t

t

(

)

(

M

) ⁄

[

⁄

⁄

0

1

f(in kHz)

x

Sampling interval , ms f z f f . Therefore Aliasing or overlap of the adjacent spectra in the sampled spectrum because f f . The sampled spectrum , U* f f ∑ f f as shown in Fig. 2. The resultant spectrum, U* is c st t f r ‘f’ s s w w c is the same figure given in option(b) |

x t

x

+

⁄

⁄

t ⁄ ⁄

[

⁄

s

s c(

⁄

⁄

]

c s ⁄

⁄

5.

⁄

⁄ s

)c s( ) π

[Ans. B] y t x t t x t t Since y(t) is periodic with period T. Therefore x t t x t t will also be periodic with period T.

|

fs M f

0 Fig. 3 |

|

|

2

3

is fourier series coefficient of signal x(t) therefore [ ] c s t s c f r

f |

f

-2

⁄

s

t

]

⁄

*

Fig. 1

⁄

+x

w r x t 0, otherwise

[Ans. B] Highest frequency of the input signal, fh = 1kHz as shown in its spectrum of Fig. 1 | | r| f |

)

-1 0 1 Fig. 2

f(in kHz)

th

th

th

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GATE QUESTION BANK

i.e.

t

w

r t π

t 6. I.

Signals and Systems

∫[ ∫

± ± ± π/

]x t t

π ∫[ ∫

π

[Ans. C] A periodic signal x(t) has always infinite energy

III.

∫x t t

∫x t t

For a0 to be purely imaginary = 2j The first integral should be zero and the second integral should not be zero. i.e. xR(t) should be either 0 or an odd function. option (D) is not true. And xI(t) should be a constant or an even function so that the integral is not zero. option (C) is true. 7.

}

]x t t

t

t

option (A) is not true. The average value a0 of a periodic signal need not be zero. Infact a0 = j2 as given in this question option (B) is not true. The complex value of a0 indicates that x(t) should be complex. Let x(t) = xR(t) +jxI(t) Where xR(t) and xI(t) are the real and imaginary parts of x(t).

II.

{

π ∫x t x t t

π ∫ |x t | ∫ |x t |

π

π ∫

t

π ∫|

|

π

8.

[Ans. A] f(t) is odd  no cosine terms and no dc. Also, by inspection, half wave symmetry no sine even harmonics  no second harmonics amplitude = 0.

9.

[Ans. D] From the given signal x(t) π rad/sec π s c

10.

[Ans. D] Observe the given periodic f(t) shown in Fig. 1 f(t) A

[Ans. D]

t

0

∫|

|

∫ Fig. 1

∫[ ∫

t

(1) Its

t]

average ∫

value

,

f t t

(2) It has even symmetry. i.e; f(t) = f(-t) th

th

th

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GATE QUESTION BANK

s t rms r s t f t (3) The function f1(t) = [f(t)-a0] has half – wave symmetry or odd harmonic symmetry: i.e., f t

f (

)

Signals and Systems

14.

[Ans. C] Given function is odd is imaginary and odd in

15.

[Ans. A]

Even harmonics are absent From observations (2) and (3) above, bn f r v u s f‘ ’ an f r v v u s f‘ ’ an f r

x t

x

s

x t

f1(t)

12.

u

u

f

t

16.

*

S

* *

|S P

s

s s

t

sr

(x t ) t

f ur r tr

sf rm

m

ry

IN 1.

[Ans. A] t Given ,

Use the F.T. pairs: t u t

*

s

| s

+

( )s

[Ans. B] x t v fu ct

y

+ putt s

s

t

+

*

( )

y

[Ans. A] Poles of the system are the roots of the | equation |S r

s

y t

f t ff t s v t is proportional to f(t) if f(t) is even. 13.

( )

s

s

[Ans. B] Since F( ) is FT of f(t) hence ∫

s

y

[Ans. C] Fourier series exists for periodic waves. Since it is a square wave, convergence occurs at most points

/ s

Convolution in time domain = multiplication of Fourier transforms t y t x t x( )

t

Fig. 2

11.

/ /

t x( )

0

s

s

+

If t replaced by ( t), (

+

is replaced by

u t

s

± th

th

th

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GATE QUESTION BANK

u t

u

Signals and Systems

t So, y t

||

||

5. In general, K t 2.

t/

[Ans. C]

||

Given x(n) = ( ) u ||

t

y

[Ans. C] From the given plot, t

x

∑ y t

t

∑ ( )

∑ ( )

Take F.T on both sides Use the following pairs and properties: Let f(t)

rst t rm mm r t

t

f t t t

6.

[Ans. A] {x t t }

t

{

c s c s

4.

7.

[Ans. A] f z; f z f z u m t fr qu cy f x t f0=0.2 Hz (HCF of are the frequency) f f rm c f f harmonic

x t

|

t

|

}

| ||

|

[Ans. C] The fourier transform of u

t s

s >0 f

u t will exist

If a e-5

th

th

th

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GATE QUESTION BANK

z

x

(

=∑ (C) y [ n ] (D) y [ n ]

z

)

y

(A)

(C)

(B)

(D)

2.

(A)

| |

(C)

| |

(B)

| |

(D)

| |

of X(z) z (A) an-1 (B) an 3.

) u

(

+ z

z +1

) u

The region of convergence (ROC) of the ztransform of x[n] (C) s

(B) s |z|

(D) Does not exist

z

)H(z)

is

the

entire

Z-plane

(Except z = 0). It can then be inferred that H (z) can have a minimum of (A) one pole and one zero (B) one pole and two zero (C) two poles and one zero (D) two poles and two zeros

.

(A) s |z|

at z = a for n ≥ 0 will be (C) nan (D) nan-1

H (z) is a transfer function of a real system. (When a signal x[n] = (1 + j)n is the input to such a system, the output is zero. Further, the region of convergence (ROC) of

ECE - 2014 (

| |

with | |> a, the residue

Given X (z) =

( tx

z y (z)=∑ y (z)= z

EE - 2008

ECE/EE/IN - 2012 ⁄ | | ⁄ u 9. If x then the region of convergence (ROC) of its Ztransform in the Z-plane will be

10.

Signals and Systems

|z|

EE - 2014 11.

12.

Let x[𝑛] = 𝑥 𝑛]. Let z be the -transform of [𝑛]. If 0.5+j0.25 is a zero of z which one of the following must also be a zero of z (A) (B) / (C) / (D) The z-transform of the sequence x[n] is given by

z

with the region of

convergence|z|

. . Then, x [2] is_______.

EE - 2007 1. The discrete – time signal x[n]

X (Z) = ∑

4.

Let

causal signal x[n]. Then, the values of x[2] and x[3] are (A) 0 and 0 (C) 1 and 0 (B) 0 and 1 (D) 1 and 1 IN - 2008 1. The region of convergence of the ztransform of the discrete-time signal x[n] = 2nu[n] will be (A) | |>2 (C) | |> (B) | |< 2 (D) | |
0. The final value of f(t) would be (A) 0 (C) (B) 1 (D)

1 2 , but nonzero for < 2 4.

Assuming zero initial conditions, the response y(t) of the above system for the input x t u t s v y (A) u t (B) u t (C) u t (D) u t

Given that F(s) is the one-sided Laplace transform of f(t), the Laplace transform of s ∫ f (A) sF(s) – f(0)

(C) ∫

(B)

(D) [F(s) – f(0)]

s

ECE - 2010 5. A continuous described by y t y t t t

time y t

LTI

system x t t

s

f t

then the initial

and final values of f t are respectively (A) 0, 2 (C) 0, 2/7 (B) 2, 0 (D) 2/7, 0 ECE - 2013 7. A system is described by the differential equation

y t

x t .

Let x(t) be a rectangular pulse given by t x t , t rw s Assuming that y (0) =0 and at t =0, the Laplace transform of y(t) is (A)

(C)

(B)

(D)

EE – 2006 1. The running integrator, given by y t

∫x t

(A) has no finite singularities in its double sided Laplace Transform Y(s) (B) produces a bounded output for every casual bounded input (C) produces bounded output for every anticasual bounded input (D) has no finite zeros in its double sided Laplace Transform Y (s)

is x t

th

th

th

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GATE QUESTION BANK

EE - 2010 Common Data for Questions 2 and 3 Given f t and t as shown below:

f(t) 1

(A) (B)

t

1

g(t) 1 0

3.

EE - 2012 5. Consider the differential equation y t y t y t t t t y wt y t | | t um r c v u

0

2.

Signals and Systems

t

5

3

t can be expressed as (A) t f t (B)

t

f(

(C)

t

f( t

(D)

t

f(

2 1

|

(C) 0 (D) 1

IN - 2006 1. Given, x t x t t xp the function x(t) is (A) xp t u t (B) xp t u t (C) t xp t u t (D) xp t u t

t u t

) IN - 2010 2. u(t) represents the unit step function. The Laplace transform of u(t- is

) )

The Laplace transform of

t is

(A) (B)

(A)

(C)

(B)

(D)

IN - 2013 3. The

(C)

discrete-time

transfer

function

is

(D) EE - 2011 4. Let the Laplace transform of a function f(t) which exists for t > 0 be s and the Laplace transform of its delayed version f t be s . Let (s) be the complex conjugate of s with the Laplace v r s t s s . If G(s) =

f

|

|

, then the inverse Laplace

transform of G(s) is (A) mpu s t (B) y mpu s t– (C) an ideal step function u(t) (D) an ideal delayed step function u(t–

(A) (B) (C) (D)

Non minimum phase and unstable. Minimum phase and unstable. Minimum phase and stable. Non-minimum phase and stable.

IN - 2014 4. The transfer function of a system is given by / s s The input to the system is ( )=sin100 t . In periodic steady state the output of the system s f u t y t s πt ). The phase angle ( ) in degree is ___________.

th

th

th

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GATE QUESTION BANK

Signals and Systems

Answer Keys and Explanations ECE 1.

[Ans. C]

v

s

s} s sF(s) has poles ± , which lie on the imaginary axis. v u t r mc t pp For the given F(s) , f(t) = s t u t ≤f ≤ Hence the final value can be taken as : ≤f ≤ 2.

3.

s s t

r

[Ans. D] poles of S.Y(s) lie on right half of Splane) s = 1 is right s – plane pole Unbounded.

s

s

s

s tx t

s

s

s

u t

s

s

s

s s

s

s

s

s s Taking inverse Laplace transform on both side y t u t 6.

[Ans. B] s

s

f t

s

s

Initial value

[Ans. A] For the given LTI system

mf t

H(s) = mf t

The frequency response is given by

ms s

s

s (

)

s (

s s

s

)

t For the input x(t) = sin (t +1) with | | s (t y t ) t r /s c Y(.) is zero for all sampling frequencies sy t 4.

[Ans. B] 0∫ f

5.

Final value theorem mf t ms s ms

7. 1

[Ans. B] x t u t

s s

s s s

s

u t

x s

With zero initial condition.

s s Now taking Laplace transform

[Ans. B] y t y t x t y t x t t t t Taking Laplace transform on both side (assuming zero initial conditions). s s s s s s s s

s y s y s y s

th

th

sy s

y s

s s

s

s s

s

th

s

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GATE QUESTION BANK

s s

t

s

Signals and Systems

f(

t

)

f(

t

)

Time shifting (delay by 2)

EE 1.

1

[Ans. B] y t

∫x t

0

t

∫

u t

∫

t

3.

t

5

[Ans. C] Form Fig. 3 in the above question, g(t) = u(t 3) u(t 5) Use L.T pair and property:

t

u t

s

u t *

4

Fig. 3

if we take x(t) as casual input ut x t u t y t

3

2

1

t

+

s f

t

s

s

s

s *

+

f r

t s

u

4.

[Ans. B] f t t

2.

[Ans. D] Time scaling (Expansion by 2) and Time shifting by 3 get g(t) from f(t) f(t) , f(t/2) and g(t) = f (

f t

) are shown

1 t

5.

s

s

s

s s |

| s | | s | ,

[Ans. D] +

Fig.1

Time scaling (expansion by 2)

y t

t

Converting to s – domain, s2y(s) – sy(0) – y (0) + 2[sy(s) – y(0)] + y(s) = 1 [s2 + 2s + 1] y (s) + 2s + 4 = 1

f(t/2) 1

0

→ s

| s

s

s G(s) t t ) The inverse LT of G(s) is a unit ideal impulse delayed by .

f(t)

1

t

s

fig. 1,2 and 3

0

→

y(s) = 1

2

find inverse lapalce transform y(t) [– 2 e–t te–t] u(t)

t

Fig. 2

= 2e–t + te–t – e–t | th

th

=2–1=1 th

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GATE QUESTION BANK

IN 1.

Signals and Systems

[Ans. A] Given, x t x t t xp t u t taking Laplace transform on both sides we get, s

s

s

s Taking inverse Laplace transform of X(s) is x t xp t u t 2.

[Ans. C] Laplace transform of u(t) =1/s Use Time shifting property: If the L.T of f(t) is F(s), Then L.T of x(t) = f(t s s f u t-

3.

s

s

[Ans. D] Transfer function is Clearly all the zeros are on the RHS of imaginary axis so the system is nonminimum phase. Also, the poles z = 0.5 is inside the unity circle thus the system is stable.

4.

[Ans. *]Range 67 to 69 P s s t ( ) c s t

t π

th

th

th

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GATE QUESTION BANK

Signals and Systems

Frequency Response of LTI systems and Diversified Topics ECE - 2006 1. A signal m(t) with bandwidth 500 Hz is first multiplied by a signal g(t) where t

∑

A uniformly distributed random variable X with probability density function y 1

2.5

f x

If S(f) is the power spectral density of a real, wide-sense stationary random process, then which of the following is ALWAYS true? (A) S ≥S f (B) S f ≥ (C) S f s f (D) ∫ S f f

t

The resulting signal is then passed through an ideal low pass filter with bandwidth 1 kHz. The output of the low pass filter would be (A) t (C) 0 (B) m t (D) m t t 2.

4.

x

2.5

(u x

u x

)

Where u(.) is the unit step function is passed through a transformation given in the figure below. The probability density function of the transformed random variable Y would be (A) f y

(u y

(B) f y (C) f y

y

ECE - 2008 Statement for Linked Answer Questions 5 &6 The impulse response h(t) of a linear time-invariant continuous time system is given by h(t) = exp( 2t) u(t), where u(t) denotes the unit step function. 5. fr qu cy r sp s f t s system in terms of angular frequency is v y

u y

(B)

(D)

The output of this system to the sinusoidal input x (t) = 2cos (2t) for all time t, is (A) 0 (B) c s t π (C) c s t π (D) c s t π

7.

{x[n]} is a real-valued periodic sequence with a period N. x[n] and X[k] form Npoint Discrete Fourier Transform (DFT) pairs. The DFT Y[k] of the sequence

)

y

(D) f y

(C)

6.

y y

(A)

y

y /

y (u y

u y

)

ECE - 2007 3. If is the autocorrelation function of a real, wide-sense stationary random process, then which of the following is NOT true? (A) |≤ (B) | (C) (D) The mean square value of the process is R(O)

y[n]= ∑

xrx

r is

(A) |X [k]|2 (B)

∑

r x

r

(C)

∑

x r x

r

(D) 0

th

th

th

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GATE QUESTION BANK

8.

The probability density function (PDF) of a random variable X is as shown below.

0

1

0

(B)

(B) (C) (D)

1

x

The corresponding cumulative distribution function (CDF) has the form (A) CDF 1

1

(A)

PDF

1

ECE - 2009 10. The 4 –point discrete Fourier transform (DFT) of a discrete time sequence { 1, 0 ,2, 3} is (A) [ 0 , 2 +2j , 2 , 2 2j] (B) [2, 2 +2j , 6,2, 2j] (C) [ 6 , 1 3j , 2 , 1 + 3j] (D) [ 6 , 1 +3j , 0 , 1 3j] 11.

A system with transfer function H(z) has impulse response h(.) defined as h[2]=1 , h[3] = 1 and h[k] = 0 otherwise. Consider the following statements. S1: H(z) is a low-pass filter S2: H(z) is an FIR filter. Which of the following is correct? (A) Only S2 is true (B) Both S1 and S2 are false (C) Both S1 and S2 are true, and S2 is a reason for S1 (D) Both S1 and S2 are true, but S2 is not a reason for S1

12.

A white noise process X(t) with two-sided power spectral density / z is input to a filter whose magnitude squared response is shown below

1 x

CDF 1

0

1 (C)

1 x CDF

1

0

1 (D)

Signals and Systems

1 x

CDF 1

| X(t)

1

0

10kHz

1 x

f | Y(t)

10kHz

f

The power of the output process Y(t) is given by (A) (B) (C) (D)

1

9.

1

|x| |x| P x xp xp is the probability density function for the real random variable X, over the entire x axis. M and N are both positive real numbers. The Equation relating M and N is th

th

th

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GATE QUESTION BANK

13.

If the power spectral density of stationary random process in a sinc-squared function of frequency, the shape of its autocorrelation is (A)

Signals and Systems

t

t

(B) t

t

(C)

(D)

ECE - 2010 14. The Nyquist sampling rate for the signal s t

is given by

(A) 400 Hz (B) 600 Hz 15.

16.

(C) 1200 Hz (D) 1400 Hz

For an N-point FFT algorithm with , which one of the following statements is TRUE? (A) It is not possible to construct a signal flow graph with both input and output in normal order (B) The number of butterflies in the m stage in N/m (C) In place computation requires storage of only 2N node data (D) Computation of a butterfly requires only one complex multiplication Consider the pulse shape s t as shown. The impulse response t of the filter matched to this pulse is

ECE - 2011 17. The first six points of the 8-point DFT of a real valued sequence are 5, 1 j3, 0, 3 j4, 0 and 3+j4. The last two points of the DFT are respectively (A) 0, 1 j3 (C) 1+j3, 5 (B) 0, 1+j3 (D) 1 j3, 5 ECE - 2013 18. The impulse response of a system is h(t)=tu(t). For an input u(t 1), the output is (A)

u t

(B)

u t

(C)

u t

(D)

u t

EC/EE/IN - 2013 19. For a periodic signal v(t)=30sin100t+10cos300t+6sin (500t π/ , the fundamental frequency in rad/s is (A) 100 (C) 500 (B) 300 (D) 1500

s t

ECE - 2014 20. Consider two real valued signals, x(t) band-limited to [ 500 Hz, 500 Hz] and y(t) band-limited to [ 1 kHz, 1 kHz]. For z(t) = x(t).y(t), the Nyquist sampling frequency (in kHz) is ______.

t

t

t

21.

t

An FIR system is described by the system function z

t

z

z

The system is th

th

th

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GATE QUESTION BANK

(A) maximum phase (B) minimum phase (C) mixed phase (D) zero phase 22.

through the linear constant-coefficient differential equation y t y t y t x t t t Let another signal g(t) be defined as t t ∫ t t If G(s) is the Laplace transform of g(t), then the number of poles of G(s) is________

Consider the periodic square wave in the figure shown. x

27. t

Let

z

z qz z z r z . The quantities p, q, r are real numbers. |r|

q

(A) also has a pole at

all . If (A) (B)

w m

r | (

)|

≤

for

, then b equals (C) / (D) /

25.

A modulated signal is y t m t c s πt where the baseband signal m(t) has frequency components less than 5 kHz only. The minimum required rate (in kHz) at which y(t) should be sampled to recover m(t) is _____

26.

A causal LTI system has zero initial conditions and impulse response h(t). Its input x(t) and output y(t) are related

300

(B) has a constant phase response over the z-plane: arg| z | = constant (C) is stable only, if it is anticausal (D) Has a constant phase response over the unit circle: arg| | = constant

For an all-pass system z

≤

EE - 2006 1. A discrete real all pass system has a pole at z = 2 300 it, therefore

. If the

zero of H(z) lies on the unit circle then r = _____ 24.

∑x

Denote this relation as DFT(x). For , which one of the following sequences satisfies DFT (DFT (x)) = x ? (A) x = [1 2 3 4] (B) x = [1 2 3 2] (C) x = [1 3 2 2] (D) x = [1 2 2 3]

pz

Consider P

The N-point DFT X of a sequence x[n], ≤ ≤ is given by √

The ratio of the power in the 7th harmonic to the power in the 5th harmonic for this waveform is closest in value to _______. 23.

Signals and Systems

EE - 2007 Statement for Linked Answer Questions 2 & 3: 2. A signal is processed by a causal filter with transfer function G(s). For a distortion free output signal waveform, G(s) must (A) Provide zero phase shift for all frequency (B) Provide constant phase shift for all frequency

th

th

th

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GATE QUESTION BANK

(C) Provide linear phase shift that is proportional to frequency (D) Provide a phase shift that is inversely proportional to frequency 3.

4.

G(z) = z + z is a low-pass digital filter with a phase characteristics same as that of the above question if (1/3) (A) (C) / (B) (D) Consider the discrete-time system shown in the figure where the impulse response of G(z) is g[0] = 0, g[1] = g[2] = 1, + +

Σ

G(z)

K

The system is stable for range of values of K (A) [ 1, ½] (C) [ 1/2, 1] (B) [ 1, 1] (D) [ 1/2, 2] EE - 2008 5. A signal x (t) = sinc( t) where constant (s c x

filter with a cut-off frequency of 20 Hz. The resultant system of filters will function as (A) An all-pass filter (B) An all-stop filter (C) A band stop (band-reject) filter (D) A band-pass filter EE - 2014 7. A sinusoid x(t) of unknown frequency is sampled by an impulse train of period 20 ms. The resulting sample train is next applied to an ideal lowpass filter with a cutoff at 25 Hz. The filter output is seen to be a sinusoid of frequency 20 Hz. This means that x(t) has a frequency of (A) 10 Hz (C) 30 Hz (B) 60 Hz (D) 90 Hz IN - 2006 1. A digital filter has the transfer function H(z) =

If this filter has to reject a

50 Hz interference from the sampling frequency signal should be (A) 50 Hz (C) (B) 100 Hz (D)

is a real

) is the input

to a Linear time invariant system whose impulse response h(t) = sinc ( t) where s real constant. If min ( t s the minimum of and and similarly max ( ) denotes the maximum of and and K is a constant, which one of the following statements is true about the output of the system? (A) It will be of the form K sinc( t) where = min ( ) (B) It will be of the form K sinc( t) where = max ( ) (C) It will be of the form K sinc( t) (D) It cannot be a sinc type of signal

Signals and Systems

2.

the input, then for the input 150 Hz 200 Hz

The spectrum of a band limited signal after sampling is shown below. The value of the sampling interval is

f(Hz) 100 0

100 150

350 400

(A) 1 ms (B) 4 ms 3.

600

(C) 2 ms (D) 8ms

A digital measuring instrument employs a sampling rate of 100 samples/second. The sampled input x(n) is averaged using the difference equation y x x x x /

EE - 2011 6. A low-pass filter with a cut-off frequency of 30 Hz is cascaded with a high-pass th

th

th

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GATE QUESTION BANK

For a step input, the maximum time taken for the output to reach the final value after the input transition is (A) 20ms (C) 80ms (B) 40ms (D) IN - 2007 4. Let x(t) be a continuous-time, real-valued signal band-limited to F Hz. The Nyquist sampling rate, in Hz, for y(t) = x (0.5t) + x(t) x(2t) is. (A) F (C) 4F (B) 2F (D) 8F

IN - 2010 8. 4-point DFT of a real discrete-time signal x[n] of length 4 is given by X[k ], n= 0,1 2, 3, and k =0, 1, 2 ,3.It is given that X[0] =5,X[1]=1+ j1, X[2]=0.5. X[3] and x[0] respectively are (A) 1 j, 1.875 (B) 1 j, 1.500 (C) 1+j, 1.875 (D) 0.1 j0.1, 1.500 9.

A digital filter having a transfer function H(z) =

y[n 1].

If y[n] = 0 for n < 0 and x[n] = [n], then y [n] can be expressed in terms of the unit step u[n] as (C) u (A) ( ) u (D) u (B) ( ) u

10.

IN - 2009 6. An analog signal is sampled at 9kHz. The sequence so obtained is filtered by an FIR filter with transfer function H[z] z . One of the analog frequencies for which the magnitude response of the filter is zero is (A) 0.75kHz (C) 1.5kHz (B) 1kHz (D) 2kHz 7.

The transfer function H(z) of a fourth order linear phase FIR system is given by z z z z . Then G(z) is (A) z z (B)

z

(C)

z

(D)

z

z z z

is implemented

using Direct Form – I and Direct Form – II realizations of IIR structure. The number of delay units required in Direct Form – I and Direct Form – II realizations are, respectively (A) 6 and 6 (C) 3 and 3 (B) 6 and 3 (D) 3 and 2

IN - 2008 5. Consider a discrete-time system for which the input x[n] and the output y[n] are related as y[n] = x[n]

Signals and Systems

H(z) is a discrete rational transfer function. To ensure that both H(z) and its inverse are stable its (A) poles must be inside the unit circle and zeros must be outside the unit circle (B) poles and zeros must be inside the unit circle (C) poles and zeros must be outside the unit circle (D) poles must be outside the unit circle and zeros should be inside the unit circle

IN - 2011 11. The continuous time signal x(t) = c s πt s πt is sampled at the rate 100 Hz to get the ∑ signal x t x t , = sampling period The signal x t is passed through an ideal low pass filter with cutoff frequency 100 Hz. The output of the filter is proportional to th

th

th

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GATE QUESTION BANK

(A) (B) (C) (D) 12.

c s c s c s s

πt πt πt πt

s s

Which one of the following statements is TRUE? (A) This is a low pass filter (B) This is a high pass filter (C) This is an IIR filter (D) This is an FIR filter

πt πt

Shown below is the pole-zero plot of a digital filter Im Z plane

Re

6th order pole at the origin

Signals and Systems

IN - 2014 13. A discrete-time signal x[𝑛] is obtained by sampling an analog signal at 10 kHz. The signal x is filtered by a system with impulse response ℎ[𝑛] = 0.5{ [𝑛]+ [𝑛 1]}. The 3dB cutoff frequency of the filter is: (A) 1.25 kHz (C) 4 .00 kHz (B) 2.50 kHz (D) 5.00 kHz

Answer Keys and Explanations ECE 1.

[Ans. B] m(f)

500 500

20kHz

f

= m(t) in time domain.

500

2.

m(t)*g(t)=m(F) G(f) G(f)

20kHz

500

[Ans. B] Sample

of

random

variable

After transformation smaple space of random variable Y=(0, 1) Hence f y y y pt ‘ ’ s t sf s

f

m f

space

f

3.

[Ans. C] Autocorrelation function is an even function

4.

[Ans. B] PSD is always a positive quantity

500 500 20kHz 20kH z pass filtering with f After low z

th

th

th

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GATE QUESTION BANK

5.

[Ans. C]

9.

[Ans. A] P P x

u t H(s) =

h(t) =

Signals and Systems

|x|

xp

xp

|x|

∫P x x 6.

[Ans. D] Input x(t) = 2cos(2t) Frequency response π

| |

r∫

Output

} x

x

r π

r

π

π 10.

x t t π

π

| |

r ∫{

x

t

{

[Ans. D] 4-point DFT of sequence {1,0,2,3} is given as [

}

][ ]

{ }

[

π

]

[

]

π 11. c s t √

√

s

t

[ c s t √ c s t

[Ans. A]

8.

[Ans. A] P f x

x

s

t]

h[n] is high pass filter & FIR filter.

π

c s t 7.

√

12. π

t t

u t u t

[Ans. A] h[2] =1 and h[3]= 1 } h [n] = {

[Ans. B] PSD of white noise / z PSD of output | f | f f | f | utput s p w r

tu t

∫

∫f x x

f f r

Integral of increasing ramp signal is increasing parabola and integral of decreasing ramp signal is decreasing parabola.

(

r|

u

f | curv

)

f th

th

th

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GATE QUESTION BANK

17.

13.

[Ans. B] PS 18.

Signals and Systems

[Ans. B] For 8-points DFT; x x x x x x it is conjugate symmetric about x[4] x x [Ans. C]

= FT

t

tu t s

s

c s

s t

[c⏟ s

s

*

πt

c s ⏟

πt ]

[Ans. A] t

[Ans. D] (A), (B) and (C) are wrong

s

t

c s

t π⁄ ]

are rational

20.

[Ans. C] Impulse response h(t) of the filter matched to the input pulse s(t) existing from t =0 to t= T is given by h(t) =s[-(t-T)] the operation of time reversal and then time delay of T should be performed on s(t) , to get h(t). This is equivalent to the following statement; scan the sketch of s(t) backward from t =T to t =0 and draw the sketch of h(t) forward from t=0 to t =T. s(t) & h(t) are shown below in Fig.

21.

fu

m

t

p r

fu

m

t fr qu

=

cy

p r

[Ans. *] Range 2.99 to 3.01 f z z f f z

z

[Ans. C] z

z z

z z

z z

z

z

z

z

z z

So, zeros are at

T

ut

+

s [

Maximum frequency in s t is 600Hz ( s t ) SO, Nyquist sampling rate 2 + z

16.

s

c s 19.

15.

s

y t

s

s

⁄s

y s [Ans. C]

u t

⁄p

s c fu ct 14.

and



T

th

and

.

When all the zeroes are inside the unit circle minimum phase th

th

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GATE QUESTION BANK



When all the zeros are outside the unit circle Maximum phase When some zeros are inside unit circle some outside unit circle mixed phase



22.

[Ans. *] Range 0.50 to 0.52 Consider one period from t t s c x t t t t

| S | 23.

Pz

(P

z z

z

qz

z

z rz

z z

r

f

z sz

r 25.

[Ans. *] Range 9.5 to 10.5 u t s y t m t c s πt to recover m(t) minimum required rate should be twice the minimum frequency of m(t) f f z

26.

[Ans. *] Range 0.99 to 1.01 y y y t x t t t y y * y t + t t s s s s s t t ∫ t t

)

z s

rz (z

) (z

z*

z )

r z

(z

) (z

s *

+

r Since, this zero lies on unit circle |z| z z S c

v

|r|

27.

r r

s

t

s s

s

+ s s

[Ans. B] x

r

r

s s s s

x t

s s s s So number of poles G(s) = 1

)

One zero is at origin. The other zero lies at z

z w

p

z

z

z

Poles of H(z) is z r p ss syst m w

)

(q

z

z

r

[Ans. *] Range z

z

z

| |

z

[Ans. B]

t

c s π c s π π m rm c | | c s π | | π

P w r f

24.

Signals and Systems

√

[

]

r √

th

[

th

]

th

x

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GATE QUESTION BANK

[

] x

[

z

z

] x

r

z z

z

z z S t syst m w st f t’s ut r most pole will lie inside the unit circle. Location of poles

[Ans. C] For causal system, if all poles are inside the unit circle then system is stable, and converse is true for anti-causal system. 2

z z

r

Now check for each Options EE 1.

Signals and Systems

±√

z √

2

√ √

If anti-causal, ROC is |z| Syst m s st . 2.

3.

4.

[Ans. C] For distortion free output phase shift must be linear function of frequency i.e. proportional to frequency this is because delay to all frequency component will be equal.

sc

t

s s t sf

y pt

(A) 5.

[Ans. A] x(t) = sinc( t h(t) =s t x(t) = s π t

[Ans. A] For distortion free output phase shift must be linear function of as will as all the frequency component must be amplified by same amount so z corresponds to frequency . While z corresponds to frequency 3 . In order to have same amplification of frequency component at ,

s π t s π t 1/

𝜔

π

π

h(t)= s π t 1/

𝜔

[Ans. A] Given g(1) =g(2) =1. Otherwise 0 i.e. g[n] = therefore G(z) = z z Therefore overall transfer function of closed loop system z z z

π

π

y(t) = h(t) * x(t) 1/

if y π

th

th

π

th

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GATE QUESTION BANK 1/

or

[Ans. C] Sinusoid x(t) frequency say f .Hz s mp

Or Y(j𝜔

So

7.

if

π

y(t) = s π t c

t

f

y(t) = s

c

t

f

z utput z u t frqu IN 1.

[Ans. D] The frequency response, f of LPF with cutoff frequency, 30 Hz is shown in Fig. 1. The frequency response f of HPF with cutoff frequency, 20 Hz is shown in Fig. 2 If these two filters are cascaded, the overall frequency response of the resultant system H(f) = H1(f) H2(f), shown in Fig. 3, represents a BPF with Bandwidth, B = 10 Hz H1(f)

0 10 20 30 40

2.

0

f

f

f

f

ms=4ms

[Ans. B] Since output y depends on input, such as no delay, delay by 1 unit, delay by 2 unit, delay by 4 unit, so it will sum all the sample after 4Ts (maximum delay), to get one sample of y[n].

f

t 4.

0

20

30

s t

Where fs is the sampling frequency i.e., X(f) is repeated at ±f ± f ± f tc From the given spectrum of Xs(f) in the question, fs = 250 s/sec S mp t rv

Fig. 2 H(f)

20

f ∑

Ts=

20 30

cy tr

[Ans. B] Let X(f) be the spectrum of the bandlimited signal, x(t). let Xs(f) be the spectrum of the sampled signal xs(t)

3.

20

z

[Ans. A] In order to reject the 50 Hz interference, sampling frequency must be as low as the noise frequency (so that they are separated in time domain and has less interference) So, f z

Fig. 1 H2(f)

30

cy

f

f

y(t) = s

30

fr qu

π

So output is of the form k sinc( t Where m 6.

Signals and Systems

ms c

[Ans. C] x(t) band limited to FHz i.e., fm = FHz y(t) = x(0.5t) + x(t) –x(2t) is x(t) – maximum frequency =

f

z

Fig. 3 th

th

th

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GATE QUESTION BANK

x(0.5t) – maximum frequency =

Signals and Systems

z

z

z z

x(2t) – maximum frequency = 2FHz. Y(t) – maximum frequency

z z

= max (F, Nyquist rate = =2 5.

S

m x frequency z

we know that for linear phase filter

[Ans. A]

for ≤ ≤ where M is the number of points filter here, Now,

Given y(n) = x(n) – y y y

y

x

Take z – transform: z [

z

]

z

H(z) = For the given x(n) = z

z

So, : : : : This is only in option (a).

Use the standard Z.T. pair: z |z| | | u z y

(

8.

[Ans. A] As signal is real, X[3] = 1 – J and

|z|

) u

∑ 6.

[Ans. B] z ( )

z c s

| (

| (

)|

√

7.

c s

s

9.

[Ans. B]

√

c s

Given H(z) =

√

s

This T.F corresponds to a 3rd order digital filter D F – I realization requires 2N delays D F – II realization or canonic realization with minimum number of delay elements requires N delays. Where N is the order of the order of the filter – I requires 6 delays D F – II requires 3 delays

)|

so, | ( )| w r π f

s

s

≤ , if

≤ π π

π f r

z

[Ans. A] z Let, z then,

z

z z

z z

th

th

th

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GATE QUESTION BANK

10.

11.

12.

[Ans. B] The discrete time system with rational transfer function , H(z) is stable if the poles of H(z) lie inside the unit circle in the z – plane. r z ts v rs / z t stable , both the poles and zeros of H(z) must lie inside the unit circle in the z – plane. [Ans. B] Highest frequency component in x(t) is 150 Hz. So, the Nyquist sampling rate is 300 Hz. But, x(t) is sampled at 100 Hz. While cos(100 πt) with frequency 50 Hz will be recovered satisfactorily after passing through the low – pass and sin(300πt) will get aliased resulting in filter output sin (100 πt). Cos (100 πt s ’t c tribute to aliasing.

z

z

)(

[z

⌊z

z

[(z

[ z

(z z

) z

]

z

z

z

⌋

z

z

z

As the impulse response, h(n) = Inverse Z.T of H(z) has only finite duration =7 samples, the given digital filter is an FIR filter. 13.

[Ans. B] Given that ( )

[

] c s

| (

√

)| √

cut ff

z

πr π

]

th

th

√

s

c s( ) |

| π

√ ⁄s mp r

s

c s

c s

√

s

c s

c s

)

) z

z

z

c s z

z ]

[Ans. D] From the given pole – zero plot of the digital filter, the system function (

Signals and Systems

√ π

r z π

⁄s mp

π

th

z

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GATE QUESTION BANK

Control Systems

Basics of Control System ECE - 2008 1. A signal flow graph of a system is given below

ECE - 2010 2. The transfer function Y(s)/R(s) of the system shown is (s)

-

(s) s

1 1

1/s

u s

1/s 1

u

1

1/s

(A) 0

(C)

(B) The set of equations that correspond to this signal flow graph is x x (A) (x ) = [ ] (x ) + x x [

[

[

[

s

=

[

x ] (x ) x

+

=

[

=

n

t

tr ns r un t on

( )

+ 4.

[

x ] (x ) x

(s)

w

( ) x x ]( ) x

(s)

s

(s)

u ] .u / x [x ] x

(D)

(s)

u ] .u / x x ( ) x

(C)

ECE - 2014 3. For the following system,

u ] .u / x (x ) x

(B)

(D)

s

( ) ( )

s

s s(s s ( ) s(s ( )

s s

) )

Consider the following block diagram in the figure. (s)

(s)

+

u ] .u / (A)

(s) s (s) (C)

(B)

(D)

tr ns r un t on

th

th

th

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GATE QUESTION BANK

EE - 2007 1. The system shown in figure below. c0

b0

b1

Σ 1/s

Σ

EE - 2010 3. As shown in the figure, a negative feedback system has an amplifier of gain 100 with 10% tolerance in the forward path, and an attenuator of value 9/100 in the feedback path. The overall system gain is approximately:

C1 Σ

Σ

Σ

Σ1/s

Control Systems

p

a

a

1

0

Can be reduced to the form X

+ Σ +

X

Y

Y

Σ

P

⁄

P

(A) 10 1% (B) 10 2%

(C) 10 5% (D) 10 10%

Z

With (A) X = s

EE - 2014 4. The closed-loop transfer function of a

,

Y= ⁄(s

(B) X = 1, Y =

) ) ⁄(s

s ( s

(C) X =

(D) X

s ⁄(s

) ⁄(s

s

The steady

)

)

s

The signal flow graph of a system is shown below. (s) is the input and (s)is the output.

,

( s

(

state error due to unit step input is________ 5.

s s

(s)

system is

s

) s

,

s

(s)

s

)

EE - 2008 2. A function y(t) satisfies the following differential equation,

()

(s)

s

Assuming n the input-output transfer

+ y(t) = (t),

( )

(s)

where (t) is the delta function. Assuming zero initial condition and denoting the unit step function by u(t), y(t) can be of the form (A) (C) u(t) (B) (D) u(t)

( )

( )

(s)

( )

(s)

( )

(s)

( )

th

(s)

th

, function

of the system is given by s s

s s

s

s s

s

s s

s

s

th

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GATE QUESTION BANK

6.

The block diagram of a system is shown in the figure (s)

(s)

s

(s)

IN - 2009 3. A filter is represented by the signal flow graph shown in the figure. Its input x(t) and output is y(t). The transfer function of the filter is

If the desired transfer function of the system is ( )

Control Systems

1

1

X(s)

k

then G(s) is

( )

(A) 1 (B) s

Y(s)

(C) 1/s (D)

1/s

1/s

1

(A)

1 s

(B)

s 1 s(s2  2)

s(s2  1) s2  2

(A) (B)

(D)

(C)

K

(

(C)

)

(D)

(

) )

1

1

1

C

The transfer function (C/R) of the system is

IN - 2007 2. A feedback control system with high K, is shown in the figure below: R(s) +

)

Y(s)

1

(C) 2

1

(

1

IN - 2011 4. The signal flow graph of a system is given below. R

1/s

(

(B)

s

1

1

(A)

IN - 2006 1. The signal flow graph representation of a control system is shown below. The Y(s) transfer function is computed as R(s)

R(s)

1/S

k

G(s)

C(s)

(D)

)

(

) (

)

(

) (

)

(

) (

)

(

)

IN - 2012 5. The transfer function of a Zero-orderHold system with sampling interval T is

H(s)

Then the closed loop transfer function is. (A) sensitive to perturbations in G(s) and H(s) (B) sensitive to perturbations in G(s) and but not to perturbations H(s) (C) sensitive to perturbations in H(s) and but not to perturbations G(s) (D) insensitive to perturbations in G(s) and H(s)

(

(A)

(

)

(C)

(B)

(

)

(D)

IN - 2013 6. The complex function tanh(s) is analytic over a region of the imaginary axis of the complex s- plane if the following is TRUE everywhere in the region for all integers n (A) Re(s) =0 (C) Im(s) ≠ (

(B) Im (s) ≠ n

th

th

(D) Im(s)≠

th

)

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GATE QUESTION BANK

7.

The signal flow graph for a system is given below. The transfer function

( ) ( )

for

this system is given as

Control Systems

IN - 2014 8. A plant has an open-loop transfer function, (s)

(s)

s

s

(A)

(C)

(B)

(D)

(s )(s )(s ) The approximate model obtained by retaining only one of the above poles, which is closest to the frequency response of the original transfer function at low frequency is

(s)

(A)

(C)

(B)

(D)

Answer Keys and Explanations ECE 1. [Ans. D]

u

2.

[Ans. B] (s)

(s) x

(s)

s

x s x

u

(s)

x x

(s)

(s) [

s (s) (s)

o (s) x x x u x x x u x x x x x u x x [ ] [ ][ ] [ ] 0u 1 x x One can denote any state by any name So, that answer is x x u [x ] = [ ] (x ) + [ ] .u / x x

(s)

s

r or 3.

]

s

(s) (s) s

(s) (s)

s

[Ans. D] When x (s) (s)

(s) s s s

y(s) x (s)

th

th

s s(s

th

)

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GATE QUESTION BANK

4.

[Ans. C]

Control Systems

These are four individual loops

(s)

(s)

s

(s)

(s)

s

s

s

s

s s s

s s All the loops touch forward paths (

(s)

(s)

(s)

(s)

s s s s s ng M son’s g n ormul (s) (s) s

(s) (s) (s) (s)

(s) (s) EE 1.

)

s

(

)s ( s) (s s) (s

( (

) s

) s

)

()

(s) Σ

[Ans. D] Σ

(s)

s

Σ

Σ

(s) xy (s) yz Comparing eg. (I) and (ii), we get s xy s s s yz s s Hence option (D) is correct

s s

Signal flow graph of the block – diagram

s ( )

2.

s

[Ans. D] Take LT on both sides,

( )

s.Y(s)+Y(s) =1  Y(s) = (  y(t) =

There are forward paths 3.

)

u(t)

[Ans. A] T.F =

th

th

=10

th

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GATE QUESTION BANK

6.

Also,

Control Systems

[Ans. B]

 overall change in T.F for G = 10 is

(s)

(s)

s

= 1%

(s)

1% 4.

[Ans. 0] (s)

s

(s)

s ( or un t st p)

s

(s)

s(s

y(t)

y st t

s(s

s

1

)

( ) ( ) ( ) ( ) So put G(s) = s ( ) ( )

( )

IN 1.

Loops: (

s

(s) (s) n

M

( ) ( ( )

)

( )

s

s

M

s

s

s

) s

∑

num

( )1

[Ans. A]

)

(

s

( )

( )

0

s

( ) s

1 C(s)

s

1

rror

s

G(s)

1/s

s

[Ans. C] Using Mason’s gain formula Paths:

s

1

R(s)

)

s

l m sy(s)

t 5.

Convert the given block diagram to signal low gr p n us M son’s rul 1

There individual loops with gains: ro p t s

( )( )( s s

) ( )( )( s s

0

1

∑ (

)

s

(s) (s)

s

M

)

s

(

M

)( )( s

.

/

.

/

s (s) (s)

( .

/

s

s

)

2. (

[Ans. C] In this case, open loop gain sensitivity is measured with respect to KG. Because here open loop gain = KG

)

s s s

(

s th

th

(

)

)

(

th

)(

)

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)

s

GATE QUESTION BANK

)( ) (

(

s ount)

6.

Control Systems

[Ans. D] t n (s)

(

)(

(

)

( or l rg s

(

s

)

)

For analytical region there should be no poles ≠ os sn os sn ≠ os ≠ ( n ) ≠

)

(

(

)

s n (s) os (s)

)

So close loop transfer function is sensitive to perturbation in H(s) but not to perturbation in G(s) for large k values.

7.

[Ans. A] u(s)

3.

s

y(s)

forward path =

. /; (

( )

=

( )

4.

s

[Ans. A]

(

)

( (

)

|

) )

Loop =

[Ans. C]

(s) =1 (

)

(

) 8. (

( 5.

[Ans. A] Writing the given open loop transfer in

) )

form of

[Ans. A] The transfer function of a Zero-orderHold system having a sampling interval T is

(

g t

.

/.

/.

/

So, at low frequencies, the approximate model equivalent to original transfer function will

)

(

th

)

th

th

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GATE QUESTION BANK

Control Systems

Time Domain Analysis ECE - 2006 1. In the system shown below, x(t) = (sin t) u(t). In steady-state, the response y(t) will be

x(t) (A) (B) 2.

3.

s √ √

y(t)

sin .t

/

(C)

sin .t

/

(D) sin t –cos t

√

ECE - 2008 6. Step responses of a set of three secondorders underdamped systems all have the same percentage overshoot. Which of the following diagrams represents the poles of three systems? (A) j

sin t

The unit impulse response of a system is (t) t For this system, the steady-state value of the output for unit step input is equal to (A) 1 (C) 1 (B) 0 (D)

(B) j

The unit-step response of a system starting from rest is given by (t) – or t . The transfer function of the system is (A)

(C)

(B)

(D)

(C) j

ECE - 2007 4. The frequency response of a linear, timeinvariant system is given by H(f)=

5.

(D)

. The step response of the system is (A) 5 (1 )u(t) (C) (1

)u(t)

(B) 5 (1

)u(t)

)u(t)

(D)

(1

j

The transfer function of a plant is (s)

(

)(

)

.

The

second-order

approximation of T(s) using dominant pole concept is (A) (B)

( (

)( )(

) )

(C) (D)

( (

7.

A linear, time-invariant, causal continuous time system has a rational transfer function with simple poles at s and s , and one simple zero at s . A unit step u(t) is applied at the input of the system. At steady state,

) )

th

th

th

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GATE QUESTION BANK

the output has constant value of 1. The impulse response of this system is (A) , xp ( t) xp ( t)- u(t) (B) , xp ( t) xp ( t) xp ( t)- u(t) (C) , xp ( t) xp ( t)- u(t) (D) , xp ( t) xp ( t)- u(t) 8.

Group I lists a set of four transfer functions. Group II gives a list of possible step responses y(t). Match the step responses with the corresponding transfer functions Group I

Control Systems

4. y(t) 1

t

(A) (B) (C) (D) 9.

P-3, Q-1, R-4, S-2 P-3, Q-2, R-4, S-1 P-2, Q-1, R-4, S-3 P-3, Q-4, R-1, S-2

The magnitude of frequency response of an underdamped second order system is 5 at 0 rad/sec and peaks to

s s

s

s

s

s Group II 1.

√

at

√ rad/sec. The transfer function of the system is,

s

(A)

(C)

(B)

(D)

ECE - 2009 10. The unit step response of an underdamped second order system has steady state value of 2. Which one of the following transfer function has these properties?

y(t)

t

2. y(t)

(A)

(C)

(B)

(D)

ECE - 2010 11. A system with the transfer function

1

( )

has an output

( )

t

3. y(t)

1 t

y(t)

os . t

x(t)

p os . t

/ for the input signal /.Then, the system

p r m t r ‘p’ s (A) √

(C) 1

(B)

(D)

√

√

ECE - 2011 12. The differential equation y

x(t)

describes

a

system with an input x(t) and an output th

th

th

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GATE QUESTION BANK

y(t). The system, which is initially relaxed, is excited by a unit step input. The output y(t) can be represented by the waveform

Control Systems

r

(s)

y

(s)

Which one of the following compensators C(s) achieves this? (C) (s ) (A) . /

(A) y(t)

(B) t

(D)

/

.

/

15.

The natural frequency of an undamped second-order system is 40 rad/s. If the system is damped with a damping ratio 0.3, the damped natural frequency in rad/s is ________.

16.

The input u(t) where u(t) is the unit step function, is applied to a system

(B) y(t)

t

.

(C)

with transfer function

y(t)

. If the initial

v lu o t output s t n t v lu of the output at steady state is _______. t

17.

(D) y(t)

The steady state error of the system shown in the figure for a unit step input is _________ (s) r(t)

(s)

(s) (t)

s

(t)

t s

ECE - 2012 13. A system with transfer function ) (s )(s (s) (s )(s )(s ) is excited y s n( t). The steady-state output of the system is zero at (A) r s (C) r s (B) r s (D) r s ECE - 2014 14. For the (s)

(

following )(

)

feedback

18.

(s)

(s) s(s

(A) 16 (B) 4

system

. The 2%-settling time of

the step response is required to be less than 2 seconds.

For the second order closed-loop system shown in the figure, the natural frequency (in rad/s) is )

(C) 2 (D) 1

EE - 2007 1. Consider the feedback control system shown below which is subjected to a unit step input. The system is stable and has th

th

th

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GATE QUESTION BANK

the following parameters kp = 4, ki = 10, n ξ 1

0

The transfer function of a system is given as

z

s +

5.

Σ

KP

s

ξ s

+

The steady state value of z is (A) 1 (C) 0.1 (B) 0.25 (D) 0

(

(A) (B) (C) (D)

+ Σ

Control Systems

)

. The system is ___________

An over damped system An underdamped system A critically damped system An unstable system

EE - 2009 6. The unit-step response feedback system with

of a unity open loop

transfer function G(s) =

Statement for Linked Answer Question Q.2 & Q.3 R-L-C circuit shown in figure,

(

)(

)

is

shown in the figure. The value of K is 1

m

0.75 0.5 0.25 0 0

2.

3.

For a step- input overshoot in the output will be (A) 0,Since the system is not under damped (B) 5% (C) 16% (D) 48% If the above step response is to be observed on the non-storage CRO, then it would be best have the as a (A) Step function (B) Square wave of frequency 50 Hz (C) Square wave of frequency 300 Hz (D) Square wave of frequency 2.0KHz

EE - 2008 4. The transfer function of a linear time invariant system is given as G(s) =

1

2 Time

(A) 0.5 (B) 2

3

4

(C) 4 (D) 6

EE - 2010 7.

For the system

(

)

, the approximate

time taken for a step response to reach 98% of its final value is (A) 1s (C) 4s (B) 2s (D) 8s EE - 2011 8. The steady state error of a unity feedback linear system for a unit step input is 0.1. The steady state error of the same system, for a pulse input r(t) having a magnitude of 10 and a duration of one second, as shown in the figure is r(t) 10

. The steady state value of

the output of the system for a unit impulse input applied at time instant t=1 will be (A) 0 (C) 1 (B) 0.5 (D) 2

t

1s

(A) 0 (B) 0.1 th

th

(C) 1 (D) 10 th

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GATE QUESTION BANK

IN - 2007 Common Data for Questions 1 & 2 The following figure represents a proportional control scheme of a first order system with transportation lag. R(s)

(s

1.

2.

at which the response to a unit step input reaches its peak is seconds 4.

The damping coefficient for the closed loop system is (A) 0.4 (C) 0.8 (B) 0.6 (D) 1

5.

The value of p is (A) 6 (B) 12

C(s)

+

Control Systems

)

The angular frequency in (s )( sradians/s )( at ) (s s which the loop phase lag becomes is (A) 0.408 (C) 1.56 (B) 0.818 (D) 2.03

)

IN - 2009 6.

A plant with a transfer function

(

)

is

controlled by a PI controller with =1 and n un ty configuration. The lowest value of that ensures zero steady state error for a step change in the reference input is (A) 0 (C) 1/2 (B) 1/3 (D) 1

The steady state error for a unit step input when the gain = 1 is (C) 1 (A) (D) 2 (B)

IN - 2008 3. If a first order system and its time response to a unit step input are as shown below, the gain K is

Statement for the Linked data Q.No.7 & Q.No.8: A disturbance input (t)is injected into the unity feedback control loop shown in the figure. Take the reference input r(t) to be a unit step.

y(t)

r(t)

(C) 14 (D) 16

(s)

y

r(t)

+

d(t)

+ 1

+

1 S(S+1)

+

y(t)

0.8

7.

If the disturbance is measurable, its effect on the output can be minimized significantly (s) To using a feedforward controller eliminate the component of the output due to (t) s n t ( ) Should be (C) √ (A) √ (D) √ (B)

t

(A) 0.25 (B) 0.8

(C) 1 (D) 4

Statement for Linked Answer Questions 4 &5 A unity feedback system has open loop transfer function G(s) =

(

)

√

Let (s) ontroll r I (t) s n t the amplitude of the frequency component of y(t) due to d(t) is

8.

.The time

th

th

th

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GATE QUESTION BANK

(A) √

(C) √

(B) √

(D) √

IN - 2011 11. The unit-step response of a negative unity feedback system with the open-loop transfer function G(s) = (A) (B)

IN - 2010 9. A unity feedback system has an open loop transfer function (s)

(

)

(s) (

(A) 0 (B) 0.5

(C) (D)

ECE/EE/IN - 2013 12. Assuming zero initial condition, the response y(t) of the system given below to a unit step input u(t) is

(A) u(t) (B) t u(t)

(s)

)

Y(s)

U(s)

A unit ramp input is applied to the system shown in the adjoining figure. The steady state error in its output is r(s)

is

.The value

of k that yields a damping ratio of 0.5 for the closed loop system is (A) 1 (C) 5 (B) 3 (D) 9 10.

Control Systems

(C)

u(t)

(D)

u(t)

(C) 1 (D) 2

Answer Keys and Explanations ECE 1.

(s [Ans. A] y(t) x(t) (s) (s) (

)

x(t)

utput

s s ( )u(t) When t = at steady state output = 1

√

√ s n(t)u(t)

y(t) 2.

s s

(t) (s)

s

√

s n .t

3.

/

[Ans. B] ,st p r spons s

[Ans. C] (t) (s) (s)

*

+

s s

4.

[Ans. B]

s

() (s) (s)

output s(s

)

)

s

(s

)s

( )

s th

th

s

.s th

/

s

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GATE QUESTION BANK

t p r spons

s

.s

w

ns

s .s s

/

s s (t) ( )u(t) Which is also the required impulse response of the system.

/ (s

s

)

s

8.

When s =

[Ans. D] Comparing the given transfer functions with

⁄ (s)

5.

s

y(t)

s

,

-u(t) s

)(s

s

ξ

s

ln

ξ s Therefore P is undamped

[Ans. D] In dominant pole concept the factor that has to be eliminated should be in time constant form (s

Control Systems

s

s

ξ

)

s ov r / (s

. s

s

)

mp

s

s

ξ

s

s rt 6.

[Ans. C] ov rs oot

p n s on ξ s

M

r ξ os Where is the angle made by pole from negative real axis. To make M same, should be the same. 7.

s

√

[Ans. C] Transfer functions (s ) (s) (s )(s )

r

mp

9.

[Ans. A]

10.

[Ans. B] Steady state value = -2 Denominator ξ ξ underdamped

11.

[Ans. B] (s)

t

or

(s)

s

s un

s (s) (s) Output, (s) v n l m s (s) s (s ) (s )(s )

mp

ξ

Input (s)

or l m

lly

 1=

√

√



P=

√

(s ) (s )(s ) th

th

th

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GATE QUESTION BANK

12.

(

[Ans. A] y y y x(t) t t Taking Laplace Transform of both sides (s) s (s) s (s) (s) (s) ( s ) (s) s s ol s r

ts

[Ans. C] (s) x(s)

15. (s

(s

)(s ) )(s )(s )

ξ Where damped natural frequency undamped natural frequency ξ damping ratio

y(s)

14.

√

16.

[Ans. C] ξ

(s s s pt on o

(s)

(s) y( ) 17.

(s)

u(t)

(s)

ξ So all poles of CLTP should lie on the left of s = line The characteristic equation is (s) (s) . We have to check whether are the roots this equation lies on me left of s = or not

o

[Ans. *] Range - 0.01 to 0.01 s (s) s x(t)

s ttl ng t m

pt on

[Ans. *] Range 38.13 to 38.19 We know √

s

(s )(s ) . / (s )(s )(s ) s For applying final value theorem system must be stable, mean all poles should lie in left half plane ) (s So (s ) r s

s)

(s )(s ) (s+1)(s+2)+0.15+5s=0 s s s s (s) pt on (s ) (s) (s) o (s ) (s )(s ) (s )(s ) (s ) s s s Hence Option C is correct

As poles are on the right hand side of splane so given system is unstable system. Only option (A) represents unstable system. 13.

Control Systems

s

(s)

s l m s y(s)

lm

s s

[Ans. *] Range 0.49 to 0.51 (s)

(s)

(s) s

( )

s (s) (s) )(s )(s s s (s)

s

)

(s) (s) (s) (s) (s) (s) (s) (s) (s) (s) * + (s) (s)

(s) (s)

(

s

)

(s) (s)

(s) th

th

(s) (s) th

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GATE QUESTION BANK

Control Systems

(s) (s) r

(s)

s s

s s

2.

[Ans. C] m

s

l m s (s)

l ms

s s s

s s +

18.

i

[Ans. C] r ns r un t on or

or

s

ξ

s

s

r un t

∫ t t Taking Laplace transform

syst m

(s)

(

s

[Ans. A] Step input

s ∫ t

(s)

s

s

(s)

(s)

(s)

s

(s) (s)

s

t (s)

ξ s

s

(s) (s) (s)

(

s

)( s

ξ s

)

s

s

+

/

√

√

√

√ √

Overshoot /.

s

Comparing with s s

s .

(s)

*

s

(s) s Steady state value of Z t s (s) t

s

(s) s

.s

(s)

(s)

I(s)

Characteristic eq.

n (s) (s) (s) (s) (s) (s) (s) (s) (s) (s) (s) (s) (s) [ ] (s) (s) (s) (s) * + (s) (s) (s)

) I(s)

(s)

I(s) EE 1.

s

⁄ √

/

= 0.163 or 16.3% th

th

th

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GATE QUESTION BANK

3.

[Ans. C]

5.

Control Systems

[Ans. C] M(s)

s s Comparing with standard form

√ √ r

M(s)

s

Setting time (t )

ξ

Square Wave

The system is critically damped 6.

[Ans. D] t y st t v lu o r spons Input is unit step So steady state error

T

For the square Wave ⁄ should be

ms For

r

(s)

(s)

n

w t

t

z ms

(s

t

t

Therefore, it would be best to have the as a square wave of 300Hz.

(s)

(s) (s)

ppl

)

(s)

s (s) (s) (s) s

t (

[Ans. A] r(t) un t mplus (t ) R(s) = 1[r(t)] = (s) (s) (s) s

)(s

s Steady state error using final value theorem t s (s)

z ms

(s) (s) (s)

(s)

rror

greater than t For z

4.

s

ξ

ms

For

s

ξ

)(

)

tt

s 7.

[Ans. C]

s s Steady state value of output, using final value theorem t s (s) s t s s

[

] ( s s rst or s .c Gain=2, T=1 sec For 98% t s

th

th

th

r syst m)

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GATE QUESTION BANK

8.

[Ans. A]

R(s)

Control Systems

Given

C(s) +

G(s)

+

, 4.

[Ans. B] (s)

(

)

n t s (s) | (s) x(t)

,u(t)

x(s)

[

s (

s

 u(t s

)-

]

s

,

) | (s)

s

√

⁄

5.

[Ans. B]

6.

[Ans. A]

-

( ) ( )

IN 1.

t n ( ) t n ( ) r s -

d(t)= (t)

(s)

n

s or r(t) u(t) os t on l rror onst nt t (s)

( √ 8.

(s

s

] ) ( )

s(s (

)

(

)

(

)

+ ) )

(s)

, (s)

(s)

y(s)-

(

)

ontroll r

So s s s (s) * y(s) * + + ) s(s ) s(s y(s) s ( ) (s) s s t

)

t (s) [Ans. D] lm

(s)

(s) wor s s s s

)

s(s

) (

[Ans. B]

s (s)

)

⁄

y(s)

OR (s) (s)

( )

(t)

s nt *

(s)

3.

(

(t) [

[Ans. B] or

) (

( )

s s [Ans. D] Output due to

7.

( )

(

/ )

(

s m

( ) Phase lag It satisfy ,

(s)

/.

m

[Ans. D] (s)

2.

.

y(s) | | x(s)

s

th

th

√

th

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GATE QUESTION BANK

9.

[Ans. D] T(s) =

(

)

= √

and

=

or 10.

Control Systems

√

or

K = 9.

[Ans. B] (s)

s(s ) For unit Ramp input, r(t)

t u(t)

lo ty rror onst nt (s) t

11.

[Ans. D] (s) (s) s The unit step response is given by (s) (t)

12.

)

s(s (

s

s

) u(t)

[Ans. B]

H(s) u(s)

⁄

Y(s)

u(s)= ⁄s [unit step i/p] ⁄s = ⁄ y(s) = H(s).U(s)= ⁄s s y(t) 0 ⁄s 1 t u(t)

th

th

th

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GATE QUESTION BANK

Control Systems

Stability & Routh Hurwitz Criterion ECE - 2006 1. The positive values of and so that the system shown in the figure below oscillates at a frequency of r s respectively are R(s) (s

(s s

) s

C(s) )

4.

The number of open right half plane poles of G (s) =

is

(A) 0 (B) 1

(C) 2 (D) 3

ECE - 2012 5. The Feedback system shown below oscillates at 2 rad/s when (s)

(s s

(s

(A) 1, 0.75 (B) 2, 0.75

(C) 1, 1 (D) 2, 2

ECE - 2007 2. If the closed-loop transfer function of a control system is given as, (s) (A) (B) (C) (D)

(

)(

)

, then it is

an unstable system an uncontrollable system a minimum phase system a non-minimum phase system

, where

is a parameter.

Consider the standard negative unity feedback configuration as shown below +

)

ECE - 2014 6. The forward path transfer function of a unity negative feedback system is given by (s)

(s )(s ) The value of K which will place both the poles of the closed-loop system at the same location, is _______. 7.

Consider a transfer function (s)

(

)

(

)

with p a positive

real parameter. The maximum value of p until which remains stable is __________

G(s)

Which of the following statements is true? (A) The closed loop system in never st l or ny v lu o (B) or som pos t v v lu s o t closed loop system is stable, but not for all positive values (C) For all positive v lu s o t los loop system is stable (D) The closed loop system is stable for ll v lu s o ot pos t v n negative

(s)

(A) K=2 and a=0.75 (B) K=3 and a=0.75 (C) K=4 and a=0. 5 (D) K=2 and a=0.5

ECE - 2008 3. A certain system has transfer function G(s) =

) s

EE - 2007 1. The system shown in the figure is U1 +

s s

Σ

s

(A) (B) (C) (D)

th

Σ

U2 +

stable unstable conditionally stable stable for input u1, but unstable for input u2 th

th

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GATE QUESTION BANK

2.

If the loop gain K of a negative feedback system having a loop transfer function ( (

) )

(C) unstable and of the minimum phase type (D) unstable and of the non-minimum phase type

to be adjusted to induce a sustained

oscillation then (A) The frequency of this oscillation must be ⁄ rad/s. √ (B) The frequency of this oscillation must be 4rad/s (C) The frequency of this oscillation

6.

The frequency response of the linear system ( ) is provided in the tabular form below. ( ) ( ) 1.3 1.2 1.0 0.8 0.5 0.3 The gain margin and phase margin of the system are (A) n (B) n (C) n (D) n

must be 4 or ⁄ rad/s √ (D) Such a K does not exist EE - 2008 3. Figure shows a feedback system where K > 0. The range of K for which the system is stable will be given by +

Σ

(A) 0 < K < 30 (B) 0 < K 390

EE - 2009 4. The first two rows of Routh's tabulation of a third order equation are as follows. s3 2 2 2 s 4 4. This means there are (A) two roots at s= ± j and one root in right half s-plane (B) two roots at s = ± j2 and one root in left half s-plane (C) two roots at s= ± j2 and one root in right half s-plane (D) two roots at s = ± j and one root in left half s-plane

Control Systems

EE - 2014 7. In the formation of Routh-Hurwitz array for a polynomial, all the elements of a row have zero values. This premature termination of the array indicates the presence of (A) only one root at the origin (B) imaginary roots (C) only positive real roots (D) only negative real roots 8.

For the given system, it is desired that the system be stable. The minimum value of for this condition is __________ (s)

(s s

(

)s

(s)

) (

)s

(

)

EE - 2011 5. An open loop system represented by the transfer function G(s) = (

(

) )(

)

is

9.

A system wth the open loop transfer function

(A) stable and of the minimum phase type (B) stable and of the non-minimum phase type

(s)

th

s(s

th

)(s

s

th

)

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GATE QUESTION BANK

is connected in a negative feedback configuration with a feedback gain of unity. For the closed loop system to be marginally stable, the value of K is_________ 10.

A single-input single-output feedback system has forward transfer function G(s) and feedback transfer function H(s). It is given that |G(s)H(s)| < 1. Which of the following is true about the stability of the system? (A) The system is always stable (B) The system is stable if all zeros of G(s)H(s) are in left half of the s-plane (C) The system is stable if all poles of G(s)H(s) are in left half of the s-plane (D) It is not possible to say whether or not the system is stable from the information given

IN - 2006 1. The range of the controller gains (Kp, Ki) that makes the closed loop control system (shown in the following figure) stable is given as (s)

(s) s

(s

Ki  20 12 K (B) Ki < 0 and Kp  i  20 12

(A) Ki < 0 and Kp
4 (B) KC> 0

)

( )

(C) KC> 2 (D) KC< 2

IN - 2010 3. The open loop transfer function of a unity gain feedback system is given by: G(s) =

( (

) )(

)

. The range of positive

values of k for which the closed loop system will remain stable is: (A) 1 10, the root at √ s t to o s pl n 4.

(

(

)

(



)

(



so

(

) (

(

)

) (

)

)

( ) or

o

st

r qu r

.(

)(

6.

[Ans. C] K=

(

)

(

) ) ( (

7.

√

s s(s )(s (s ( s(s

(s)

)

[Ans. B] (s ) s (s ) For unity feedback, equation is 1+ G(s) = 0 s s s

√

[Ans. D]

(s) 5.

) /

)

)

S=

(

v lu

=0

s s S= √ RL exists at 3.41 (

)

But b = 20 is not the required value of b because it will cancel out an open-loop pole

) (

) )(

(

[Ans. C] (s)

s

or

[Ans. C] T(s)=

Control Systems

) ))(s ( )(s )

))

(s)

characteristic

th

th

th

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GATE QUESTION BANK

Control Systems

Angle at which RLD approaches to zeros is the angle of arrival for zeros. , (

)

0

. /1

t nt [ * [

( )

t n t nt t n

8.

( )] ( )]+ ( )

[Ans. ] Int r s t on o symptot s sum o pol s sum o z ros num r o pol s num r o z ro ( ) ( )

th

th

th

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GATE QUESTION BANK

Control Systems

Frequency Response Analysis Using Nyquist plot ECE - 2006 1. The open-loop transfer function of a unity-gain feedback control system is given by, G(s) =

(

)(

. The gain

)

margin of the system in dB is given by (A) 0 (C) 20 (B) 1 (D) 2.

Consider

two

(s)

and

transfer (s)

G(s)

4.

Which of the following statement is true? (A) G(s) is an all-pass filter (B) G(s) has a zero in the right-half plane (C) G(s) is the impedance of a passive network (D) G(s) is marginally stable

5.

The gain and phase margins of G(s) for closed loop stability are (A) 6 dB and (C) 6 dB and (B) 3 dB and (D) 3 dB and

functions, . The

3-dB bandwidths of their frequency responses are, respectively

3.

(A) √

√

(B) √

√

(C) √

√

(D) √

√

The Nyquist plot of ( ) ( ) for a closed loop control system, passes through ( ) point in the GH plane. The gain margin of the system in dB is equal to (A) Infinite (B) greater than zero (C) less than zero (D) zero

ECE - 2011 Common Data Question 6 and 7: The input – output transfer function of a ( )

plant

(

)

. The plant is placed

in a unity negative feedback configuration as shown in the figure below. r

u

(s)

y s(s

)

l nt

ECE - 2009 Common Data for Question 4 and 5 The Nyquist plot of a stable transfer function G(s) is shown in the figure. We are interested in the stability of the closed loop system in the feedback configuration shown

6.

The signal flow graph that DOES NOT model the plant transfer function ( ) is (A) 1

u

1/s

1/s

1/s

100 y

Im

(B)

u

1/s

1/s

1/s

100

y

Re

th

th

th

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GATE QUESTION BANK

Control Systems

(D)

(C)

1/s

u

1/s

1/s

100

y

⁄

(D) 1/s

1/s

u

7.

1/s

100

y

The gain margin of the system under closed loop unity negative feedback is (A) 0 dB (C) 26 dB (B) 20 dB (D) 46 dB

EE - 2006 1. Consider the following Nyquist plots of loop transfer functions over = 0 to = . Which of these plots represents a stable closed loop system? (1)

Im R =

8.

For the transfer function ( ) the corresponding Nyquist plot for the positive frequency has the form (A)

Re 1

Im

(2)

5

=

Re

(B) (3)

Im

j5

= Re

(C)

(4)

Im

= ⁄

1

(A) (B) (C) (D)

th

Re

(1) only all, except (1) all, except (3) (1) and (2) only

th

th

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GATE QUESTION BANK

EE - 2007 2. If x = Re G(j ), and y = Im G(j ) then for , the Nyquist plot for G(s) = 1 / s(s+1) (s+2) becomes asymptotic to the line (A) x = 0 (C) x = y (B) x =

Control Systems

(A)

Im 3/4 Re

(D) d = y / √

EE - 2009 3. The polar plot of an open loop stable system is shown below. The closed loop system is

𝛚=0 𝛚=0

(B)

Im

Imaginary

Real

Re 3/4

(A) Always stable (B) Marginally stable (C) Unstable with one pole on the RHS splane (D) Unstable with two poles on the RHS s-plane 4.

(C)

The open loop transfer function of a unity feedback system is given by ( )⁄ G(s) = s The gain margin of this system is (A) 11.95dB (C) 21.33dB (B) 17.67dB (D) 23.9dB

𝛚=0

Im

Re

1/6

Im

(D)

EE - 2010 5. The frequency response of (s) )(s )- plotted in the ,s(s complex ( ) plane (for ) is

Re

𝛚=0 1/6

th

th

th

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GATE QUESTION BANK

EE - 2011 6. A two-loop position control system is shown below. Motor s(s

IN - 2009 4. A unity feedback control loop with an open transfer function of the form

(s)

(s)

having a transfer function

s Tacho-generator

by G(s) = ( 5.

G(s) = 2.

3.

(

)(

6.

The gain margin of the system is (A) 0.125 (C) 0.5 (B) 0.25 (D) 1

(C) 125√ (D) 125√

v lu o or t mp ng r t o to be 0.5, corresponding to the dominant closed-loop complex conjugate pole pair is (A) 250 (C) 75 (B) 125 (D) 50

IN - 2012 7. The open loop transfer function of a unity negative feedback control system is given by G(s) =

)

The phase crossover frequency of the system in radians per second is (A) 0.125 (C) 0.5 (B) 0.25 (D) 1

is inserted

)

(A) 250√ (B) 250√

⁄

Statement for Linked Answer Questions 2 &3 Consider a unity feedback system with open loop transfer function

√

The value of K for the phase margin of the system to be 45° is

(s)

(C) (D)

√

IN - 2011 Common Data for Questions 5 and 6 The open-loop transfer function of a unity negative feedback control system is given

IN - 2008 1. For the closed loop system shown below to be stable, the value of time delay TD (in seconds) should be less than

⁄ ⁄

)

into the loop, the phase margin will become (A) (C) (B) (D)

The gain k of the Tacho-generator influences mainly the (A) peak overshoot (B) natural frequency of oscillation (C) phase shift of the closed loop transfer function at very low ) frequencies ( (D) phase shift of the closed loop transfer function at very high ) frequencies (

(A) (B)

(

has a gain crossover frequency of 1 rad/s and a phase margin of . If an element

)

(s)

Control Systems

(

)(

)

. The gain margin of the

system is (A) 10.8 dB (B) 22.3dB

(C) 34.1dB (D) 45.6dB

IN - 2014 8. The loop transfer function of a feedback control system is given by (s) (s)

)( s ) s(s Its phase crossover frequency (in rad/s), approximated to two decimal places, is __. th

th

th

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GATE QUESTION BANK

Control Systems

Answer Keys and Explanations ECE 1.

6.

[Ans. D] 𝐹or option (D).

[Ans. D]

( )

(s)

(s )(s or g n m rg n t n ( ) t n

(

( )

)

t n

⁄

. /

7.

[Ans. C]

)

(s) (s)

) s(s ( ) t n phase cross-over

=0 o (s)

For

[ ]

Gain margin =

log

t n (

0 1 r

Using Root locus of G(s)

) (

(

root locus will cut the when k

) (

3.

M

= 0 axis only 0 1

8.

[Ans. D] At ( GM = 20 log

4.

[Ans. B]

5.

[Ans. C] ( At log

|

(

) (

)

)

( ) (

)

)

log

[Ans. A] Assuming no.of open loop poles in the RHS of s – plane = P = 0 Complete nyquist plots

|

Im

) (

)

GM = 20

| (

At

|

EE 1.

)

)

[Ans. A] ( ) =5 So ( ) is a straight line parallel to axis.

)

) (

(

)

(

M n

[Ans. C] BW depends only on denominator and is equal to √(

)

s

)

(

So gain margin =

frequency,

ut s (

2.

(s)

s not tr ns r un t on o

) (

)

PM = No.of encirclements = N = 0 th

th

th

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GATE QUESTION BANK

N=P–Z=0

Control Systems

Hence the system is unstable So option (A) is correct

Hence system is stable.

2.

[Ans. B]

Imj

(

) (

)

( (

)(

symptot to x = 3.

Two anti – clockwise encirclement N=2 P–Z=N=2

(

)

) )(

)

s

[Ans. D] 2 clockwise encirclement of 1 + j0 = 0; N = 2 =2 Z = number of closed loop ploes in RHS Hence system is unstable

4.

[Ans. D] Open loop transfer function, G(s) = Put s =

Hence system is unstable. Im

(

)

At phase crossover frequency ( phase of OLTF is ( )

Two clockwise encirclement of – 1 Hence N = 2

|

| s lw ys (

(

Im

or ny v lu o

) n t

Hence the system is unstable

)

(p

s

ross r qu n y)

)

Gain margin = 20log

| (

)|

) )

((

log 5.

[Ans. A] (

)

As

( ((

)

(

G(

) )

(

))

)

s ( ) -0 + Option (A) satisfies above

Two clockwise encirclement of – 1 Hence N = 2 Z=2 th

th

th

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GATE QUESTION BANK

6.

[Ans. A] T(s) =

2. (

)

r

s

(

(

)

(

ξ

)

)

M is dependent on K (A) is correct is independent of K (B) is not correct T(J ) = (

[Ans. C] ( ) t n ( ) 

)

(

)

(

( )

Control Systems





3.

[Ans. A] ( )

4.

[Ans. A] M o (s)

r

t n

.

(

)

s

/

( )≈ ( )≈

( ) n ( ) r p rt lly orr t

( [Ans. C]

(s) = s

(s)

)

)

t low r qu n s t g r qu n s

IN 1.

t n ( )

t n (

( (

PM of

y(s)

5.

t n √ ) √ )

(

√

)

(s) =

(

)

[Ans. B] PM = 45°= 180° t n

(s)

. /) 

t n

. /

rad/sec

( )

y(s)

. /

|(

)

|

√ G(s) = For system to be stable G(s) should lie in the left of ( 1, 0) For finding the critical/ marginal value apply phase condition w

6.

[Ans. A] r t r st

(s ) s s s ( The dominant poles are given by ) s s (

w

or (s) (s) w

qu t on

s

s

(

)

)

s |

s

|

w

So w

For stable system

or

7.

[Ans. C] = 15 r/s √ th

th

√ th

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GATE QUESTION BANK

G.M in dB = 8.

log 0

Control Systems

1

[Ans. *]Range 0.30 to 0.34 (s) (s)

s(s

)( s

)

or (s) (s)

t n

t n ( )

(

t n (

)

)

As t n So

=1 r s r s

th

th

th

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GATE QUESTION BANK

Control Systems

Frequency Response Analysis Using Bode Plot ECE - 2006 Statement for Linked Answer Questions 1 &2 Consider a unity-gain feedback control system whose open-loop transfer function is

ECE - 2010 4. For the asymptotic Bode magnitude plot shown below, the system transfer function can be M gn tu

( ) 1.

The value of 'a' so that the system has a phase-margin equal to is approximately equal to (A) 2.40 (B) 1.40

2.

(C) 0.84 (D) 0.74

With the value of 'a' set for a phasemargin of the value of unit-impulse response of the open-loop system at second is equal to (A) 3.40 (C) 1.84 (B) 2.40 (D) 1.74

(A)

(C)

(B)

(D)

ECE - 2014 5. The phase ( )

(

margin )(

)(

in )

degrees

calculated

of

using

the asymptotic Bode plot is ________. ECE - 2007 3. The asymptotic bode plot of a transfer function is as shown in the figure. The transfer function G(s) corresponds to this Bode plot is |G(

6.

In a Bode magnitude plot, which one of the following slopes would be exhibited at high frequencies by a 4th order all-pole system (A) (C) (B) (D)

7.

The Bode asymptotic magnitude plot of a minimum phase system is shown in the figure.

)|(dB)

60 20 dB/decade 40 40 dB/decade 20

0

0.1

1

10

2 0

( (

100

) )

60 dB/decade

(A) (B)

(

)( (

)(

(C)

) )

(D)

(r (

)(

(

)(

)

⁄s) n log s l

If the system is connected in a unity negative feedback configuration, the steady state error of the closed loop system, to a unit ramp input, is_________.

)

th

th

th

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GATE QUESTION BANK

EE - 2006 1. The Bode magnitude plot of ( )(

H (j ) = ( (

(A)

) )

Control Systems

EE - 2008 2. The asymptotic Bode magnitude plot of a minimum phase transfer function is shown in the figure.

is

)

( 40

)

40 dB/ decade

(dB) 20

20

20 dB/ decade (r s) (log scale)

0.1 0

+1

+3

+2

Log ( )

20

0 dB/ decade

0

(B)

(

This transfer function has (A) Three poles and one zero (B) Two poles and one zero (C) Two poles and two zeros (D) One pole and two zeros

) 40 20

0

+1

+2

+3

Log ( )

EE - 2009 3. The asymptotic approximation of the log-magnitude vs frequency plot of a system containing only real poles and zeros is shown. Its transfer function is 40 dB / dec

(C)

(

)

60 dB / dec

80 dB 40

20

0

+1

+2

+3

r

Log ( )

0.1

(A)

(D)

( )(

(

(B) (

2 5

(

) ( )(

(C)

) )

( )( (

(

(D)

)

s

25

(

) ) ) )(

)

) 40

EE - 2014 4. The Bode magnitude plot of the transfer

20

function (s) 0 +1

+2

+3

( .

)( /(

) ).

/

is shown

below: Note that -6 dB/octave = -20 dB/decade. The value of is_______

Log ( )

th

th

th

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GATE QUESTION BANK

The gain and phase margin of the system are (A) 0 dB, 180° (C) 0 dB, 10° (B) 3.88 dB, 170° (D) 3.88 dB, 10°

t v t v

Control Systems

t v t v

t v

t v (r

⁄s )

IN - 2008 2. The Bode asymptotic plot of a transfer function is given below. In the frequency range shown, the transfer function has dB

5.

6.

For the transfer function (s ) ( ) s(s )(s s ) The values of the constant gain term and the highest corner frequency of the Bode plot respectively are (A) 3.2, 5.0 (C) 3.2, 4.0 (B) 16.0, 4.0 (D) 16.0, 5.0

+ 20dB / decade 20dB / decade 0dB / decade

log

(A) (B) (C) (D)

The magnitude Bode plot of a network is shown in the figure (

) lop

log

The maximum phase angle and the corresponding gain respectively, are (A) 30° and 1.73dB (B) 30° and 4.77dB (C) + 30° and 4.77dB (D) + 30° and 1.73dB IN - 2006 1. A unity feedback system has the following open loop frequency response: (r s ) ( ) ( ) 2 7.5 3 4.8 4 3.15 5 2.25 6 1.7 8 1 10 0.64

3 poles and 1 zero 1 pole and 2 zeroes 2 poles and 1 zero 2 poles and 2 zeroes

IN - 2010 3. The asymptotic Bode magnitude plot of a lead network with its pole and zero on the left half of the s-plane is shown in the adjoining figure. The frequency at which the phase angle of the network is maximum (in rad/s) is

r s (log scale)

(A) (B)

(C)

√

(D)

√

IN - 2013 4. The Bode plot of a transfer function G(s) is shown in the figure below.

th

th

th

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GATE QUESTION BANK

Control Systems

Gain (dB)

The gain (20 log|G(s)|) is 32 dB and -8 dB at 1 rad/s and 10 rad/s respectively. The phase is negative for all . Then G(s) is 40 32

(A)

(C)

(B)

(D)

20

5.

The discrete – time transfer function is

0 1

10

-8

(r

s)

(A) (B) (C) (D)

100

Non – minimum phase and unstable. Minimum phase and unstable. Minimum phase and stable. Non – minimum phase and stable.

Answer Keys and Explanations ECE 1.

4.

[Ans. A]

[Ans. C] | ( )| M w r

G(s) = ( (

.

/

(

)

.

/

(

)

)

)

t n (

5.

)

[Ans. *] Range 42 to 48 (

)

M t n ( √

2.



) 

√



= 0.84

[Ans. C] (s)

(

)

 g(t)

u(t)

t u(t)

(

3.

[Ans. D] (s) o

s(s

)(

plot s ( r t’s typ log

) ) orm syst m so

)

( )( )( ) From the plot frequency at which gain is 0 dB is r s ( ) M

At t = 1, g(t)

6.

[Ans. A] In a transfer function if all are poles if we plot the BODE diagram, then an each and every corner frequency we have to introduce a line of slope and hence on the 4th frequency the slope of line will become – 80 dB/sec and will continue upto infinity

|

| K = 100

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GATE QUESTION BANK

7.

[Ans. *] Range 0.49 to 0.51 From the Bode plot given, we can see that there are 2 break frequencies at = 2 and = 10 rad/sec. Also it is a type 1 system. So, the transfer function is ( ) ( ) ( ) ( ) At = 1 rad/sec, (given)

2.

[Ans. C] Initial slope is – 40 dB/decade, it means there are double pole at origin. Slope changes from dB/decade to dB/decade. It means there is a zero. Slope changes from dB/decade to 0 dB/decade at some other frequency. It means there is one more zero. Therefore transfer function has two poles and zeros.

20 log | |

3.

[Ans. B]

log ( ( In s domain, ( (s) ( (

) )

)

l m s (s) (

log

s) s)

(s s(s (s lm s

) ) )

/

) )

[Ans. *] Range 0.7 to 0.8 For initial dotted slope

[Ans. A] (

/

/.

(s )(s

s (s

( )(

log )

s .

)

)

( .

4.

(

( s ) s )( s )

s (

So,

EE 1.

Control Systems

(

)

)

log

)

( )

log (

( )( ) Corner frequencies are 1, 10 & 100 rad/sec ( ) ( )

log

)

log .

(s) .

log

/.

/.

/

/.

/

log (

)

5.

[Ans. A] (s)

(s )(s

s(s (

log

)(

s( log log log

log log log

(

) s

s )

s) .

/

)

n m x mum r qu n y th

)

th

th

r s

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GATE QUESTION BANK

6.

Control Systems

[Ans. C]

.

(

/

(s)

)

.

/

has 2 poles and 1 zero s s

3.

t n

[Ans. B] This condition is only for phase lead N/W

t n

=√ (

4.

)

(

)

(

√

[Ans. B] 32 dB

)

1

10

√ t n √

. /

√

√ √

√

g n

to Is 1 dec are change & change is (G) is 40dB lop s r r pol s s or g n

√

. / √

g n IN 1.

t n

√

log √

So, G(s)= = 32 dB (given)

[Ans. D]  M r s  M

log

(

=

) 5.

[Ans. D] z z For minimum phase system, all poles and zeros must lie inside the unit circle. For stable system, all poles must be inside the unit circle. For the system, zero is at 2 pole is at 0.5. This system is stable but non – minimum phase.

[Ans. C] Compare with Bode magnitude plot of standard transfer function. . (

(z)

/

) .

. /|

log

= 3.88 dB 2.

log

/

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th

th

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GATE QUESTION BANK

Control Systems

Compensators & Controllers ECE - 2006 1. The transfer function of a phase-lead

Z -

(s)

compensator is given by

+

where . The maximum phase-shift provided by such a compensator is (A) ⁄ (C) ⁄ (B) ⁄ (D) ⁄

Group I

ECE - 2007 2. A control system with PD controller is shown in the figure. If the velocity error constant Kv =1000 and the damping ratio ξ t n t v lu s o p and KD are r

(A) (B) (C) (D) 3.

+

Σ

s

s(s

)

Kp=100, KD=0.09 Kp=100, KD=0.9 Kp=10, KD=0.09 Kp=10, KD=0.9

The open-loop transfer function of a plant is given as 0 (s)

1. If the plant is

operated in a unity feedback configuration, Then the lead compensator that can stabilize this control system is (A) (B)

(

)

(

)

(C) (D)

(

)

(

)

Group II 1. PID controller 2. Lead compensator 3. Lag compensator (A) Q – 1, R – 2 (C) Q – 2, R – 3 (B) Q – 1, R – 3 (D) Q – 3, R – 2

y

ECE - 2009 5. The magnitude plot of a rational transfer function G(s) with real coefficients is shown below. Which of the following compensators has such a magnitude plot? (

) 20dB

log

ECE - 2008 4. Group 1 gives two possible choices for the impedance Z in the diagram. The circuit elements in Z satisfy the condition R2C2 >R1C1. The transfer function represents a kind of controller. Match the impedances in Group I with the types of controllers in Group II.

(A) (B) (C) (D)

Lead compensator Lag compensator PID compensator Lead – lag Compensator

ECE - 2010 6. A unity negative feedback closed loop system has a plant with the transfer function G(s) =

and a controller

(s) in the feed forward path. For a unit step input, the transfer function of the

th

th

th

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GATE QUESTION BANK

controller that gives minimum steady state error is (A)

(s)

(B)

(s)

(C)

(s)

(D)

(s)

(

)(

)

(

)(

)

s

ECE - 2012 Statement Linked answer Questions 7 and 8 The transfer function of a compensator is given as (s) . 7.

8.

Gc(s)is a lead compensator if (A) (B) (C) (D)

Control Systems

(D) A lag-lead compensator that provides an attenuation of 20 dB and phase lead of 450 at the frequency of 3 rad/s EE - 2008 2. The transfer function of two compensators are given below (s ) s (s ) (s ) Which one of the following statements is correct? (A) C1 is a lead compensator and C2 is a lag compensator (B) C1 is a lag compensator and C2 is a lead compensator (C) Both C1 and C2 are lead compensators (D) Both C1 and C2 are lag compensators IN - 2006 1. The transfer function of a position servo

The phase of the above lead compensator is maximum at (C) √ r s (A) √ r s (D) s (B) √ r s √ r

system is given as G (s) =

1 . A first s(s  1)

order compensator is designed in a unity feedback configuration so that the poles of the compensated system are placed at – 1  j1 and 4. The transfer function of the compensated system is

EE - 2007 1. The system 900/s(s+1)(s+9) is to be compensated such that its gain-crossover frequency becomes same as its uncompensated phase-crossover frequency and provides a 450 phase margin. To achieve this, one may use (A) A lag compensator that provides an attenuation of 20 dB and a phase lag of 450 at the frequency of √ rad/s (B) A lead compensator that provides an amplification of 20dB and a phase lead of 450 at the frequency of 3 rad/s (C) A lag-lead compensator that provides an amplification of 20 dB and a phase lag of 450 at the frequency of √ rad/s.

(A)

(

(C)

)

(D)

(B)

(

)

(

)

Common Data Questions 2, 3, 4 The following figure describes the block diagram of a closed loop process control system. The unit of time is given in minute ontroll r (s)

r(s)

th

th

(

)

th

stur

l nt

n

(s)

m(s) s

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(s)

GATE QUESTION BANK

2.

The digital implementation of the controller with a sampling time of 0.1 minute using velocity algorithm is ( ) m( )

* ( )

∑ ( )+

( ) m( )

* ( )

∑ (

( ) m( )

m(

( ) m( )

m(

4.

motor is given as

( ) ( )

. When

connected in feedback as shown below, the approximate value of that will reduce the time constant of closed loop system by one hundred times as compared to that of the open – loop system is

)+

(

)-

(

)-

R(s)

(s)

(s)

) ( )

, 3.

ECE/EE/IN - 2013 6. The open – loop transfer function of a dc

) , ( )

Control Systems

Suppose a disturbance signal d(t)= sin 0.2t unit is applied. Then at steady state, the amplitude of the output e(t) due to the effect of disturbance alone is (A) 0.129 unit (C) 0.529 unit (B) 0.40 unit (D) 2.102 unit

(A) 1 (B) 5

(C) 10 (D) 100

IN - 2014 7. Consider the control system shown in figure with feed forward action for rejection of a measurable disturbance d(t). The value of K, for which the disturbance response at the output 𝑦( ) is zero mean, is:

The control action recommended for reducing the effect of disturbance at the output(provided that the disturbance signal is measurable) is (A) cascade control (B) P-D control (C) ratio control (D) feedback-feed forward control

(t)

y(t)

r(t)

s

(A) 1 (B) 1

IN - 2007 5. A Cascade control system with proportional controllers is shown below.

(C) 2 (D) 2

(s) (

)(

)

( s

)

(s)

Theoretically, the largest values of the gains and that can be set without causing instability of the closed loop system are: (A) 10 and 100 (C) 10 and 10 (B) 100 and 10 (D) n

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GATE QUESTION BANK

Control Systems

Answer Keys and Explanations ECE 1.

4.

[Ans. B]

[Ans. D] M xp s s t (s) t n t n From maximum phase shift

s s s

t ( (

,

( (

) ) -

(

(t ) ) s (s) (s )(s ) (s) s The above equation is a PID controller

)

)

or

√ t n (√ ) 2.

t n

√

s ( ) ( ) ( ) ( ) The above equation is a lag compensator. ( )

[Ans. B] (s)

( (s m m

)

s (s) (s) (

) ) )

(s

comparing Eq.(1) with standard 2nd order equation

5.

[Ans. D] ( ) plot shows presence of 2 poles & 2 zeroes in Bode equivalent plot

6.

[Ans. D] m

√

s

ξ

3.

S.E(s)= 7.

[Ans. A]

(s )(s ) Only option (A) is satisfies.

s

t n

s (s )(s ) The lead compensator C(s) should first

(s) (s)

m s

(

)

for (D)

t n

for phase lead t n

term

(s (s

( ) ( ))

(

[Ans. A]

(s)

stabilize the plant i.e. remove

s

should be positive

t n

) Option (A) & (C) satisfies, it may be observed have be observed that option (C) will have poles and zero in RHS of s – plane, thus not possible (not a practical system) it can be concluded that option (A) is right

)

th

th

th

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GATE QUESTION BANK

8.

[Ans. A] y ( )

t n

Control Systems

A lag lead compensator that provides an attention of 20dB and phase lead of 45 at the frequency of 3rad/sec is used.

t n

For maximum phase shift t

2.

[Ans. A] t n

t n

is always positive

t n

t n

is always negative

⁄

sl IN 1. √ r EE 1.

s

)(s

)

r

)|

|

. .

s

s(s

)(s

) (

)|

)

|

/ /

M

log

M

log (

| (

) ( )

(s (s

(s)

)

Negative GM implies that the system is unstable. should become gain cross over frequency, r s , the magnitude should be 0dB. o m t m gn tu ‘ ’ t r s a lag compensator which gives an attenuation of 20dB(before compensation the magnitude is 20dB) and to obtain 45 phase margin at r s a lead compensator with a phase lead of 45 is used.

)

(s) ) (s) s(s ol s o t omp ns t syst m r g v n s s s (s ) t (s) (s ) r t r st qu t on )(s ) (s ) s(s ) ,(s -(s ) ( )s ( )s s (s )(s ) s s s s ( )

√

) (

( (

|

(s)

( )

(s) (s) (s) (s)

√ | (

s l g omp ns tor

[Ans. C] Let the T.F of the compensator be o t omp ns t syst m

[Ans. D] s(s

n

2.

) )

[Ans. A] v n (s) {

m(s)

(

)} s

{ (s) m( ) s

th

* ( ) pl

th

(s)}

s

∑ ( )+

tr ns orm o

th

(t) t

I(s) s

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GATE QUESTION BANK

3.

( ) 0 s For stable system all elements of first column should be greater than zero. By substituting all the given options in o nt o ’ w ll g v n g t v v lu except option D.

[Ans. B] The given diagram can be reduced as (s)

(s)

(s)

s

r ns r un t on (s)

( )

( )

( )

( )

Control Systems

6.

[Ans. C] Open loop transfer function of a dc motor as ( ) ( )

( )

R(s)

( )

(

(s)

(

(s)

)

d(t)= sin 0.2t or g tt ng mult ply ng |

tor Topic: P controller with unity feed back Formula: For first order system loop

|

|

|

√

(

( )

transfer function is

)

with

o mpl tu o output (t) n t l mpl tu

. Now for

( )

closed loop overall transfer function is given by .

[Ans. B] (PD controller can be used)

( )

/ .

/ (

5.

)

Dividing numerator and denominator by 10

[Ans. D] (s) (

)(

)

( s

)

(s)

ow

)( s

((s

)

)( s

( ) ( )

.

/

o

(s)

(s)

comparing

( )

( )

( )

4.

(s)

(s)

)

)

( y omp r ng rom ormul ) In Question given that time constant of closed loop system is

times of

( ) ( )

((

)(

)

)(

)

Characteristics equation is ( )s s s ( ) For stability, according to R-H array 6 (6+3k2) s ( 11 s ( ) 0 s

so .

)

/

approximate value

th

th

th

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GATE QUESTION BANK

7.

Control Systems

[Ans. D] y(s)

(s)

(

(s)

y(s)) [

s

]

y(s) (s) (s) [

(s) [ s s

(s) [

]

] ]

Zero means at

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th

th

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GATE QUESTION BANK

Control Systems

State Variable Analysis ECE - 2006 1. A linear system is described by the following state equation ̇ (t)

̇ (t)

(t)

0

ECE - 2007 2. The state space representation of a separately excited DC servo motor dynamics is given as, +

0

10 1

0

The Eigen value and eigenvector pairs ( v ) for the system are

1

The state-transition matrix of the system is os t s nt (A) 0 1 s n t os t s nt ] (B) [ – os t s nt os t – os t s n t] (C) [ s nt os t os t s nt (D) 0 1 s nt os t

*

3.

4.

( )

(C)

(B)

(D)

x(0) = 0

1, then the system response is

x(t) = 0

1. If the initial state vector

of the system changes to x (0) = 0

0

1/

(B) .

0

1/ and .

0

1/

(C) .

0

1/ and . 0

1/

(D) .

0

1/ and . 0

1/

(B) 0

1 1

(C) 0

1

(D) 0

1

ECE - 2009 5.

Consider the system A=0

= Ax + Bu with

1 and B = 0 1 , where p and q

are arbitrary real numbers. Which of the following statements about the controllability of the system is true? (A) The system is completely state controllable for any nonzero values of p and q (B) Only P = 0 and q = 0 result in controllability (C) The system is uncontrollable for all values of p and q (D) We cannot conclude about controllability from the given data

1 u;

Common Data Questions 3 and 4 Consider a linear system whose state space representation is ̇ (t) = Ax (t). If the initial state vector of the system is

1/ and .

(A) 0

of the motor is

(A)

0

The system matrix A is

Where is the speed of the motor, is the armature current and u is the armature voltage. The transfer function ( )

(A) .

ECE - 2010 Common Data for Questions 6 and 7 The signal flow graph of a system is shown below.

1,

(s)

s

s

(s)

then the system response becomes x (t) = 0

1.

th

th

th

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GATE QUESTION BANK

6.

The state variable representation of the system can be (A) (B) (C) (D)

7.

ẋ

0

1x

y

,

-x

ẋ

0

1x

y

,

-x

ẋ

0

1x

y

,

ẋ

0

y

,

0 1u 0 1u 0 1u

-x 1x

0 1u

-x

The transfer function of the system is (A)

(C)

(B)

(D)

ECE - 2011 8. The block diagram of a system with one input and two outputs 𝑦 and 𝑦 is given below. u

ECE - 2012 9. The state variable description of an LTI system is given by x ẋ x [ẋ ] [ ][ ] [ ]u x ẋ x - [x ] y , x Where y is the output and u is the input. The system is controllable for (A) a1≠ 2 =0, a3 ≠ (B) a1 =0, a2 ≠ 3≠ (C) a1=0, a2 ≠ 3 =0 (D) a1 ≠ 2 ≠ 3 =0 ECE - 2013 Statement for Linked Answer Questions 10 and 11: The state diagram of a system is shown below. A system is described by the state – variable equations ̇ u y u u

y

s

10. y

s

1

A state space model of the above system in terms of the state vector and the output vector 𝑦 ,𝑦 𝑦 - is , -x

(B) ẋ

,

(C) ẋ

0

(D) ẋ

0

, -u y

-x

, -u 1x

1x

, y

̇

0

y

,

̇

0 ̇

(C) ,

-x

0 1x 11.

,

̇

0

y

,

y

0

1u

0

1u

0

1u

0

1u

u 1 -

u 1

-

u 1

-

u

The state transition matrix of the system shown in the figure above is (A) 0 (B) 0

th

-

0

y (D)

1

,

y

-x

0 1u y

0 1u y

(B)

0 1x

1

1

1

1

The state – variable equation of the system shown in the figure above are (A)

(A) ẋ

Control Systems

th

(C) 0

1

t t

(D) 0

1

th

1 t

1

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GATE QUESTION BANK

ECE - 2014 12. Consider the state space model of a system, as given below x ẋ x [ẋ ] [ ][ ] [ ]u x ẋ x - [x ] y , x The system is (A) controllable and observable (B) uncontrollable and observable (C) uncontrollable and unobservable (D) controllable and unobservable 13.

or x

s

s

x

x

[

1

(B) [

]

(C) [

]

(D) [ 16.

y

]

0 1 x(t) s

(A) 0

s

x

0 1 x(t) nx

Consider the state space system expressed by the signal flow diagram shown in the figure. u

Control Systems

]

The state transition matrix (t) of a x ẋ system [ ] 0 1 0x 1 is ẋ t (A) 0 (C) 0 1 1 t t (B) 0 (D) 0 1 1 t

EE - 2006 1. For a system with the transfer function (

H(s) =

)

, the matrix A in the

state space form ẋ = Ax + Bu is equal to The corresponding system is (A) always controllable (B) always observable (C) always stable (D) always unstable 14.

15.

An unforced linear time invariant (LTI) system is represented by x ẋ [ ] 0 1 0x 1 ẋ If the initial conditions are x (0)=1 and x ( ) t solut on o t st t equation is (A) x (t) x (t) (B) x (t) x (t) (C) x (t) x (t) (D) x (t) t x (t) The state equation of a second-order linear system is given by ẋ (t) x(t) x( ) x or x

0

1 x(t)

0

(A) [

]

(B) [

]

(C) [

]

(D) [

]

EE - 2008 Statement for Linked Answer Questions 2 and 3 The state space equation of a system is described by ẋ = Ax + Bu, y = Cx, where x is state vector, u is input, y is output 1 , B = 0 1, C = ,

and A = 0 2.

-

The transfer function G(s) of this system will be

(A)

1 n

(

(B) th

( th

(C)

)

(D)

) th

(

)

(

)

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GATE QUESTION BANK

3.

A unity feedback is provided to the above system G(S) to make it a closed loop system as shown in figure.

r + ( t )

y ( t )

G(s)

Σ

For a unit step input r(t), the steady state error in the output will be (A) 0 (C) 2 (B) 1 (D) EE - 2009 Common Data Questions: 4 & 5 A system is described by the following state and output equations ()

x (t)

x (t)

()

x (t)

u(t)

4.

5.

EE - 2013 Common Data Questions 7 and 8 The state variable formulation of a system is given as x ẋ [ ] 0 1 0x 1 0 1 u x ( ) ẋ x - 0x 1 x ( ) n y , 7. The system is (A) Controllable but not observable (B) Not controllable but observable (C) Both controllable and observable (D) Both not controllable and not observable 8.

The response y(t) to a unit step input is (A) (B)

u(t)

y(t) = x1(t) Where u(t)is the input and y(t) is the output

(C) (D) EE - 2014 9. A system matrix is given as follows. [

The system transfer function is (A)

(C)

(B)

(D)

The state transition matrix of the above system is (A) [

]

(B) 0

1

(C) 0

1

(D) 0

0 (A) (B) (C) (D)

1

]

The absolute value of the ratio of the maximum eigenvalue to the minimum eigenvalue is _____ 10.

The second order dynamic system u

t

y the matrices P, Q and R as follows: 0

1

0 1

1

EE - 2010 6. The system ̇

Control Systems

, The system has the following controllability and observability properties: (A) Controllable and observable (B) Not controllable but observable (C) Controllable but not observable (D) Not controllable and not observable

with 0 1 is

stable and controllable stable but uncontrollable unstable but controllable unstable and uncontrollable th

th

th

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GATE QUESTION BANK

11.

The state transition matrix for the system x ẋ [ ] 0 1 0x 1 0 1 u is ẋ (A) 0 (B) 0

(C) 0

1 t

1

(D) 0

t

Control Systems

IN - 2008 2. The state space representation of a ̇= 0

system is given by

1

y = [1 0]x. The transfer function

t 1

Consider the system described by following state space equations x ẋ [ ] 0 1 0x 1 0 1 u ẋ x - 0x 1 y , If u is unit step input, then the steady state error of the system is (A) 0 (C) 2/3 (B) 1/2 (D) 1

IN - 2006 1. The state-variable representation of a plant is given by ̇ = Ax + Bu , y = Cx. Where x is the state, u is the input and y is the output. Assuming zero initial conditions, the impulse response of the plant is given by (A) exp (At) (B) xp , (t - )] Bu () d (C) C exp (At) B (D) xp , (t - )] Bu () d

( ) ( )

of

the system will be (A)

12.

1 X + 0 1u,

(C)

(B)

(

(D)

)

IN - 2009 3. A linear time-invariant single-input single-output system has a state space model given by

=Fx + Gu; y = Hx

Where F =0

1; G=0 1; H =,

-.

Here, x is the state vector, u is the input, and y is the output. The damping ratio of the system is (A) 0.25 (C) 1 (B) 0.5 (D) 2 IN - 2011 4. The transfer function of the system described by the state-space equations x ẋ [ ] 0 1 0x 1 0 1 u ẋ x - 0x 1 is y , (A) (C) (B)

th

(D)

th

th

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GATE QUESTION BANK

Control Systems

Answer Keys and Explanations ECE 1.

6. [Ans. A] ∅(t) = L *( I ( I

)

) +

0

 ( I

ẋ

)

) s nt 1 os t

0

 s

ẋ ẋ y

x u… … ( ) x x … … ( ) (x ẋ ) (x x x ) y x … … ( ) From equation 1, 2, 3 the state variable representation of the system is

(s)

I (s)

I (s) (s) u(s)

u(s)

( ) ( )

(

∅  ∅ ∅(t)

1x

𝑦

,

-

0 1u

[Ans. C] ,sI

∅ ∅ ] ∅ ∅ ∅ ∅

, &

∅ ∅

-0

, s

10 1

s (s

)

]

s s

) ) s

s s

-[

1 )

-

s

∅ ∅

0

(( I (s )(s find eigen vector

s s Method II It can also be solved by applying the M son’s g n ormul

&

[Ans. D] From above, , I 0

5.

0

[Ans. A] Let A = [

4.

ẋ

)

7. 3.

x ẋ

(s)

(s) ) (s)

(s) x

u (s) )(s

s

s

1

[Ans. A]

t  (s

ẋ

(s)

1

( os t (t) = 0 s nt

2.

[Ans. B]

-

0

8.

1

[Ans. B] (s) u(s) s (s) x (s) x (s) u(s) s x (s) y (s) u(s) s x (s) Similarly x (s) y (s) u(s) s u(s) so sx (s) x (s) u(s) y (s) x (s) sx (s) x (s) u(s)

1

[Ans. C] , 

-

p p 0q q1 p q  un ontroll

l

p, q

th

th

th

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GATE QUESTION BANK

y (s) x (s) ẋ (t) x (t) u(t) y (t) ẋ (t) x (t) u(t) y (t) x ẋ [ ] 0 1 0x 1 0 1 u(t) ẋ x x u ,y y -

rom qu st on y 9.

x (t) x (t)

,

-

y=,

From the graph we can find that ,u x - ( ) ẋ ẋ x u… ( ) n ( ),x ẋ ẋ ẋ x ẋ From eq. (i) put value of ẋ ( x ẋ x u) ẋ x x u…… ( ) s m l rly y ẋ ( ) y x x u … ( ) ( ẋ x x ) ( u) [ ] ( ẋ x x ) u In matrix form x ẋ [ ] 0 1 0x 1 0 1 u ẋ x ẋ 0 1 0x 1 0 1 u And x , - 0x 1 u y -x u y ,

0 1

x ] [x ] + [ ]u x x - [x ] x

A=[

]& B = [ ]

So, AB = [ =*

][ ] = [ ] +

So, the controllability matrix, h ⌈

⌉ =[

]

11.

[Ans. A] ,s I

Determinant should be non-zero So, a1a2 (– a2) ≠ 1≠ a2 ≠ a3 may be 0 10.

y

u

[Ans. D] x *x + = [ x

Control Systems

0

-

s0

1

s -

0

sI

s

[Ans. A] For state variable form, we have to find number of integral (1/s) in the graph Assign output of these integral with state variable & input of these integral with derivative of state variable. 1. Then find out the relation between these derivative in terms of state variable and input 2. Similarly we can write the relation between output and these variable and input using state flow graph 3. Lets assume & as state variable

1

1

s

,sI

0

s s

1

(s

sI

)

) s [(s ] Take inverse Laplace transform both side ,sI

s

-

[(s 0

th

th

)

s

]

1

t

th

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GATE QUESTION BANK

12.

[Ans. B] [

]

2

[ ]

, Controllable matrix ,

[

]

l

,

[Ans. B] Applying linearity property 0

o x(t) the state 16.

[Ans. D] (t)

0

0

1

[

, I

-

0

s

s s

EE 1.

x 1 0x 1

s ]

1

t

2

3 1t

0 0

1 1

t

3

(s) (s)

(s) ( )

( )

( )

( )

(s) (s) (s) s x t

… ) 0

1

[Ans. B]

x ( ) x ( )

t

t

0

(t)

0

1

s

The rank of S is 3 So the system is always controllable

t

1

s

+

(I

]

) -

[s

(t)

[Ans. C] ẋ [ ] 0 ⏟ ẋ

1

]

,( I s 0

The state controllable matrix is , ̇ ̇

14.

0 1

[

-[ ]

*

1

o

[Ans. A] From the state diagram, equations are ̇ [ ̇ ][ ][ ] [ ] ̇ 𝑦

15.

0 1

so o s rv

1

x (t) x ( ) x (t) x ( )

]

n

2 0

-

[

…

3

[

Rank is less than 3, so uncontrollable Observability matrix o , ( )

13.

Control Systems

(s

).

) s /

s s s (s) s x sx (s) u(s)

t

2

s

(s s

x (s)

Replacing s by x t

…

t

th

x t x t th

ẋ

x t

x

u(t) … ( )

x

th

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3]

GATE QUESTION BANK

x ẋ x t Replacing eq.(i) ẋ x x x u(t) ẋ x x x u(t) ẋ x ẋ x ẋ x x x u(t) x ẋ [ẋ ] [ ] [x ] [ ] u(t) x ẋ o

2.

[

3.

,

- (0

s

5.

-

[Ans. B] 0 ( I

)

( I

)

0

s

(

m

(

s

( ))

(

(

0

1

x (t)

y(t)

,

x (t) -[ ] x (t)

y o

,

-&

, I

-

0

s s

s 0

s

s (s )(s r ns r un t on , I -

]

1

[Ans. C] , AB]  | ≠ controllable (s) ( I ) & (t) ( (s )) has exponential with positive power Unstable

7.

[Ans. A] , M

1 ontroll

,

M

8.

-

( I

)

, I

-

0

0

s

1

*, I

l

-

(s)+

1

s

1 )

l

ot o s rv

[Ans. A] Y(t) ,

1

s

s

6.

0

1

s

0

t(M ) ≠

0

]

s

)

t(M ) 1

)

s

)

0 1

y(t)

)(s

)

[Ans. C] Selecting x (t) and x (t) as state variables x (t) ẋ (t) x (t) x (t) u(t) t x (t) ẋ (t) x (t) u(t) t ẋ (t) x (t) [ ] 0 ] 0 1 u(t) 1[ ẋ (t) x (t) ẋ

, I

(s

[s m

0

1

s s

[Ans. A]

o

1

[

1 )0 1

s

s

1 s 0 1 (s )(s ) (s ) , -0 1 s (s )(s ) s s (s )(s ) s s ,

]

s 4.

0

[Ans. D] (s)

Control Systems

[s

] s

th

th

th

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GATE QUESTION BANK

-

[, I x (t) [ ] x (t)

9.

)

[s(s

)]

,, I

s [(s * I

(s)-

-

[

12.

]

)(

(s)

0 ,

)

-

) )-

,( (

(s)

-

[Ans. C] 0

1

or ontroll ,

0 1

0

-

11.

1 2.

l

[Ans. C] 0

s

1+

1 s 0 1

s

)

s

rror

lm

(s)

l ms (

1

l m u(t) s

s(s

y(t) s

)

)

[Ans. A] (s) ( I

)

[

1 (

)

th

(sI

→

(

)

(s)

th

) s

)

) (s)

y(t)

 ( I

I

(sI

(t) → (sI

( I

State transition matrix s I 0 1 s s 0 s I (s )

0

s

s y st t

u(t) (s)

l

ot o s rv

1

[Ans. C]

1

0

s

t

(s) (s)

≠ ontroll For observability:

y(s) (s) )

(s)

s

s(s

l -

,

IN 1.

s

y(s) u(s)

v n

1

t

- *0 s

,

I

,

10.

0

[Ans. A]

|

,(

+

]

s

( I

[Ans. *] Range 2.9 to 3.1 Characteristic equation |

)

r ns r un t on

x (t) -[ ] x (t)

,

Y(t)

s(s

(s)]

Control Systems

)

]

(

(

)

.,

-0

)

0

1

-0

., 1/

th

)

1 0 1/ (s)

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GATE QUESTION BANK

3.

Control Systems

[Ans. B] The poles of the s/y are obtained from the s/y matrix, f det (SI – F )= 0 0

1

S(S + 2) + 4 = 0

4.

[Ans. A] T(s) = ( I , (

) -

(

)

0

10 1

)

th

th

th

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GATE QUESTION BANK

Analog Circuits

Diode Circuits - Analysis and Application ECE - 2006 1. For the circuit shown below, assume that the zener diode is ideal with a breakdown voltage of 6 Volts. The waveform observed across R is

ECE - 2007 2. The correct full wave rectifier circuit is (A)

Input Output

~

i

(B)

(A)

6V

Input

6V

(B)

Output

(C)

12V

(C)

12V Input Output

6V (D)

(D)

Input

6V

Output

3.

For the Zener diode shown in the figure, the Zener voltage at knee is 7 V, the knee current is negligible and the Zener dynamic resistance is 10 . If the input voltage ( ) range is from 10 to 16 V, the th

th

th

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GATE QUESTION BANK

Analog Circuits

output voltage ( ) range from

. The small signal input where

.

i

(A) 7.00 to 7.29V (B) 7.14 to 7.29 V

~

(C) 7.14 to 7.43 V (D) 7.29 to 7.43 V

ECE - 2008 4. In the following limiter circuit, an input voltage = 10sin100 t is applied. Assume that the diode drop is 0.7V when it is forward biased. The Zener breakdown voltage is 6.8V.The maximum and minimum values of the output voltage respectively are

6.

The bias current (A) 1 mA (B) 1.28 mA

7.

The ac output voltage (A) (B) (C) (D)

8.

A Zener diode, when used in voltage stabilization circuits, is biased in (A) reverse bias region below the breakdown voltage (B) reverse breakdown region (C) forward bias region (D) forward bias constant current mode

1k D1

through the diodes is (C) 1.5 mA (D) 2 mA is

D2 Z

(A) 6.1V, 0.7 V (B) 0.7 V, 7.5 V

6.8 V

(C) 7.5 V, 0.7 V (D) 7.5 V, 7.5 V

ECE - 2009 5. In the circuit below, the diode is ideal. The voltage V is given by

(A) min ( (B) max (

) )

(C) min ( (D) max(

ECE - 2012 9. The diodes and capacitors in the circuit shown are ideal. The voltage v(t) across the diode is

~

) )

ECE - 2011 Statements for Linked Answer Questions 6 and 7 In the circuit shown below, assume that the voltage drop across a forward biased diode is 0.7V. The thermal voltage

(A) (B) i

th

th

–1

(C) 1 – (D) 1 – i

th

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GATE QUESTION BANK

ECE/EE/IN - 2012 10. The I-V characteristics of the diode in the circuit given below are i

(A) (B) (C) (D)

sin (sin (sin 0 for all t

Analog Circuits

i i

)/2 )/2

{ ECE - 2014 13. In the figure, assume that the forward voltage drops of the PN diode and Schottky diode are 0.7 V and 0.3 V, respectively. If ON denotes conducting state of the diode and OFF denotes nonconducting state of the diode, then in the circuit,

i

The current in the circuit is (A) 10 mA (C) 6.67 mA (B) 9.3 mA (D) 6.2 mA ECE/EE/IN - 2013 11. In the circuit shown below, the knee current of the ideal Zener diode is 10 mA. To maintain 5 V across RL, the minimum value of RL in  and the minimum power rating of the Zener diode in mW, respectively , are

(A) both and are ON (B) is ON and is OFF (C) both and are OFF (D) is OFF and is ON 14.

100

ILoad 10V

RL

VZ=5V

(A) 125 and 125 (B) 125 and 250 12.

The diode in the circuit shown has = 0.7 Volts but is ideal otherwise. If = 5sin(𝜔𝑡)Volts, the minimum and maximum values of (in Volts) are, respectively,

(C) 250 and 125 (D) 250 and 250

A voltage 1000 sin Volts is applied across YZ. Assuming ideal diodes, the voltage measured across WX in Volts, is

(A) 5 and 2.7 (B) 2.7 and 5

(C) 5 and 3.85 (D) 1.3 and 5

1k W

Y Z

15. X

1k

th

The figure shows a half-wave rectifier. The diode D is ideal. The average steadystate current (in Amperes) through the diode is approximately __________

th

th

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GATE QUESTION BANK

+

Analog Circuits

+

D1 D2

i z

~

Vi

V0

RL =

5V

10V

-

16.

(A)

Two silicon diodes, with a forward voltage drop of 0.7 V, are used in the circuit shown in the figure. The range of input voltage for which the output voltage is

10

10

(B)

(A) (B) (C) (D)

5 5

EE - 2006 1. What are the states of the three ideal diodes of the circuit shown in figure?

(C) 10 5 5 10

10V

5A

(D) (A) (B) (C) (D)

ON, OFF, ON, OFF,

OFF, ON, OFF, ON,

OFF OFF ON ON

10

10 2.

Assuming the diodes D1 and D2 of the circuit shown in figure to be ideal ones, the transfer characteristics of the circuit will be th

th

th

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GATE QUESTION BANK

Analog Circuits

(B)

EE - 2007 3. The three – terminal linear voltage regula r i e e a l a resistor as shown in the figure. If Vin is 10 V, what is the power dissipated in the transistor? +10V

(C) RL Vin 6.6V Zener diode 0

(A) 0.6 W (B) 2.4 W

(D)

(C) 4.2 W (D) 5.4 W

EE - 2008 4. The equivalent circuits of a diode, during forward biased and reverse biased conditions, are shown in the figure. +

+

0.7 V

5.

In the voltage doubler circuit shown in he igure he wi h ‘S’ i l e a Assuming diodes D1 and D2 to be ideal, load resistance to be infinite and initial capacitor voltages to be zero, the steady state voltage across capacitors C1 and C2 will be

+

~

𝜔𝑡 S

~ If such diodes is used in clipper circuit of figure given above, the output voltage (v0) of the circuit will be

i

(A) (B) (C) (D)

(A)

= 10V, = 10V, = 5V, = 5V,

= 5V = 5V = 10V = 10V

EE - 2009 6. The following circuit shown has a source voltage Vs as shown in the graph. The current through the circuit is also shown th

th

th

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GATE QUESTION BANK

Analog Circuits

a

10V

15V

15 10

(A) 4V (B) 5V

Vs (Volts)

5 0

EE - 2011 8. A clipper circuit is shown below.

-5

1k

-10 -15

0

100

200 300 Time (ms)

400

D

~

1.5 Current (mA)

(C) 7.5V (D) 12.12V

5V

1.0 0.5

Assuming forward voltage drops of the diodes to be 0.7 V, the input-output transfer characteristics of the circuit is

0 -0.5 -1.0 -1.5

(A) 0

100

200 300 Time (ms)

400

The element connected between a and b could be

4.3

a

4.3 a

a

(B) 10

a

4.3

EE - 2010 7. Assuming that the diodes in the given circuit are ideal, the voltage is

4.3

th

th

th

10

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GATE QUESTION BANK

Analog Circuits

dynamic resistance of the diode at room temperature is approximately,

(C)

5.7

1.7V 0.7 0.7 5.7

(A) (B) (D) 10

10

(C) (D)

IN - 2011 2. Assuming zener diode has currentvoltage characteristics as shown below on the right and forward voltage drop of diode is 0.7 V, the voltage in the circuit shown below is

EE - 2014 9. The sinusoidal ac source in the figure has an rms value of

√

. Considering all

possible values of , the minimum value of in to avoid burnout of the Zener diode is __________

I

2.7V

√

V

0.7V

~ (A) 3.7 V (B) 2.7 V

10.

Assuming the diodes to be ideal in the figure, for the output to be clipped, the input voltage must be outside the range

~ (A) (B)

(C) 2.2 V (D) 0 V

IN - 2014 3. For the circuit shown in the figure assume ideal diodes with zero forward resistance and zero forward voltage drop. The current through the diode in mA is ___________.

(C) (D)

IN - 2008 1. In the circuit shown below, the ideality factor of the diode is unity and the voltage drop across it is 0.7V. The th

th

th

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4.

Analog Circuits

For the circuit shown in the figure, the ra i r ha β =0.7 V, and the voltage across the Zener diode is 15 V.The current (in mA) through the Zener diode is ___________.

Answer Keys and Explanations ECE 1.

[Ans. B] When , zener diode is in forward bias so When , zener diode is in reverse bias, so when , zener diode will be OFF and . When , zener diode will be in breakdown region and .

2.

[Ans. C] Option – C, Circuit makes Full –wave rectifier.

3.

[Ans. C] For zener

When

4.

[Ans. C] When , is forward bias and are OFF, and When is +10V, are ON and zener diode is in reverse bias so V

5.

[Ans. A] When , diode is ON and V = When , diode is OFF and V = 1 V So V = min (

6.

[Ans. A]

7.

[Ans. B]

and

r

r

When

, current

~ th

th

r

th

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GATE QUESTION BANK

10.

Analog Circuits

[Ans. D]

i

8.

[Ans. B] Zener diode operated in reverse in breakdown region.

i i r Since diode will be forward biased voltage across diode will be 0.7 V i

Stabilized

9.

11.

[Ans. A]

[Ans. B] i

~

5v

40 mA

10 V

la per

ea

e i ier

10 mA

he ex i e y he la pi g section clamp the positive peak to 0 volts and negative peak to 2 volts. So whole cos ( i l wer y 1 volts

When zener starts will be held @ 5V. i

pu

When zener starts the current drawn from supply is 50 mA at 10 V p wer drawn = 500 mW This should be dissipates in circuit power i ipa e i Resistance = Remaining 250 mW has to be dissipated by zener assuming worst bad ze er h ul e ra e r

th

th

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GATE QUESTION BANK

12.

[Ans. D]

14.

Analog Circuits

[Ans. C]

1k y i tz

i

1k w

X

When Z is relatively positive to y Both D1& D2 are & l a Gets shorted.

When

positive maximum

i

When y is relatively positive to Z all Diode Biased urre ll w always. 13.

[Ans. D]

l

When

negative minimum

i

Assume both the diode ON. Then circuit will be as per figure (2)

15.

w p is off and hence

i le

i

[Ans. *] Range 0.08 to 0.12 For average steady state case, capacitor is open circuited S i

th

a pere

th

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GATE QUESTION BANK

16.

[Ans. D]

3.

Analog Circuits

[Ans. B] So current through Current through 1k ,

i

So

mA

mA

= 0.5893 Power dissipated in transistor i

i

r hi W

4.

i

[Ans. A] When diode is OFF, the equivalent circuit is shown as follows: i

i

EE 1.

i

r hi

[Ans. A] Let is ON, OFF and equivalent circuit

10 V

i

~ i

OFF, then

i.e. when is max i.e., so diode never conducts and it is always OFF. So, i

5A

In this case,

5.

10A

[Ans. D] When , is ON and charges upto 5V and When , is ON and charges by -10 V and So in steady state, and

A So voltage across diode So is in reverse bias i.e. it is OFF. Voltage across diode V it is also in Reverse bias so OFF. Voltage across diode it is in forward bias and ON.

2.

[Ans. A] When

, ,

and

is ON and

6.

is OFF so . is OFF, so

[Ans. A] It is

are OFF, so When

OFF so,

mA, and diode conducts , diode will be reverse bias and I will become zero.

th

th

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GATE QUESTION BANK

7.

Analog Circuits

Hence for output to be clipped input should not be lie inside this range

[Ans. B] Right side diode is off So ,

8.

9.

[Ans. C] When output will follow input, because zener diode and normal diodes are off When V Zener diode forward bias and V When V Diode is forward bias and

IN 1.

[Ans. B] Dynamic resistance r

r 2.

[Ans. C] First Assume that the diode does not reach reverse breakdown. So circuit becomes

[Ans. *] Range 299 to 301

i

10.

[Ans. B]

S S This is less than reverse breakdown voltage of diode . So our assumption is correct

i

i

3.

[Ans. 10] Diode goes to forward biased due to 10 V and 8V across diode and current through diode is

4.

[Ans. *] Range 40 to 43

i

i

Given circuit is union of both the above circuit for range

th

th

th

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Analog Circuits

β

Nodal at

th

th

th

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Analog Circuits

AC & DC Biasing-BJT and FET ECE - 2006 1. An n- channel depletion MOSFET has following two points on its curve: (i) d (ii) Which of the following Q-points will give the highest trans-conductance gain for small signals. (A) (B) (C) (D) ECE - 2007 2. The DC current gain ( ) of a BJT is 50. Assuming that the emitter injection efficiency is 0.995, the base transport factor is (A) 0.980 (C) 0.990 (B) 0.985 (D) 0.995 3.

For the BJT circuit shown, assume that the of the transistor is very large and = 0.7 V. The mode of operation of the BJT is

The value of current is approximately (A) 0.5 mA (C) 9.3 mA (B) 2 mA (D) 15 mA ECE - 2011 5. For a BJT, the common – base current g i α .98 d he c ec r b e junction reverse bias saturation current . . This BJT is connected in the common emitter mode and operated in the active region with a base drive current . The collector current for this mode of operation is (A) 0.98 mA (C) 1.0 mA (B) 0.99 mA (D) 1.01 mA 6.

10 kΩ

2V

9.

In the circuit shown below, for the MOS transistor, and the threshold voltage . The voltage at the source of the upper transistor is 6V

1 kΩ

5V

(A) cut-off (B) saturation

W/L = 4

(C) normal active (D) reverse active W/L = 1

ECE - 2010 4. In the silicon BJT circuit shown below, assume that the emitter area of transistor Q1 is half that of transistor Q2.

(A) 1 V (B) 2 V

th

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(C) 3 V (D) 3.67 V

th

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ECE - 2013 7. In the circuit shown below, the silicon npn transistor Q has a very high value of . The required value of R2 in k to produce IC = 1mA is

10.

Analog Circuits

For the MOSFETs shown in the figure, the threshold voltage | | = 2 V and ( )

.

. The value of

(in mA) is ______.

VCC 3V

R1 60 k

IC

Q R2

(A) 20 (B) 30 8.

RE 500 

(C) 40 (D) 50

11.

In a MOSFET operating in the saturation region, the channel length modulation effect causes (A) an increase in the gate – source capacitance (B) a decrease in the transconductance (C) a decrease in the unity –gain cutoff frequency (D) a decrease in the output resistance

ECE - 2014 9. For the n-channel MOS transistor shown in the figure, the threshold voltage is 0.8 V. Neglect channel length modulation effects. When the drain voltage =1.6 V, the drain current was found to be 0.5 mA. If is adjusted to be 2 V by changing the values of R and , the new value of (in mA) is

For the MOSFET shown in assume W/L = 2, = 2.0 V, 100 / and = transistor switches from region to linear region when is__________

the figure, 0.5 V. The saturation (in Volts)

EE - 2006 1. Consider the circuit shown in figure. If the of the transistor is 30 and ICBO is 20 nA and the input voltage is + 5 V, then transistor would be operating in +12V . Ω

Ω

Q Ω

(A) 0.625 (B) 0.75

(C) 1.125 (D) 1.5

12V

(A) saturation region (B) active region th

th

th

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Analog Circuits

(C) breakdown region (D) cut – off region 2.

A TTL NOT gate circuit is shown in fig. Assuming . of both the transistor, if . then the states of the two transistors will be, +5V

(A) 0 mA (B) 3.6 mA

.

(C) 4.3 mA (D) 5.7 mA

EE - 2010 5. The transistor circuit shown uses a silicon transistor with = 0.7V, and a dc current gain of 100. The value of is +10 V

(A) (B) (C) (D)

d e er e e er e d

d d e er e

50 k

10 k

EE - 2007 3. The common emitter forward current g i f he r i r h w i F = 100.

100 

+10V Ω

Ω

(A) 4.65V (B) 5V 6.

(C) 6.3V (D) 7.23V

Figure shows a composite switch consisting of a power transistor (BJT) in series with a diode. Assuming that the transistor switch and the diode are ideal , the I-V characteristic of the composite switch is

Ω

The transistor is operating in (A) Saturation region (B) Cutoff region (C) Reverse active region (D) Forward active region

V

+

I

EE - 2008 4. Two perfectly matched silicon transistors are connected as shown in the figure. Assuming the of the transistors to be very high and the forward voltage drop in diodes to be 0.7 V, the value of current I is

I

(B)

(A)

V

V

th

th

I

th

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GATE QUESTION BANK I

I

(D)

(C) V

V

IN - 2006 1. The biasing circuit of a silicon transistor is shown below. If  = 80, then what is VCE for the transistor? +12V

EE - 2011 7. The transistor used in the circuit shown be w h f d is negligible. 15k 1k

Analog Circuits

RB

RC

Ω

2.2k

Ω

8

D = 0.7V .

(A) 6.08 V (B) 0.2 V

(C) 1.2 V (D) 6.08 V

IN - 2007 2. In the circuit shown below,

.

.

If the forward voltage drop of diode is 0.7V, then the current through collector will be (A) 168 mA (C) 20.54 mA (B) 108 mA (D) 5.36 mA EE - 2014 8. The transistor in the given circuit should always be in active region. Take = 0.2 V, = 0.7 V. The maximum value of in . Which can be used, is__________

.

The f he r respectively, (A) 19 and 2.8 V (B) 19 and 4.7 V (C) 38 and 2.8 V (D) 38 and 4.7 V 3.

i

r

d

are,

The three transistors in the circuit shown below are identical, with = 0.7 V and .

th

th

th

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Analog Circuits

IN - 2014 6. In the figure, transistors T and T have identical characteristics. of transistor T is 0.1 V. The voltage is high enough to put T in saturation. Voltage of transistors T T and T is 0.7 V. The value of ( ) in V is ___________.

10 V Ω 2V Ω

9

The voltage (A) 0.2 V (B) 2 V

T

is (C) 7.4 V (D) 10 V

IN - 2010 4. The matched transistors and shown in the adjoining figure have =100. Assuming the base-emitter voltages to be 0.7V, the collector-emitter voltage V2 of the transistor is

(A) 33.9V (B) 27.8V

T

T

(C) 16.2V (D) 0.7V

IN - 2013 5. In the transistor circuit as shown below, the value of resistance RE in k is approximately, +10V

1.5k 15k .

(A) 1.0 (B) 1.5

6k

IC=2.0 mA . VCE=5.0 V RE

Vout

(C) 2.0 (D) 2.5

th

th

th

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GATE QUESTION BANK

Analog Circuits

Answer Keys and Explanations ECE 1.

5.

[Ans. D] α

[Ans. D] From the below graph it is clear that a increase conductance i.e. slope of graph is increased.

α .98 .98 . 6.

9 .

[Ans. C] The transistor which has

12mA

and

6 V

So that transistor in saturation region. The transistor which has

2.

Transfer characteristic of n-channel DMOSFET.

Drain is connected to gate So that transistor in saturation

[Ans. B]

The current flow in both the transistor is same

α

( ) (

α base transport factor × emitter injection efficiency b e r

r f c r

( ) (

)

.99 .98

3.

)

[Ans. B] Given is large so d Assuming BJT is in active Applying KVL in Base. Emitter loop . . . 9.8 w .9

8

7.

[Ans. C]

.

So BJT is in saturation 4.

[Ans. B] Assuming . .

~

d

.

i ce is very large

=1mA

So,

eg igib e

So, = 2.04 mA

ge dr

2 mA th

th

cr th

e i er f

re i

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r

GATE QUESTION BANK

11. . w

Analog Circuits

[Ans. *] Range 1.4 to 1.6 Initially the transistor is in saturation [

. .

.

( )]

.

.

w

r

gi e circ i

r .

.

r .

b i

.

i g

i

. .

. .

.8 .

8.

[Ans. D]

9.

[Ans. C] ere

EE 1. i

r i

.

[Ans. B] Let the device is in saturation and . V

regi

.

Collector current .

.

.8

For saturation

.

Base voltage . current through 100

. .

.8

. .

. .

10.

. .

= 5.3mA

= 0.178 mA re i

ce

.

= 0.127 mA c rre hr .

.

[Ans. *] Range 0.88 to 0.92

gh

re i

ce

= 0.29 mA Base current

. 9 . = 0.163 mA So this is less than it means device is not in saturation. It is in active region.

NMOS is in saturation region . . .9

2.

[Ans. C] When . he will be in reverse active mode .i.e. Reverse on and will be ON.

3.

[Ans. D] We assume BJT is in active region applying KVL in base emitter circuit . i g th

th

th

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8.

[Ans. *] Range 22 to 23 To calculate max, take end condition . i . . i

9. 9 .

Ti i 4.

Analog Circuits

.9

i

ci e c i e regi

. .

. .8

[Ans. B]

.

. IN 1.

[Ans. B] .

. 9 9 . and drop across diode = 0.7 . . . i

So device is not active region. Let the device is in saturation and . V . .9

ery rge

As both transistors are perfectly matched and . of both transistors. . mA 5.

.9 8

. .

[Ans. A]

So device is in saturation, so

.

. 9.

2.

. .

[Ans. A]

.

6.

[Ans. C] When V will be + ve both transistor and diode will be on making V across them Zero and Current I will be flow and when V is -ve both will be off offering infinite resistance so current I will be Zero.

7.

[Ans. D] Transistor is in Saturation region . . .

.

. . th

th

th

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Apply KVL .

Analog Circuits

. 8 For matched transistor .

.

di is same for both the transistors) Now apply KVL in the outer port of transistor . 8 .8

9 9. Apply KVL to output terminals 5. .9 3.

9. .

[Ans. A] 10 V

. .8

2m A

.

Ω

V=IR=1.5k × 2mA=3V

15k

[Ans. C]

=10-3=7V

10V

gi e

2V 6k

gi e

Using KVL in the left hand transistor, .

Ω

be .

6.

gi e

[Ans. *] Range 5.5 to 5.8 9

Now again apply KVL to the collector resistor of the right and transistor T .

. .

.

. 4.

[Ans. B] rre hr .

T

T .

gh

re i

.

ce

.

i

Apply KVL on path 1 9 . .

. .

=1.66mA

. th

th

th

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Tr

i

. r ide ic

y

h .

Analog Circuits

– . . .98

. .

th

th

th

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Analog Circuits

Small Signal Modeling of BJT and FET ECE - 2006 Common Data Questions 1, 2, 3 In the transistor amplifier circuit shown in the figure below, the transistor has the following parameters: h h The capacitance can be assumed to be infinite.

defined to be

2.

3.

In the figure above, the ground has been shown by the symbol Under the DC conditions, the collector-toemitter voltage drop is (A) 4.8 Volts (C) 6.0Volts (B) 5.3 Volts (D) 6.6Volts

ECE - 2008 4. Two identical NMOS transistors M1 and M2 are connected as shown below. is chosen so that both transistors are in saturation.The equivalent g of the Pair is

of the of the

Statement for Linked Answer Questions 5 and 6 In the following transistor circuit,

If is increased by 10%, then collectorto-emitter voltage drop (A) increases by less than or equal to 10% (B) decreases by less than or equal to 10% (C) increases by more than 10% (D) decreases by more than 10% The small-signal gain of the amplifier ⁄ is (A) 10 (C) 5.3 (B) 5.3 (D) 10

.

The equivalent g of the pair is (A) the sum of individual g transistors (B) the product of individual g transistors (C) nearly equal to the g of M1 (D) nearly equal to g ⁄g of M2

~

1.

at constant

, and

and all the

capacitances are very large.

5.

The value of DC current is (A) 1mA (C) 5mA (B) 2mA (D) 10mA

6.

The mid-band voltage gain amplifier is approximately (A) 180 (C) 90 (B) 120 (D) 60 th

th

th

of

the

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ECE - 2009 7. A small signal source () A +B i is applied to a transistor amplifier as shown below. The transistor has =150 and h =3k .Which expression best approximates ( )?

(B) The input resistance decreases and the magnitude of voltage gain increases (C) Both input resistance and the magnitude of voltage gain decrease (D) Both input resistance and the magnitude of voltage gain increase

12V

3k

100k

V0(t) 100 nF Vi(t) 100nF 20k

(A) (B) (C) (D)

( ( ( (

)= )= )= )=

μF

900k 

ECE - 2012 9. The current ib through the base of a silicon npn transistor is 1 + 0.1 cos(10000 t) mA . At 300 K, the r in the small signal model of the transistor is i

(

)

i

(

Analog Circuits

i

)

i i

ECE - 2010 8. The amplifier circuit shown below uses a silicon transistor. The capacitors and can be assumed to be short at signal frequency and the effect of output resistance can be ignored. If is disconnected from the circuit, which one of the following statements is TRUE?

(A) (B)

Ω Ω

(C) (D)

Ω Ω

ECE - 2013 10. The small-signal resistance(i.e., dVB/dID) in k offered by the n – channel MOSFET M shown in the figure below, at a bias point of VB =2V is (device data for M: device transconductance parameter 2 kn μnCox(W/L) 4 μ / , threshold voltage VTN=1V and neglect body effect and channel length modulation effects) ID

VB

~

M

(A) The input resistance increases and the magnitude of voltage gain decreases

(A) 12.5 (B) 25 th

th

(C) 50 (D) 100 th

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Analog Circuits

ECE - 2014 11. For the amplifier shown in the figure, the BJT parameters are and thermal voltage = 25 mV. The voltage gain ( / ) of the amplifier is _____ .

(A) g (B) μF

μF

F

12.

(C) g (D)

14.

A BJT in a common-base configuration is used to amplify a signal received by a 50 antenna. Assume kT/q = 25 mV. The value of the collector bias current (in mA) required to match the input impedance of the amplifier to the impedance of the antenna is __________

15.

For the common collector amplifier h w i he figu e he JT h high negligible = 0.7 V. The ( ) and maximum undistorted peak-to-peak output voltage (in Volts) is

In the circuit shown, the PNP transistor has = 0.7 Vand = 50. Assume that = 100 k . For to be 5 V, the value of (in k .) is __________

μF μF

13.

Consider the common-collector amplifier in the figure (bias circuitry ensures that the transistor operates in forward active region, but has been omitted for simplicity). Let be the collector current, be the base-emitter voltage and be the thermal voltage. Also, g are the small-signal transconductance and output resistance of the transistor, respectively. Which one of the following conditions ensures a nearly constant small signal voltage gain for a wide range of values of ? th

th

th

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GATE QUESTION BANK

ECE/EE/IN - 2012 1. The voltage gain below is

(A) 0.967 (B) 0.976

of the circuit shown

~

(A) (B)

(C) (D)

EE - 2014 2. The magnitude of the mid-band voltage gain of the circuit shown in figure is (assuming h of the transistor to be 100)

Analog Circuits

(C) 0.983 (D) 0.998

IN - 2008 2. For a single stage BJT common base amplifier (A) current gain as well as voltage gain can be greater than unity (B) current gain can be greater than unity but voltage gain is always less than unity (C) voltage gain can be greater than unity but current gain is always less than unity (D) current gain as well as voltage gain is always less than unity 3.

In the amplifier circuit shown below, assume VBE he f he transistor and the values of C1 and C2 are extremely high. If the amplifier is designed such that at the quiescent point

h

its VCE =

~

where VCC is the power supply

voltage, its small signal voltage gain |

|

will be

(A) 1 (B) 10

(C) 20 (D) 100

IN - 2006 1. An amplifier circuit is shown below. Assume that the transistor works in active region. The low frequency smallsignal parameters for the transistor are g = 20 mS, 0 = 50, =, =0. What is the voltage gain,

( ) of the

amplifier? (A) 3.75 (B) 4.5

+Vcc

Ω

Vi

~

Ω

(C) 9 (D) 19

IN - 2009 Common Data Questions 4 and 5 The circuit shown in the figure uses three identical transistors with = 0.7V and Gi e : = = Ω th

th

th

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GATE QUESTION BANK

kT/ = 25mV. The collector current of transistor 2mA.

4.

The bias voltage at the base of the transistor is approximately (A) 9.3V (C) 10.3V (B) 10.0V (D) 11.0V The small signal voltage gain of the circuit is (A) 20 (C) 20 (B) 40 (D) 40

is

+12V

5. R1

R2

Q1

Analog Circuits

Q2 Q3 R3 12V

Answer Keys and Explanations ECE 1.

So, [Ans. C] Under DC conditions capacitor as open. By KVL,

)

g

will act

( ) Which is equal to the g of

but μ

(

.

[Ans. A] Equivalent circuit

5.

= 6V 2.

[Ans. B] in increased by 10%, i.e. By KVL,

4μ i e ge By KVL, in E – B loop, e e e 3.

[Ans. A]

4.

[Ans. C] Both transistor carry same current and both are in saturation

[Ans. D]

6.

Voltage gain u

th

i

th

e

(

)(

)

ge

th

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GATE QUESTION BANK

11.

Analog Circuits

[Ans. *] Range – 240 to – 230 From the model i

i

(

7.

[Ans. B] The e is

i

e

we f

u u

) e

i

ge

h h (

)

i

For the calculation of gain we need to calculate . so DC analysis is required

8.

[Ans. A] The moment is disconnected from the circuit, h ( h ) but with capacitor, h which also reduces voltage gain So, increases and decreases.

9.

[Ans. C]

g

i = 1 + 0.1 cos ( Calculate Solution: [

) = 3V

whe e

i he he

ge

e u e

44

]

4

10.

[Ans. B] μ L

(

) μ

(

L μ 4 4

L

) i

(

) (

i

)

(

i

4 th

th

th

)i

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GATE QUESTION BANK

Analog Circuits

i i i i

μ

g

i

e

i i

i

e

4 4 he e e figu e i i ( ) i i ( )i ( ) ) ( i ( ) i ( ) i

f

4 4

e

Now assuming that collector current g emitter terminal. g

is very large, the flows through the

g

4 4

g g

12.

So if g constant g

[Ans. *] Range 1.04 to 1.12 (

)

(

)

which is nearly

14.

[Ans. *] Range 0.49 to 0.51 For CB configuration

15.

[Ans. *] Range 9.39 to 9.41

4 4 13.

[Ans. B] This is a common-collector configuration.

i e eg i e can vary ( )i between 12V in the positive direction and 0V in the negative direction biased around So, positive excursion of output voltage = 4 Negative excursion of output voltage

i u

` Replacing the transistor equivalent model, we get

with

T

–

So, undistorted peak to peak output voltage 4 4 th

th

th

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GATE QUESTION BANK

Analog Circuits

Apply miller’s theorem ge i

u e i i

i e eg i e can vary ( )i between 12V in the positive direction and 0V in the negative direction biased around So, positive excursion of output voltage = 4 Negative excursion of output voltage

i

i ( ( ) i i ( ( ) i

[Ans. D] In DC Analysis capacitor will behave as an open circuit

i

(

)

whe e

So, undistorted peak to peak output voltage 4 4 EE 1.

i

~

4

) ) 4

(

(

)

4

Consider input section

i

~

i i

4

i

(i i

i )

i

)i

i

i (

i

(

i i i

i (

(

)

4

(

)

) 2.

)i

[Ans. D] g

whe e

g Small signal analysis of circuit

i g and Q point to be in the middle of the load line

i

~

(

)

i

g

th

th

th

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)

GATE QUESTION BANK

IN 1.

Analog Circuits

By Voltage division rule [Ans. D] Apply KVL to the base emitter circuit

Ω ~

Ω

Vi

w

AC equivalent circuit for

L

he

e

e i e

i ui

model

i

i

4 g

G i

~

|

|

4 | |

①

4.

[Ans. A] Emitter

voltage

of

transistor,

g L i Where i i i

10 V

① i i

(i i (i (

i

g

)

So i )

g

5.

) i ( (i [ 2.

3.

[Ans. B] When output is taken between one collector and ground then

)i )

g i i i

g

(i e

i g

e

e g i )

and if take the output between two collectors then differential gain. g collector resistance

]

[Ans. C] For CB, current gain is close to unity while voltage gain is very high.

g

/ and here output is taken between one collector and ground therefore gain g

[Ans. A]

4

th

th

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GATE QUESTION BANK

Analog Circuits

BJT and JFET Frequency Response ECE - 2010 Common Data for Questions 1 and 2 Consider the common emitter amplifier shown below with the following circuit parameters: g A/V, ,

RD 10 k

C V0

Vi

M RL 10 k

and (A) 8 (B) 32

(C) 50 (D) 200

ECE - 2014 4.

1.

The resistance seen by the source v is (A) (C) (B) (D)

2.

The lower cut off frequency due to (A) 33.9 Hz (C) 13.6 Hz (B) 27.1 Hz (D) 16.9 Hz

A cascade connection of two voltage amplifiers and is shown in the figure. The open-loop gain , input resistance , and output resistance for and are as follows:

The approximate overall voltage gain / is __________. is

ECE - 2013 3. The ac schematic of an NMOS common – source stage is shown in the figure below, where part of the biasing circuits has been omitted for simplicity. For the nchannel MOSFET M, the transconductance gm =1mA/V, and body effect and channel length modulation effect are to be neglected. The lower cutoff frequency in Hz of the circuit is approximately at

IN - 2011 1. The amplifier shown below has a voltage gain of an input esistance of and a lower 3-dB cut-off frequency of 20 Hz. Which one of the following statements is TRUE when the emitter resistance is doubled?

th

th

th

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GATE QUESTION BANK

Analog Circuits

v

v

~

(A) Magnitude of voltage gain will decrease (B) Input resistance will decrease (C) Collector bias current will increase (D) Lower 3-dB cut-off frequency will increase

th

th

th

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GATE QUESTION BANK

Analog Circuits

Answer Keys and Explanations ECE 1.

[Ans. B] Equivalent model of the given circuit is shown below

h

~

[ IN 1.

h

]

[Ans. A] The small signal model is

g The resistance seen by the source h

h

2.

h

[Ans. B] Lower cut-off frequency due to

h

f h increases

3.

c

w

ec eases ec ease

f f 4.

doubled Zi

ou e ec eases owe frequency decreases as increases s inc eases

[Ans. A] w

h

ec eases

≅8

[Ans. *] Range 34 to 34.72

p ifie

p ifie

th

th

th

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GATE QUESTION BANK

Analog Circuits

Feedback and Oscillator Circuits ECE - 2006 1. The input impedance ( ) and the output ( ) impedance of an ideal transconductance (voltage controlled current source) amplifier are (A) (C) (B) (D)

ECE - 2011 4. In the circuit shown below, capacitors and are very large and are shorts at the input frequency. is a small signal input. The gain magnitude | ⁄ | at 10 Mrad/s is 5V 2k𝛀

ECE - 2007 2. In a transconductance amplifier, it is desirable to have (A) A large input resistance and a large output resistance (B) A large input resistance and a small output resistance (C) A small input resistance and a large output resistance (D) A small input resistance and a small output resistance ECE - 2009 3. In the circuit shown below, the op-amp is ideal the transistor has Decide whether the feedback in the circuit is positive or negative and determine the voltage V at the output of the op-amp

1nF

μH

2k𝛀

2.7V 2k𝛀 ~

(A) Maximum (B) Minimum

(C) Unity (D) Zero

ECE - 2014 5. In the ac equivalent circuit shown in the figure, if is the input current and is very large, the type of feedback is

10V

V

(A) (B) (C) (D)

+

(A) voltage-voltage feedback (B) voltage-current feedback (C) current-voltage feedback (D) current-current feedback

Positive Feedback V=10V Positive Feedback V= 0V Positive Feedback V=5V Positive Feedback V=2V th

th

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GATE QUESTION BANK

6.

The feedback topology in the amplifier circuit ( the base bias circuit is not shown for simplicity) in the figure is

Analog Circuits

(A) 5

10 (B) 10 5

~ (C)

(A) Voltage shunt feedback (B) Current series feedback (C) Current shunt feedback (D) Voltage series feedback 7.

5

10

The desirable characteristics of a transconductance amplifier are (A) high input resistance and high output resistance (B) high input resistance and low output resistance (C) low input resistance and high output resistance (D) low input resistance and low output resistance

EE - 2006 1. A relaxation oscillator is made using OPAMP as shown in figure. The supply voltages of the OPAMP are 12V. The voltage waveform at point P will be

(D)

10

5

EE - 2009 2. The nature of feedback in the op-amp circuit shown is +6V 2

K

R1 Vin C

6V

R2

(A) (B) (C) (D)

_ + 2kΩ

P 10kΩ

~

Current - Current feedback Voltage - Voltage feedback Current - Voltage feedback Voltage - Current feedback

10kΩ

th

th

th

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GATE QUESTION BANK

Analog Circuits

EE - 2014 3. In the Wien Bridge oscillator circuit shown in figure, the bridge is balanced when μF

If the lower and upper trigger level voltages are 0.9 V and 1.7 V, the period (in ms), for which output is LOW, is_________

( )

IN - 2007 1. A FET source follower is shown in the figure below:

√

+ 15 V

( ) ( )

μF

√

( ) 4.

1

An oscillator circuit using ideal op-amp and diodes is shown in the figure.

The nature of feedback in this circuit is (A) Positive current (B) Negative current (C) Positive voltage (D) Negative voltage ECE/IN - 2013 2. In the feedback network shown below , if the feedback factor k is increased , then the The time duration for +ve part of the cycle is and for ve part is . The value of 5.

Vin

V1

A0

vf=kvout

k

Vout

will be _______________

A hysteresis type TTL inverter is used to realize an oscillator in the circuit shown in the figure.

(A) Input impedance increases and output impedance decreases th

th

th

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GATE QUESTION BANK

Analog Circuits

(B) Input impedance increases and output impedance also increases (C) Input impedance decreases and output impedance also decreases (D) Input impedance decreases and output impedance increases

Answer Keys and Explanations ECE 1.

6.

[Ans. B] Current Series O/P is taken at collector which is voltage emitter node will be current and resistor is grounded Series

7.

[Ans. A]

[Ans. D] Ideal transconductance amplifier has infinite input and output resistance.

2.

[Ans. A]

3.

[Ans. D]

EE 1. 2

[Ans. A] Output will be either at . When output will be at diode connected to resistance will be on making voltage at point P equal to 6V. When output is at diode connected to 2 resistance will be on making voltage at point equal to .

2

( Refer Diagram below)

5v V +

-

+ -

[Ans. B] It is voltage – voltage feedback.

3.

[Ans. C] Barkhausen criteria for oscillation (for positive feedback)

+

+

4.

2.

[Ans. A] In the parallel RLC Ckt μHand F √

√

A = open loop gain B = feedback factor =

So that for a tuned amplifier, gain is maximum at resonant frequency 5.

(

)

(

)

||

[Ans. B]

th

th

th

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GATE QUESTION BANK

Analog Circuits

2

(

)(

)

(

2

Real and imaginary part on LHS and RHS are same so imaginary part = 0

√

For (+Ve) cycle capacitor charges from 2 2 Apply KCL at node ( ) 4.

[Ans. *] Range 1.2 to 1.3

(

(

)

) 2

( 2

~

) )

( (

)

(

)

(

)

2

2 2 2

2 [

]

For (- Ve) cycle capacitor discharges from 2 2 Apply KCL at node ( ) (

( K th

th

)

)

2 ( 2 ) th

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GATE QUESTION BANK

( 2 ) (

2

(

IN 1.

)

[Ans. D] Feedback in this circuit is drain voltage as negative.

) 2 (

2

2.

)

2

[Ans. A] Given connection is voltage series F/B. increases and decreases

2 ] 2

[

Analog Circuits

y q 2 2 [ 5.

(

)

2 ]

[Ans. *] Range 0.62 to 0.66

During the charging of capacitor () ( ) ( ( ) ( )) (

)

() During discharging of capacitor ()

( )

()

(

(

( ( )

( ))

)

) (

)

th

th

th

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GATE QUESTION BANK

Analog Circuits

Operational Amplifiers and Its Applications ECE - 2006 1. For the circuit shown in the following figure, the capacitor C is initially uncharged. At t , the switch S is closed. The voltage across the capacitor at t millisecond is

ECE - 2007 4. For the Op-amp circuit shown in the figure, is 2 kΩ

1 kΩ 1V 1 kΩ

1 kΩ

k

In the figure shown above, the OP AMP is supplied with ±15V. (A) 0 Volts (C) 9.45 Volts (B) 6.3 Volts (D) 10 Volts

(A) (B) 5.

KΩ

k

(A) (B) (C) (D)

k

k

2.

3.

(C) 0.5 V (D) 0.5 V

In the Op-amp circuit shown, assume that the diode current follows the equation I = exp (V/ ). For = 2 V, = and for = 4 V, = . The relationship between and is

Statement for Linked Answer Questions 2 and 3 A regulated power supply, shown in figure below, has an unregulated input (UR) of 15 Volts and generates a regulated output . Use the component values shown in the figure. k

2V 1V

=√ =e = ln 2 = ln 2

Statement for Linked Answer Questions 6 and 7 Consider the Op-Amp circuit shown in the figure.

In the figure above, the ground has been shown by the symbol The Power dissipation across the transistor Q1, shown in the figure is (A) 4.8 Watts (C) 5.4 Watts (B) 5.0 Watts (D) 6.0 Watts If the unregulated voltage increases by 20%, then power dissipation across the transistor Q1 (A) Increases by 20% (B) Increases by 50% (C) Remains unchanged (D) Decreases by 20% th

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GATE QUESTION BANK

6.

7.

The transfer function

(s)/

(A)

(C)

(B)

(D)

(s) is

10.

Analog Circuits

Consider the Schmidt trigger circuit shown below k

If

sin t and sin t , then the minimum and maximum values of (in radians) are respectively (A) –π/ and π/ (C) –π and (B) and π/ (D) –π/ and

k k

A triangular wave which goes from 12V to 12V is applied to the inverting input of the OPAMP. Assume that the output of the OPAMP swings from +15V to 15V.The voltage at the non-inverting input switches between (A) 12V and +12 V (B) 7.5V and 7.5V (C) 5V and +5V (D) 0V and 5V

ECE - 2008 8. Consider the following circuit using an ideal OPAMP. The I-V characteristics of the diode is described by the relation I= (e

) where m and V is the voltage across the diode (taken as positive for forward bias) D

4k

= 1 100k

ECE - 2009 11. In the following astable multivibrator circuit, which properties of t depend on ?

For an input voltage the output voltage is (A) 0V (C) 0.7V (B) 0.1V (D) 1.1V 9.

The OPAMP circuit shown below represents a

(A) (B) (C) (D)

(A) (B) (C) (D)

High pass filter Low pass filter Band pass filter Band reject filter

th

Only the frequency Only the amplitude Both the amplitude and frequency Neither the amplitude nor the frequency

th

th

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GATE QUESTION BANK

Analog Circuits

ECE - 2010 12. Assuming the OP AMP to be ideal, the voltage gain of the amplifier shown below is

13.

(A)

(C)

(

)

(B)

(D)

(

)

The transfer characteristic for the precision rectifier circuit shown below is (assume ideal OP AMP and practical diodes)

ECE - 2011 14. The circuit below implements a filter between the input current i and the output voltage v . Assume that the opamp is ideal. The filter implemented is a

i

(A) (B) (C) (D)

low pass filter band pass filter band stop filter high pass filter

ECE/EE/IN - 2013 15. In the circuit shown below what is the output voltage (Vout) if a silicon transistor Q and an ideal op – amp are used?

th

th

th

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GATE QUESTION BANK +15 V

Analog Circuits

Q

1 k k

k Vout .7

5V

(A) (B) 16.

15 V

15 V 0.7 V

(C) +0.7 V (D) +15 V

19.

In the circuit shown below the op – amps are ideal. The Vout in Volts is 1 k

1 k +15 V

+15 V

Vout

1 k -15 V

+1V

-15 V

(A) (B)

1 k

1 k

20. (A) 4 (B) 6

(C) 8 (D) 10

ECE - 2014 17. In the low-pass filter shown in the figure, for a cut-off frequency of 5 kHz , the value of (in k ) is ________

k

18.

In the circuit shown, the op-amp has finite input impedance, infinite voltage gain and zero input offset voltage. The output voltage is

(C) (D)

In the differential amplifier shown in the figure, the magnitudes of the commonmode and differential-mode gains are and , respectively. If the resistance is increased, then

n

In the voltage regulator circuit shown in the figure, the op-amp is ideal. The BJT has .7 and β and the zener voltage is 4.7 V. For a regulated output of 9V, the value of R (in Ω is _________

(A) increases (B) common-mode rejection ratio increases (C) increases (D) common-mode rejection ratio decreases

th

th

th

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GATE QUESTION BANK

21.

Assuming that the Op-amp in the circuit shown is ideal, is given by

Analog Circuits

(A)

(B)

22.

(A)

(C)

(B)

(D)

The circuit shown represents (C)

12V 0.7V

(A) a bandpass filter (B) a voltage controlled oscillator (C) an amplitude modulator (D) a monostable multivibrator

(D)

EE -2006 1. For a given sinusoidal input voltage, the voltage waveform at point P of the clamper circuit shown in figure will be

0.7V 12V

EE -2007 2. The circuit shown in the figure is

~ oad r

Vin

(A) A voltage source with voltage (B) A voltage source with voltage (C) A

current

source

with

current

.

th

th

th

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GATE QUESTION BANK

(D) A

current

source

with

current

Analog Circuits

(D)

. 3.

IC 555 in the adjacent figure is configured as an astable multivibrator. It is enabled to oscillate at t =0 by applying a high input to pin 4. The pin description is: 1 and 8 – supply; 2- trigger; 4- reset; 6 – threshold; 7 – discharge. The waveform appearing across the capacitor starting from t = 0, as observed on a storage CRO is

4.

The switch S in the circuit of the figure is initially closed. It is opened at time t=0. You may neglect the Zener diode forward voltage drops. What is the behaviour of for t ? +10V

+

10K

+10 V

k

8 7

+ IC 555

10K

S

10V

0.01

2, 6 C

4

5.0V

k

3

5.0V

k 1 10V

(A) It

(A)

makes a to at t makes a to at t makes a to at t makes a to at t

(B) It (C) It (D) It (B)

transition . s transition . 7 s transition . s transition . 7 s

from from from from

EE -2008 5. The block diagrams of two types of half wave rectifiers are shown in the figure. The transfer characteristics of the rectifiers are also shown within the block. P

(C)

Q

V0

V0

0

Vin

V0

Vin

V0

Vin 0

Vin

It is desired to make full wave rectifier using above two half – wave rectifiers. The resultant circuit will be

th

th

th

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GATE QUESTION BANK

Analog Circuits

(A) 5V 2.5V t(sec) (B) t(sec)

0 2.5V 5V (C) 5V 0

t(sec)

5V (D) 5V 0 5V

t(sec)

Statement for Linked Answer Questions 7 and 8 A general filter circuit is shown in the figure: 6.

A waveform generator circuit using OPAMPs is shown in the figure. It produces a triangle wave at point ‘ ’ with a peak to peak voltage of 5V for = 0 V.

If the voltage is made + 2.5 V, the voltage waveform at point ‘ ’ will become

7.

If R1 = R2 = RA and R3 = R4 = RB, the circuit acts as a (A) all pass filter (C) high pass filter (B) band pass filter (D) low pass filter

8.

The output of the filter in Q.7 is given to the circuit shown in below figure

th

th

th

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GATE QUESTION BANK

Analog Circuits

/ k z

i

rms

pamp

The gain vs frequency characteristic of output (Vo) will be (A)

k

Gain

0

~

(A) πm (B) πm (C) 1 π m (D) πm

(𝛚)

leading by 0 leading by 0 leading by 900 lagging by 0

(B) 10.

Gain

0

An ideal op-amp circuit and its input waveform are shown in the figures. The output waveform of this circuit will be

(𝛚)

(C)

Gain 6V k

0

(𝛚) +

(D)

3V

k

k

Gain

0

(𝛚) t

EE -2009 9. The following circuit has k C = 10 F . The input voltage is a sinusoid at 50Hz with an rms value of 10V. Under ideal conditions, the current is from the source is th

th

t

th

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GATE QUESTION BANK

Analog Circuits

(A)

t

t

(B) t t

t

t

t

t

t t

(C) EE -2010 11. Given that the op-amp is ideal, the output voltage is

(D)

(A) 4V (B) 6V

(C) 7.5V (D) 12.12V

EE -2011 12. For the circuit shown below,

13.

the CORRECT transfer characteristic is

th

A-Low pass filter with a cut-off frequency of 30Hz is cascaded with a high pass filter with a cut-off frequency of 20Hz. The resultant system of filter will function as , (A) An all-pass filter (B) An all-Stop filter (C) A band stop (band -reject) filter (D) An band-pass filter

th

th

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GATE QUESTION BANK

Analog Circuits

ECE/EE/IN -2012 14. The circuit shown is a (|

|)

utput

ntput

𝛚

(A) Low pass filter with f

rad/s

(B) High pass filter with f

rad/s

(C) Low pass filter with f

rad/s

(D) High pass filter with f

rad/s

𝛚

/ /

EE -2014 15. Given that the op-amps in the figure are ideal, the output voltage is

(|

|)

𝛚

(A) (B) 16.

(C) (D)

/

In the figure shown, assume the op-amp to be ideal. Which of the alternatives gives the correct Bode plots for the transfer function

?

/ /

k

𝛚

/ /

th

th

th

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17. (|

Analog Circuits

An operational-amplifier circuit is shown in the figure.

|)

𝛚

The output of the circuit for a given input is

/ /

( 𝛚

/

)

(

)

/

(

) or

(|

18.

|)

The transfer characteristic of the Op-amp circuit shown in figure is

𝛚

/ /

/

𝛚

/

th

th

th

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GATE QUESTION BANK

2.

Analog Circuits

When the switch S2 is closed the gain of the programmable gain amplifier shown in the following Figure is

k k

(A) 0.5 (B) 2

IN - 2006 1. If the value of the resistance R in the following figure is increased by 50%, then voltage gain of the amplifier shown in the figure will change by

(C) 4 (D) 8

3.

The potential difference between the input terminals of an op-amp may be treated to be nearly zero, if (A) The two supply voltages are balanced (B) The output voltage is not saturated (C) The op amp is used in a circuit having negative feedback (D) There is a dc bias path between each of the input terminals and the circuit ground

4.

A dual op-amp instrumentation amplifier is shown below. The expression for the output of the amplifier is given by

k k Vin

R

(A) 50% (B) 5% (C) 50% (D) negligible amount



(A) v0  1 



th

th

R2   (v  v 1 ) R1  2

th

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GATE QUESTION BANK



(B) v0  1 

2R2   (v  v 1 ) R1  2

8.

In the circuit shown in the following figure , the op-amp has input bias current n and input offset voltage m . The maximum dc error in the output voltage is

 2R2 (C) v0  (v  v 1 ) R1 2



(D) v0  1 



Analog Circuits

2R1   (v  v 1 ) R 2  2

k k

5.

An astable multivibrator circuit using a 555 IC is given in the following figure. The frequency of oscillation is 8

I=5 mA

4

+

3

(A) 1.0 mV (B) 2.0 mV

Output 6 2

555 discharge

7

(C= 0.1 uF)

1

(A) 20 kHz (B) 30 kHz

(C) 2.5mV (D) 3.0 mV

(C) 40 kHz (D) 45 kHz

IN - 2007 9. When light falls on the photodiode shown in the following circuit, the reverse saturation current of the photodiode changes from to . k

Statement for linked answer questions 6 and 7 In the Schmitt trigger circuit shown below, the Zener diodes have VZ (reverse saturation voltage) = 6V and VD (forward voltage drop) = 0.7V

.

6.

If the circuit has the input lower trip point (LTP)=0V, then the value of (A) 0.223 (B) 2.67

Assuming the op – amp to be ideal, the output voltage, of the circuit. (A) does not change (B) changes from 1 V to 2V (C) changes from 2 V to 1 V (D) changes from 1 V to 2V 10.

is given as

Consider the linear circuit with and ideal op-amp shown in the figure below. 1

(C) 4.67 (D) ∞

2

’

’

kΩ

7.

The input upper trip point (UTP) of the Schmitt trigger is (A) 1.5 V (C) 2.42V (B) 2.1 V (D) 7V

+

Vi

Vo

The Z-parameters of the two port feedback network are kΩ th

th

th

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GATE QUESTION BANK

and = amplifier is (A) + 110 (B) + 11 11.

kΩ. The gain of the

Analog Circuits

(D) o/ i 4 dB

(C) 1 (D) – 120

1dB 20 dB/decade

Consider the circuit shown below

f 330pF 22k Ω

21.9kHz

22k Ω 330pF

12.

10k Ω

In the circuit shown below the switch (S) is closed whenever the input voltage is positive and open otherwise.

7kΩ

R

R

The correct frequency response of the circuit is (A)

. Vin

o/ i 4 dB

The circuit is a (A) Low pass filter (B) Level shifter (C) Modulator (D) Precision rectifier

1 dB 20dB/decade f

21. 9kHz

(B)

13.

/ 0 dB 3 dB

Consider the triangular wave generator shown below.

40dB/decade R kΩ

Input

1 kΩ 10 kΩ

f

21. 9kHz

(C)

o/ i

Assume that the op amps are ideal and have ± 2 V power supply. If the input is a ± 5 V, 50 Hz square wave of duty cycle 50%, the condition that results in a triangular wave of peak to peak amplitude 5 V and frequency 50 Hz at the output is (A) RC = 1 (C)

4 dB 1 dB 40 dB/decade

21. 9kHz

Output

f

(B)

th

(D)

th

th

5

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GATE QUESTION BANK

14.

A 555 astable multivibrator circuit is shown in the figure below

Analog Circuits

Noise

Square Wave

7V 8V

KΩ

3V RA

KΩ

RB

output

-7V

threshold

Which one of the following is an appropriate choice for the upper and lower trip points of the Schmitt trigger to recover a square wave of the same frequency from the corrupted input signal (A) ± 8.0 V (C) ± 0.5 V (B) ± 2.0 V (D) 0V

ground

is shorted, the waveform at

is

(A) VC 2/3 VCC cc cc 1/3VCC 0

16.

t

(B)

V ccC 2/3 VCC cc cc 1/3 VCC

cc

The figure shows a signal op-amp differential amplifier circuit . k

k

0

(C)

t

-3V

̅̅̅̅̅̅̅̅̅ tr gger

If

6V

1V -1V

reset discharge

t

cc VC

m

+

+ +

k k

cc

m

cc

2/3 VCC

Which of the following statement about the output is correct? (A) m (B) m m (C) m v m (D) m

1/3 VCC 0

t

(D) VC cc 2/3 VCC 1/3 VCC 0

15.

t

The input signal shown in the figure below is fed to a Schmitt trigger. The signal has a square wave amplitude of 6 V p-p. It is corrupted by an additive high frequency noise of amplitude 8 V p-p.

th

th

th

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GATE QUESTION BANK

Common Data Question for Q.No. 17, 18 & 19 Consider the op-amp circuit shown in the figure below:

21.

-+

(A) 10V (B) 10.5V

-

22. -

18.

19.

A differential amplifier shown below has a differential mode gain of 100 and a CMRR of 40dB. If . and . , the output V0 is

k

k

17.

Analog Circuits

If . . and 7 and the op-amp is ideal , the value of the is (A) k (C) k (B) k (D) k Let sin πf t and K . The Op-amp has a slew rate of . / s with its other parameters being ideal .The values of and f for which the amplifier output will have no distortion are, respectively, (A) 0.1 V and 300kHz (B) 0.5 V and 300kHz (C) 0.1 V and 30kHz (D) 0.5 V and 30kHz

(C) 11V (D) 15V

The op-amp circuit shown below is that of a

k

k

k

(A) Low-pass filter with a maximum gain of 1 (B) Low-pass filter with a maximum, gain of 2 (C) High-pass filter with a maximum gain of 1 (D) High-pass filter with a maximum gain of 2 23.

Let and k . Assume that the op-amp is ideal except for a non-Zero input bias current. What is the value of for the output Voltage of the op-amp to be Zero? (A) . k (C) k (B) . k (D) k

In the op-amp circuit shown below the input voltage vin is gradually increased from 10V to +10V. Assuming that the output voltage vout saturates at 10V and +10V,vout will change from

k

IN - 2008 20. An ideal op-amp has the characteristics of an ideal (A) Voltage controlled voltage source (B) Voltage controlled current source (C) Current controlled voltage source (D) Current controlled current source

k

(A) 10V to +10V when vin (B) 10V to +10V when vin (C) +10V to 10V when vin (D) +10V to 10V when vin

th

th

th

1V 1V 1V 1V

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GATE QUESTION BANK

24.

For the op-amp circuit shown below approximately equal to

is

27.

Analog Circuits

The input resistance of the circuit shown in the figure, assuming an ideal op-amp, is 2R 3R

R

k

k

(A) (B)

10V 5V

(A) R/3 (B) 2R/3

(C) +5V (D) +10V

IN - 2009 25. The circuit shown is the figure is

28.

(C) R (D) 4R/3

In the circuit shown in the figure, the switch S has been in Position 1 for a long time. It is then moved to Position 2. Assume the Zener diodes to be ideal. The time delay between the switch moving to Position 2 and the transition in the output voltage is osition

k

k

osition

(A) (B) (C) (D) 26.

An all-pass filter A bandpass filter A highpass filter A lowpass filter

In the circuit shown, the Zener diode has ideal characteristics and a breakdown voltage of 3.2V. The output voltage for an input voltage = +1V is closest to

(A) 5.00ms (B) 8.75ms

.7k

ener diode

.7k

ener diode

(C) 10.00ms (D) 13.75ms

IN - 2010 29. In the ideal opamp circuit given in the below figure, the value of Rf is varied from

k

1k to 100k . The gain G = ( ) will

k

k k

k

(A) (B)

10V 6.6V

(C) (D)

(A) (B) (C) (D)

5V 3.2V

th

remain constant at +1 remain constant at -1 vary as -( /10,000 ) vary as ( /10,000) th

th

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GATE QUESTION BANK

30.

An active filter is shown in the below figure. The dc gain and the 3dB cut-off frequency of the filter respectively, are, nearly

k

Analog Circuits

IN - 2011 Statement for Linked Answer Questions 33and 34 and in the circuit shown below are matched n-channel enhancement mode MOSFETs operating in saturation mode, forward voltage drop of each diode is .7 reverse leakage current of each diode is negligible and the op-amp is ideal k

R1 = 15.9 k , R2 = 159 k , C1 = 1.0nF (A) 40dB, 3.14 kHz (B) 40dB, 1.00 kHz (C) 20dB, 6.28 kHz (D) 20dB, 1.00 kHz Common Data for Questions: 31 & 32 A differential amplifier is constructed using an ideal op-amp as shown in the adjoining figure. The values of R1 and R2 are 47k and 470k respectively.

31.

32.

The input impedances seen looking into the terminals V1 and V2, with respect to ground, respectively are (A) 47k and 43k (B) 47k and 47k (C) 47k and 517k (D) 517k and 47k

m

33.

The current (A) 1 mA (B) .5 mA

34.

For the computed value of current output voltage is (A) 1.2V (C) 0.2V (B) 0.7V (D) 0.7V

35.

The ideal op-amp based circuit shown below acts as a Ω

in the circuit is (C) m (D) 2 ma

Ω

0.5 µF

, the

0.5 µF

kΩ 1 µF

V1 and V2 are connected to voltage sources having an open circuit output of +1V each and internal resistances of 13k and 3k respectively. The output voltage V0 is (A) 0V (C) 1.5V (B) 0.15V (D) 10V

(A) (B) (C) (D)

th

low-pass filter high-pass filter band-pass filter band-reject filter

th

th

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GATE QUESTION BANK

36.

The value of shown below is

of the series regulator

40 V(DC) kΩ

(C) 6.7 V and 4.7 V (D) 5.3 V and 3.3 V 38.

kΩ 6 V(DC)

(A) 24 V (B) 28 V

πt

Assuming base-emitter voltage of 0.7V and β of transistor , the output voltage in the ideal opamp circuit shown below is 5V kΩ

(C) 30 V (D) 32 V

The transfer characteristics of the circuit drawn below is observed on the oscilloscope used in XY mode. The display on the oscilloscope is shown on the right hand side. is connected to the X input with a setting of 0.5 V/div, and is connected to the Y input with a setting of 2 V/div. The beam is positioned at the origin when is zero.

sin

37.

1kΩ

Analog Circuits

Ω

5V

1V

(C) (D)

(A) – 1V (B) 1/3.3V

V V

IN - 2013 39. The operational amplifier shown in the circuit below has a slew rate of 0.8 Volts / s. The input signal is . sin( t . The maximum frequency of input in kHz for which there is no distortion in the output is

~V

470k

22k

0.25sin t

~

V0

(A) 23.84 (B) 25.0 40. Assuming that the op-amp is ideal and the zener diodes have forward biased voltage drop of 0.7V, the values of reverse breakdown voltages of and are, respectively. (A) 3.3 V and 5.3 V (B) 4.7 V and 6.7 V th

(C) 50.0 (D) 46.60

The circuit below incorporates a permanent magnet moving coil milli – ammeter of range 1 mA having a series resistance of 10k. Assuming constant diode forward resistance of 50, a forward diode drop of 0.7 V and infinite reverse diode resistance for each diode , the reading of the meter in mA is

th

th

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GATE QUESTION BANK

Analog Circuits R2

mA

10 k

R1

X

10 k

Z Y

~

5V, 50Hz

V0

R3

V0

V1

(A) 0.45 (B) 0.5 41.

R4

(C) 0.7 (D) 0.9 Fig. a +5V

A signal Vi(t)=10+10sin 100πt + 10 sin 4000πt + 10sin 100000πt is supplied to a filter circuit (shown below) made up of ideal op–Amp. The least attenuated frequency component in the output will be

R

R X

. 2k 1k

~

.

Y

.

Strain Gage

750

R

. V1(t)

(A) 0 Hz (B) 50 Hz

Differential Amplifier

R

V0(t)

(C) 2 kHz (D) 50kHz

Statement for Linked Answer Questions 42 and 43 A differential amplifier with signal terminals X, Y, Z is connected as shown in Fig. (a) below for CMRR measurement where the differential amplifier has an additional constant offset voltage in the output. The observations obtained are : when Vi= 2V, V0=3mV, and when Vi=3V, V0=4mV.

42.

43.

th

Fig. b Assuming its differential gain to be 10 and the op – amp to be otherwise ideal , the CMRR is (C) (A) (B) (D)

The differential amplifier is connected as shown in Fig. (b) above to a single strain gage bridge. Let the strain gage resistance vary around its no – load resistance R by ± 1%. Assume the input impedance of the amplifier to be high compared to the equivalent source resistance of the bridge, and the common mode characteristic to be as obtained above. The output voltage in mV varies approximately from (A) +128 to – 128 (C) +122 to – 122 (B) + 128 to – 122 (D) +99 to – 101

th

th

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Z

GATE QUESTION BANK

IN - 2014 44. For the op-amp shown in the figure, the bias currents are = 450 nA and =350 nA. The values of the input bias current ( ) and the input offset current ( ) are:

47.

For the given low-pass circuit shown in the figure below, the cutoff frequency in Hz will be ___________. . 7 k

~

k . 7

k

(A) = 800 nA, =50 nA (B) = 800 nA, =100nA (C) = 400 nA, =50nA (D) = 400 nA, 100nA 45.

Analog Circuits

48.

The amplifier in the figure has gain of -10 and input resistance of 50 k . The value of and are

k

The figures show an oscillator circuit having an ideal Schmitt trigger and its input-output characteristics. The time period (in ms) of t is___________.

output

t

k nput

(A) (B) (C) (D) 46.

k

k k

k k

k k

k

Assuming an ideal op-amp in linear range of operation, the magnitude of the transfer impedance

in

of the

current to voltage converter shown in the figure is ___________. k

k . k

i

th

th

th

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GATE QUESTION BANK

Analog Circuits

Answer Keys and Explanations ECE 1.

4.

[Ans. C] 2 kΩ

[Ans. D] Initially the switch is closed at t= 0 Then circuit is The circuit is integrator circuit ∫

1 kΩ 1V

dt

1 kΩ

pply K at node dv k dt ( k

2.

)

(

k k k k ( ) k

[Ans. D]

1 kΩ

) . k ) . k

( .

k

k

. k

5.

[Ans. D] Applying KCL

k

exp (

k Volt across 24 k due to virtual ground concept. So voltage across 12k is 3V

)

exp (

k

ln ow

) i

k ln ln

watts 3.

[Ans. B] Unregulated voltage increases by 20% i. e. New regulated voltage = 18V

6.

k k ln

[Ans. A]

watts increases

th

th

th

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GATE QUESTION BANK

(

)

(

)

(

)

(

)

10. .

Analog Circuits

[Ans. C] k k

j

k

j j

j

j s s s s 7.

From KCL

s

k

s s

[Ans. C] s s

k

k

.. Threshold depends on output So, when

s s s s

When

s s tan tan tan inimum value of π at maximum value of at

∞

11.

[Ans. A]

12.

[Ans. C] a

8.

[Ans. B] (e

⁄

c

)

Where

Voltage across diode = 60 mV Voltage across 4k resistor s k m Total voltage is m m 9.

b

and k

Op-amp is ideal, so it will satisfy the virtual ground property a c So, we can redraw the circuit as

m .

[Ans. B] At low frequency, capacitor is open and inductor short so, At high frequency capacitor is short and inductor open so, so it is low pass filter.

Circuit is similar to standard inverting amplifier th

th

th

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GATE QUESTION BANK

[ 13.

(

Analog Circuits

)]

[Ans. B] i

i

i

i Hence given circuit is a high pass filter

i

15.

[Ans. B]

k .7

Redraw the given circuit Case (1) 5V

i For non-zero value of off, so i

, diode

must be by virtual short equivalent connection for transistor is

Then

t 0 t .7 When , conducting so =0V 14.

both

diode

are 16.

0.7V

[Ans. C] ain of

stage k ( ) k

[Ans. D] At Hence circuit can be redrawn as below

(

1k -2V

k i

m

k ) k

m

i

1k

Vout

i

+1V

-15 V 1k

1k

t ∞ ∞ Hence circuit can be redrawn as below

Gain of stage II =

th

th

*

th

+

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GATE QUESTION BANK

17.

[Ans. *] Range 3.1 to 3.26 ut off frequency f

20.

18.

[Ans. B] β e h β is inversily proportional to Re if increases decreases

π

πf . k z

.

Analog Circuits

if

[Ans. *] Range 1092 to 1094

21.

decreases CMRR increases

[Ans. D]

k

k

a .7

Op-Amp ideal so it will satisfy the property of virtual ground .7 ere .

k .7k .7 .7k .7k . 7

(

22.

[Ans. C]

[Ans. D] The circuit shown in the figure has positive feedback. So it can be either a oscillator or multivibrator. So option A and C are cancelled out Now, a voltage controlled oscillator is usually implemented using a artley’s oscillator where the feedback is like shown below.

Due to virtual ground

must have inductors in the feedback circuit. Since the given circuit has no inductors, it has to be a multivibrator.

. 19.

)

.7

o

th

th

th

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GATE QUESTION BANK

EE 1.

2.

5. [Ans. D] When is positive, diode will be OFF, so When is negative, diode will be ON, so

Analog Circuits

[Ans. B] From the transfer characteristic of the rectifier P is for for

will be –ve and V will be ve and .7

[Ans. D] Voltage at non inverting terminal, From the transfer characteristics of the rectifier Q is for for

Due to virtual ground current through Load, r r So this circuit acts as a current source with current

3.

[Ans. A] An astable multi-vibrator is providing pulses as given below.

But in this case initial voltage at capacitor is zero so it starts from zero also charging time will be larger (normally) than discharging time but it is made equal by using a diode. 4.

[Ans. D] It is limited circuit It makes transition from +5V to 5V (

e

⁄

For full wave rectifier output is W ‘ ’ must be connected to inverting and ‘ ’ must be connected to non-inverting terminal of the op-amp 6.

[Ans. A] When . , O/P will be clamped by a dc value of 2.5 V

7.

[Ans. C]

)

(Voltage across 100k ) . sec t . 7 s

th

th

th

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GATE QUESTION BANK

Analog Circuits

i Let,

only inverting terminal

magnitude i

ere

π

s s is only on non-inverting terminal

Let,

(

π mA and it will be lagging by 90°

) 10. (

(

v

[Ans. D] k

)

)

k

Putting the value of (

, we get k

)

s

ere (

When When

)

upto t after t

ve ve

( ) o it is high pass filter 8.

11.

[Ans. B]

[Ans. D]

( 12. C

)

[Ans. D] First section is differential amplifier having gain off 1.

. . . =

. .

So gain frequency characteristics will be as option (D) 9.

[Ans. D] Voltage at inverting terminal Output is

So th

th

th

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GATE QUESTION BANK

(

15.

)

econd stage

Analog Circuits

[Ans. B]

schmitt trigger

x

y

x x y y Applying superposition theorem When y = 0 x

x

when x y/ 13.

[Ans. D] Low Pass Filter

y y

High Pass Filter

x from equation

16.

and

[Ans. A] k i

20Hz

30Hz

Pass band

It is a band pass filter. 14.

[Ans. B] V= =0 = =-

f

= =

= High pass Fitter with f

s rad /sec

th

th

th

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GATE QUESTION BANK

17.

Analog Circuits

[Ans. D]

Since positive feedback is there at the output of op-amp 1 depending on output will vary between and At the output of op-amp ⁄ Gain of op-amp is Hence net output will vary between and

IN 1.

[Ans. D] Voltage gain So it does not depend on R.

2.

[Ans. B] When switch (

18.

is closed, )

[Ans. C] 3.

[Ans. C] When Op-Amp is used in ve feedback then voltage difference between the terminals is treated as zero.

4.

[Ans. A] The equivalent circuit is as follows:

-

When is on is off

(

)

(

.

)

when is on is off

------------- (1)

and ⟹

*

+

From equation (1) ⟹

*

+

⟹ =( th

th

) th

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GATE QUESTION BANK

5.

[Ans. B] It is a sawtooth wave generator across capacitor C.

8.

Analog Circuits

[Ans. D] DC output Voltage that is output offset voltage, (

)

So maximum offset (dc voltage) at the output will be, (

t T

d o dt d dt d dt

T

T

T

) m m m

m

T 9.

[Ans. B] k

. a

T f

T 6.

k z The given circuit is pply K at ‘a’

[Ans. C] .7 =0 .

When When When reverse circuit of the photodiode changes from 100 to , the output voltage change from 1V to 2V

.

I= I

. =0 .

1.5 =

= 4.47 7.

10.

[Ans. C] 6.7V I 1.5V

I=

.

.

=I =

.

=

. = .

.

[Ans. D] Given that the imp. Parameters are k k k k k k k Then the given op-amp circuit is

.

.

= 2.45V

th

th

th

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GATE QUESTION BANK k

k

12.

Analog Circuits

[Ans. D] R

R k

.

k a Vin b

pply K

at ‘a’

pply K

at ‘x’

t

Case-1 When ve half cycle then switch (s) is closed and circuit will become

ain 11.

R

R

[Ans. B] .

p Vin k

k p

( )

k

( 7k

Case -2 When ve half cycle then switch s is open and circuit will become

The given circuit is a second order low pass filter therefore it has a dB/decade and it has 3 dB cut-off frequency as f

)

.

R

R

.

π√ Vin

. k z It has a decade form 0 dB to frequency range.

3 dB in mid

th

th

th

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GATE QUESTION BANK

No circuit flow through left side Op-Amp due to open switch so

through resistor . But is shorted out. So discharge time constant .So capacitor will instantly discharge so correct output is in option (A)

= So, the circuit as a rectifier (fullwave) 13.

[Ans. B] O/P of first OPAMP t

15.

t dt, where

∫

k

and O/P of second OPAMP t t ∫

K

t dt

∫

t dt

input is , 50Hz square wave. O/P should be triangular wave of . 50Hz means 20msec, so in 10msec O/P should charge from . to . , when .

Analog Circuits

[Ans. B] The UTP value selected in such a way that to recover a square wave of same frequency from the input signal When The output is When and When We can recover a square wave of same frequency from the corrupted input signal when UTP and LTP value are

16.

[Ans. B] k

t

∫

dt k

m

+

+ +

k

k

m

-

-

14.

[Ans. C]

Apply KCL at KΩ

RA

node

.

reset discharge

KΩ

RB

.

out ̅̅̅̅̅̅̅̅̅ tr gger

.

. . . . . . . Now, Apply KCL at

threshold ground

Capacitor C charges toward through and until in raises upto 2/3 . This voltage is the threshold voltage of pin 6 which drives comparator 1 to trigger the flip-flop so that the output at pin 3 goes low. In addition, the discharge transistor is driven on. Causing the output at pin 7 to discharge the capacitor

.

Node

. For an ideal op-amp . . . . .7 . m

th

th

th

7

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GATE QUESTION BANK

17.

[Ans. B]

Analog Circuits

K .

k

k

--

20.

[Ans. A]

21.

[Ans. B]

K K

. K

+

.

and .

-

d

-

Apply the KCL at the inverting node

[ Put,

7

7

.

[

. 7

.

] .

.

and

. 22.

]

.

.

[Ans. D] At low frequency, C is open and At high frequency C is short and (

k 18.

.

.

)

so it is a high pass filter with max. gain of 2.

[Ans. C] sin πf (

)

23.

sin πft d | | dt

πf cos πft

[Ans. D] When and is connected to inverting terminal, output will saturate to +10V.

πf To avoid distortion, Slew rate . f

|

| πf

π f 7 7. 7 So when f k z and Then f 7 7.7 So option (C) is correct. 19.

k

.

k

nd

.

oltage divider rule

When crosses 1 V, So, output will not change to . So, output will change from to when .

[Ans. B] In order to avoid the effect of the bias current, resistance at +ve terminal must be equal to dc resistance seen from –ve terminal by replacing all the sources by their internal resistances. Therefore, k K K th

th

th

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GATE QUESTION BANK

24.

[Ans. B] Equivalent circuit k

Analog Circuits

. Now first open the zener diode then circuit will be k

k k k

k

k

By neglecting current through, 100K compare to current through 105 resistance,

. Similarly, by neglecting current through K compare to current through resistance,

So zener will be on and circuit will be as i

k .

i k

.7

k

m

.7 .7

.

o .

will lie between and . Because zener will be on then current 1 mA will be divided in two parts and output will be – 10V < < 3.2V f output then i . m i . m oi i hence this is not possible When . Then i . m . . i . m

.

5.2 V 25.

[Ans. A] s

s

s [

]

So it is all - pass filter. 26.

[Ans. B] Zener diode is in reverse bias. The equivalent circuit is,

here i

k

i

i

k k

th

th

th

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GATE QUESTION BANK

27.

[Ans. A]

Analog Circuits

at time t , when will switch to 5V.

2R

. ,

e ⁄ .7 ms

. t

3R

becomes

R

29.

[Ans. A]

k

Input resistance

a b

⃗

Voltage at inverting terminal, By KCL at node A,

k

and pply K

at a

By KCL at inverting terminal

By equ (1) and (2), Gain = 28.

[Ans. B] When the switch is at position 1, capacitor will be charged by +30V and . so voltage at non –inverting terminal will be 2.5V. When switch is moved to position 2, capacitor will start discharging and when . . will switch to 5V. Equivalent circuit

30.

[Ans. D] DC gain = 20 log | | = 20 dB 3-dB cut-off frequency =

31.

= 1 kHz

[Ans. C] 7 k 7k

k nvertinf terminal

7k 7 k

s

Input impedance seen looking into the terminals and with respect to ground Apply superposition theorem put So voltage at By virtual ground concept voltage at

s

s

c

s voltage i t e

at

i t inverting

e

⁄

terminal

⁄

th

th

th

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GATE QUESTION BANK

So input impedance seen from respect to ground is 7k Now put The input impedance from respect to ground 7 7 k 7k 32.

with

Since, and are all matched transistors So, say Then m or . m lso . m

with

[Ans. B] Now the effective circuit is 7 k

34.

[Ans. A] will be reverse biased while will be forward biased o (Forward voltage drop of ) .7 . .

35.

[Ans. A] The ideal op-amp based circuit is shown in figure 1. with T-network at the input and feedback paths,. Note the specific relations between resistances and capacitances used. It can be shown that this circuit acts as a double integrator and hence it is a low pass filter specifically transfer function

k

7k

Analog Circuits

7 k

Now voltage at point P is 7 7 By virtual ground capacitor, voltage at point Q is 7

k

Now apply KCL at point Q k

.

.

7 k . .

33.

igure

[Ans. B] k

s s

s

s .

or .

m

. sec

s

. s s The frequency response is given by j

which shows the gain falls by

40 dB per decade Clearly

m

th

th

th

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GATE QUESTION BANK

36.

[Ans. C]

When input voltage cycle) everse biased } orward iased .7 .7 .

k

k

38.

Using voltage divider rule

[Ans. D] When input voltage cycle) Forward biased Reverse biased

(During

(During

ve

[Ans. C] .7 . m β

or so

or 37.

Analog Circuits

ve

m

k

.7

ow

39.

[Ans. A] .

f

1kΩ

.

π

π

.

Where f .7 .7 .7 .7

or

. .

40.

[Ans. A]

41.

[Ans. C] For Op-Amp Analysis

.

k z

sin t .

sin t

.7

th

th

th

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GATE QUESTION BANK

s s

T.

Analog Circuits

. k z

7 d

. k z

.

d d Lower 3 dB dominant frequency is 0.5 kHz. Total response

ere one pole

s OP-Amp

k

and reject response

is low pass response;

east attenuated frequency k z

Low frequencies z z k z

f . k z

k

.

k z

Only option (C)is lying in this range Option (C) is correct

k z Op-Amp Analysis

42.

[Ans. C] from given data m m Solving (1) & (2)

s s

T.

s s *

+

*

+

43.

[Ans. B] 5V

*

j

+* *

+

j

R

+

R

High pass response R High frequencies

[

. k z

d

7

]

For variation is varies from 12.48mV to 12.56mV

. th

th

th

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GATE QUESTION BANK

Analog Circuits

m , given from bridge For change . m =127.8 mV For change op amp . m m . m m . m O/P varies from 7. m to . m

k

i . 47.

[Ans. *]Range 15 to 16 It is second-order low pass system. t’s high cut-off-frequency f

π√ ere

44.

c

[Ans. D] In an OPAMP,

f

45.

n

π√ π

n

48.

[Ans. B] Given k and gain For an inverting amplifier, Gain = -10 / k Then k

f . 7

c

nput bias current n / input offset current n

.

k k

k

. 7 . 7

.

z

[Ans. *] Range 8.0 to 8.5 T

ln [ ln [

/

] ]

. Time period .

46.

.

. m secs

[Ans. 0.6] k

k

i

. k

c

i d

virtual ground K i

at ode

i

k oi i . ki

. k ki ki . k ki . k ki ki . ki ki

k

k

i

ki k

th

th

th

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GATE QUESTION BANK

Analog Circuits

Power Amplifiers EE - 2007 1. The input signal Vin shown in the figure is a 1 KHz square wave voltage that alternates between +7V and 7V with a 50% duty cycle. Both transistors have the same current gain, which is large. The circuit delivers power to the load resistor RL. What is the efficiency of this circuit for the given input? Choose the closest answer.

(A) 46% (B) 55%

EE - 2009 2. Transformer and emitter follower can both be used for impedance matching at the output of an audio amplifier. The basic relationship between the input power Pin and output power Pout in both the cases is (A) Pin = Pout for both transformer and emitter follower (B) Pin > Pout for both transformer and emitter follower (C) Pin < Pout for transformer and Pin = Pout for emitter follower (D) Pin = Pout for transformer and Pin < Pout for emitter follower

(C) 63% (D) 92%

Answer Keys and Explanations EE 1.

[Ans. C] load voltage = ±6.3 V So efficiency

2.

[Ans. D] For emitter follower

th

th

th

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GATE QUESTION BANK

Digital Circuits

Number Systems & Code Conversions ECE-2006 1. A new Binary Coded Pentary (BCP) number system is proposed in which every digit of a base-5 number is represented by its corresponding 3-bit binary code. For example, the base-5 number 24 will be represented by its BCP code 010100. In this numbering system, the BCP code 100010011001 corresponds to the following number in base-5 system (A) 423 (C) 2201 (B) 1324 (D) 4231 2.

The number of product terms in the minimized sum-of-product expression obtained through the following K-map is (where, “d” denotes don’t care states) 1 0 0 1 0 d 0 0 0 0 d 1 1 0 0 1 (A) 2 (C) 4 (B) 3 (D) 5

ECE-2007 3. X = 01110 and Y = 11001 are two 5-bit binary numbers represented in two’s complement format. The sum of X and Y represented in two’s complement format using 6 bits is, (A) 100111 (C) 000111 (B) 001000 (D) 101001 ECE-2008 4. The two numbers represented in signed 2’s complement form are P = 11101101 and Q = 11100110. If Q is subtracted from P, the value obtained in signed 2’s complement form is (A) 100000111 (C) 11111001 (B) 00000111 (D) 111111001

ECE-2014 5. The number of bytes required to represent the decimal number 1856357 in packed BCD (Binary Coded Decimal) form is __________. EE-2007 1. The octal equivalent of the HEX number AB.CD is (A) 253.314 (C) 526.314 (B) 253.632 (D) 526.632 EE-2014 2. A cascade of three identical modulo-5 counters has an overall modulus of (A) 5 (C) 125 (B) 25 (D) 625 3.

Which of the following is an invalid state in an 8-4-2-1 Binary Coded Decimal counter (A) 1 0 0 0 (C) 0 0 1 1 (B) 1 0 0 1 (D) 1 1 0 0

IN-2006 1. A number N is stored in a 4-bit 2’s complement representation as a3 a2 a1 a0 It is copied into a 6-bit register and after a few operations, the final bit pattern is a a a a a 1 The value of this bit pattern in 2’s complement representation is given in terms of the original number is N as (A) 32 a3 + 2N + 1 (C) 2N – 1 (B) 32 a3 – 2N – 1 (D) 2N + 1 IN-2008 2. The result of (45)10 – (45)16 expressed in 6-bit 2’s complement representation is, (A) 011000 (C) 101000 (B) 100111 (D) 101001

th

th

th

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GATE QUESTION BANK

IN-2009 3. The binary representation of the decimal number 1.375 is, (A) 1.111 (C) 1.011 (B) 1.010 (D) 1.001

Digital Circuits

IN-2011 4. The base of the number addition operation 24 + true is (A) 8 (C) (B) 7 (D)

system for the 14 = 41 to be 6 5

Answer Keys & Explanations ECE 1.

2.

3.

4.

5.

[Ans. D] 100 010 4 2

011 3

001 1

[Ans. A] 1

0

0

1

0

d

0

0

0

0

d

1

1

0

0

1

1 byte = 8 bit so Here d = 7 No. of bits = 28

[Ans. C] x = 01110 y = 11001 ) x y( = 100111 Carry discard it 00111 in 6 bits will be 000111 [Ans. B] igned 2 s complement of P = 11101101 o P = 00010011 igned 2 scomplement of = 11100110 P = P (2 s complement of ) = 00010011 11100110 11111001 ) = 00000111 2 s complement of (P

byte = EE 1.

= 3 5= 4

[Ans. B] Hex number (AB.CD) ⏞ 1010 ⏞ 1011 ⏞ 1100 ⏞ 1101 For finding its octal number, we add one zero in both extreme and group 3 bit together 010 ⏟ 101 ⏟ 011 ⏟ 110 ⏟ 011 ⏟ 010 ⏟ quivalent octal number 253 632

2.

[Ans. C] Overall modulus = 5 = 125

3.

[Ans. D] BCD counter counts up to 1001

IN 1.

[Ans. D] Given number is a a a a in 2’s complement form. We know that in 2’s complement form if we copy MSB at left of MSB any times the number remains unchanged. So a a a a = a a a a a a = When we left shift a number by 1 bit then it is multiplied by 2, a a a a a 0 = 2 Now, a a a a a 0 1=a a a a a 1=2 1

[Ans. *] Range 3.9 to 4.1 A decimal digit is represented by 4 bit in BCD format, so for a decimal number with digits requires 4d bit and th

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GATE QUESTION BANK

2.

[Ans. C] (45) (45)

= ( 24)

3.

[Ans. C] 0.375 × 2 = 0.750 0.750 × 2 = 1.5 1.5 × 2 = 1.0 Hence answer is 1.011

4.

[Ans. B] Let the base is x, Here (24) (14) = (41) (4 x 2 x) (4 x = (4 x 4 + 2x + 4 + x = 4x + 1 x=7

Digital Circuits

= (101000)

1 x) 1 x )

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Digital Circuits

Boolean Algebra & Karnaugh Maps ECE-2007 1. The Boolean expression ̅̅ ̅ ̅ ̅ ̅̅ be minimized to ̅̅ ̅ (A) ̅̅ ̅ (B)

̅ ̅ ̅

̅ ̅ can

(A) X (B) Y

̅ ̅̅

̅

(C)

ECE-2014 5. The Boolean expression ( ̅̅̅̅̅̅̅̅̅̅̅ ( ̅) ̅ simplifies to

Consider the Boolean function, F(w, x, y, z) = wy + xy + ̅ xyz + ̅ ̅ y + xz + ̅ ̅ ̅. Which one of the following is the complete set of essential prime implicates? (A) ̅̅ (C) ̅ ̅̅ (B) (D) ̅̅

7.

For an n-variable Boolean function, the maximum number of prime implicants is (A) ( ) (C) (B) (D) ( )

ECE-2009 2. If X = 1 in the logic equation

EC/EE/IN -2012 3. In the sum of products function ( ) ∑( ), the prime implicants are ̅ (A) ̅ ̅ ̅ ̅̅ ̅ (B) ̅ (C) ̅ ̅ ̅ ̅̅ ̅ (D) ̅ ̅ ̅

(C) XY (D) X+Y

6.

(D)

[X+Z{ ( +X ) }] { + ( X + Y)} =1 then (A) Y = Z (C) Z = 1 (D) Z = 0 (B) Y =

̅)

)(

EE-2010 Statement for Linked Answer Questions1 and 2 The following Karnaugh map represent a function .

F YZ

ECE-2013 4. In the circuit shown below, Q1 has negligible collector – to – emitter saturation voltage and the diode drops negligible voltage across it under forward bias. If Vcc is +5 V, X and Y are digital signals with 0 V as logic 0 and as logic 1, then the Boolean expression for Z is

X YZ 0

01

11

10

1

1

1

0

X

00 01 11 1 0 0 1 0 1 1 1 A minimized form of the function is (A) (C)0 1 0 1 (B) (D)

1. R1 Z R2 X

00

F 10 0 0 0

Q1 Diode

2.

Which of the following circuits is a realization of the above function ?

Y

(A) XY

(C) X

(B)

(D) th

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GATE QUESTION BANK

(A)

Digital Circuits

( )

X F (A)

Y Z

X

(B) X

(B) X

F Y Z

(C) X

F Y Z F

(C) X

Y Z

(D) X

4.

The SOP (sum of products) form of a Boolean function is (0,1,3,7,11), where inputs are A,B,C,D (A is MSB, and D is LSB). The equivalent minimized expression of the function is )( ̅ ̅)( ̅ (A) (̅ )( ̅ ) ̅ ̅ ̅ ̅ ̅ )( )( (B) ( )( ) )( ̅ ̅ )( ̅ ̅ ) (C) (̅ )( ̅ ̅)( ̅ ̅)( ̅ (D) (̅ )( )

F Y Z

(D) Y X

( )

F

Y Z

F Z F

EE-2014Y Z 3. Which of the following logic circuits is a realization of the function F whose Karnaugh map is shown in figure

( )

( )

IN-2006 1. Min-term (Sum of Products) expression for a Boolean function is given as follows. f(A, B, C) = m (0, 1, 2, 3, 5, 6) where A is the MSB and C is the LSB. The minimized expression for the function is (A) A + (B  C) (C) (B  C) (B) (A  B) + C (D) IN-2007 2. A logic circuit implements the Boolean function F = ̅ . Y + X .̅ . ̅ . It is found that the input combination X = Y = 1 can never occur. Taking this into account, a simplified expression for F, is given by (C) X + Y (A) ̅ + ̅. ̅ (B) X + Z (D) Y + X. ̅

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GATE QUESTION BANK

3.

Let X and Y = be unsigned 2-bit numbers. The function F = 1 if X > Y and F = 0 otherwise. The minimized sum of products expression for F is (A) + ̅ .̅ .̅ ̅ + ̅ + ̅ (B) ̅ ̅ ̅ ̅ (C) + + ̅ + ̅ .̅ + ̅ (D)

Digital Circuits

IN-2011 6. For the Boolean expression ̅̅ ̅ ̅̅ ̅ ̅ ̅, the minimized Product of Sum (PoS) expression is ( (A) ̅) ( ̅) ̅ ) (B) ( ) (̅ ̅ (C) ( )( ̅) (D)

̅

IN-2008 4. The minimum sum of products form of the Boolean expression ̅̅̅ ̅̅ ̅ Y = ̅̅ ̅ ̅ ̅ ̅ ̅̅ ̅ (A) Y = P ̅ + ̅ ̅ (B) Y = P ̅ ̅ ̅ (C) Y = P ̅ ̅ ̅ ̅ ̅ (D) Y = ̅ ̅ IN-2009 5. The minimal sum-of-products expression for the logic function f represented by the given Karnaugh map is PQ RS 00 01 11 10 00 0 1 0 0

01 11 10

0

1

1

1

1

1

1

0

0

0

1

0

(A) QS + (B) (C)

+ +

(D)

th

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GATE QUESTION BANK

Digital Circuits

Answer Keys & Explanations ECE 1.

4.

[Ans. B] X Y Z 0 0 0

[Ans. D] K-map corresponding to given Boolean expression CD AB

01

00

00

11

10

1 1

01 11

1

0

1

1

1

0

0

1

1

0

Comments Transistor off diode ON Transistor off diode rev biases Transistor ON diode rev biases Transistor ON diode rev biased

1

10

So, ̅Y=Z ̅̅

̅

̅̅

̅̅ ̅

OR ̅

̅

[

]

̅

5. ̅̅

[Ans. A] ( )(

̅̅

( ̅̅

̅̅̅̅ ̅ 6.

2.

[Ans. D] yz

[Ans. D] (

*

wx

̅))+ [

(

(

̅̅

)]

1

1

By putting X = 1 (

* [

̅) (̅̅̅̅̅̅̅̅̅̅̅ ̅) ̅ ̅ ̅) ̅̅̅̅ ̅

))+ [

(

̅(

)]

]

1

1

1

1

1

1

1

1

1

1

xz y

So, P.I. are y, zx, ̅ ̅ 3.

[Ans. A] (

)

̅̅ ̅ ̅

∑(

YZ 1

)

7.

[Ans. D] Maximum P.I will occurs at condition like

̅ 1

1

1 1 ̅ ̅ ( ) So prime implicants are ̅ and ̅ .

1 1

1 1

1 1

1

i.e., no grouping at all so ) So, (

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GATE QUESTION BANK

EE 1.

2.

[Ans. D] ̅

[Ans. B] X

01 11 10

0

1

1

1

0

1

0

0

1

0

F = ̅ ̅ + YZ 2.

3.

4.

[Ans. D] From the figure it is clear that, two NAND ̅ and now two gates generate the ̅ ̅ and inputs AND gates with inputs ̅ Y and Z is used to generate two terms of SOP form and now OR gate is used to sum them and generate the F. [Ans. C] ̅̅ ̅̅ [Ans. A] ̅ ̅ (̅

K – map YZ 00 X 0 0

1

[by consensus theorem]

̅ )( ̅

̅)(

̅)(

x

x

0

X F=Y+X [Ans. D] F = 1 if X > Y, so following will be K – map of function F.

00

00 0

01 0

11 0

10 0

01

1

0

0

0

11

1

1

0

1

10

1

1

0

1

̅ +

F= 4.

A

00

01

11

10

0

1

1

1

0

1 0

1

0

1

F 0 0 1 1 1 0 x x

10 1

1

̅)

[Ans. C] From K – map

BC

Z 0 1 0 1 0 1 0 1

01 11 0 1

̅

3.

IN 1.

̅̅

Truth table: X Y 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1

YZ 00

Digital Circuits

̅ ̅̅̅ +

̅̅̅

[Ans. A]

By K – map

f = ̅ + B ̅ + ̅C =̅+B C th

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GATE QUESTION BANK PQ RS

5.

00 00 1

01 0

11 0

10 1

01

0

0

0

0

11

0

0

0

0

10

1

1

1

1

[Ans. A] PQ RS 00 Q S 00 0

6.

Digital Circuits

01 1

11 0

10 0

1

1

01

0

1

11

1

1

1

0

10

0

0

1

0

[Ans. A] ̅̅ ̅ ̅̅ ̅ ̅ ̅̅ ̅ ̅( ̅ ) ( ̅ ̅̅ ̅ ̅ ̅] ̅̅ [ ̅ )( ̅̅ [( ̅)] ̅̅ ( ̅) ̅̅ ̅ ̅( ̅) ̅ (̅ )(̅ ) ( ̅)( ̅) Alternative method:

0

00 1

01 0

11 0

10 1

0

1

0

1

1

(

̅)(

̅ ̅)

̅)

th

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Digital Circuits

Logic Gates ECE-2007 1. The Boolean function Y =AB + CD is to be realized using only 2 – input NAND gates. The minimum number of gates required is (A) 2 (C) 4 (B) 3 (D) 5

5.

Match the logic gates in Column A with their equivalents in Column B. Column B Column A

ECE-2008 2. The logic function implemented by the following circuit at the terminal OUT is

Q

(A) P NOR Q (B) P NAND Q 3.

X

2

R

3

S

4

P-2, Q-4, R-1, S-3 P-4, Q-2, R-1, S-3 P-2, Q-4, R-3, S-1 P-4, Q-2, R-3, S-1

P Q Y R

Z

(A) Two or more of the inputs P, Q, R are “0” (B) Two or more of the inputs P, Q, R are “1” (C) Any odd number of the inputs P, Q, R is “0” (D) Any odd number of the inputs P, Q, R is “1”

Y R (A) (B) (C) (D)

Q

ECE-2011 6. The output Y in the circuit below is always “1” when

(C) P OR Q (D) P AND Q

Which of the following Boolean Expressions correctly represents the relation between P, Q, R and ? P Q

1

(A) (B) (C) (D)

OUT P

P

= (P OR Q)XOR R = (P AND Q) XOR R = (P NOR Q) XOR R = (P XOR Q) XOR R

ECE-2010 4. For the output F to be 1 in the logic circuit shown, the input combination should be A

B F C

(A) (B) (C) (D)

A=1, B=1, C=0 A=1, B=0, C=0 A=0, B=1, C=0 A=0, B=0, C=1 th

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GATE QUESTION BANK

Digital Circuits

ECE-2012 7. In the circuit shown 5 Volts A C

(A) (B) (C) (D)

B Y C

10. A

B

̅̅ (

(A) (B)

̅ )

(̅

(C) (D)

4 1024 3 1024 E 2 1024 E

B. C. D.

In the circuit shown in the figure, if the expression for is

̅

(A) (B)

̅

(C) (D)

input S1 S0

0C00H 2C00H 1800H 3800H 0500H 3500H 0800H 2800H

0FFFH, 1C00H 2FFFH, 3C00H 1FFFH, 2800H 3FFFH, 4800H 08FFH, 1500H 38FFH, 5500H 0BFFH, 1800H 2BFFH, 3800H

̅

XOR

1 11 10 01 00

0,

̅

EE-2007 1. A, B, C and D are input bits, and Y is the output bit in the XOR gate circuit of the figure below. Which of the following statements about the sum S of A, B, C, D and Y is correct? A XOR B

1024

A15

A.

8 bit data bus

E

A10 A11 A12 A13 A14

y̅ y̅̅ y y

̅) ̅

ECE-2013 8. There are four chips each of 1024 bytes connected to a 16 bit address bus as shown in the figure below. RAMs 1, 2, 3 and 4 respectively are mapped to addresses

A0 A9

̅y ̅y̅ ̅y̅ ̅y̅̅

Y

E

1FFFH, 3FFFH 2FFFH, 4FFFH 18FFH, 58FFH 1BFFH, 3BFFH

ECE-2014 9. The output F in the digital logic circuit shown in the figure is

C XOR D (A) S is always either zero or odd (B) S is always either zero or even (C) S = 1 only if the sum of A, B, C and D is even (D) S = 1 only if the sum of A, B, C and D is odd

EE-2009 2. The complete set of only those Logic Gates designated as Universal Gates is (A) NOT, OR and AND Gates (B) XNOR, NOR and NAND Gate (C) NOR and NAND Gates (D) XOR, NOR and NAND Gates th

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EE-2011 3. The output Y of the logic circuit given below is

(A) 1 (B) 0

Digital Circuits

average value of the output voltage as function of τ for 0

(A)

t

( )?

Vav 5V

(C) X (D) ̅

τ

T/2

IN-2006 1. All the logic gates in the circuit shown below have finite propagation delay. The circuit can be used as a clock generator, if

(B)

Vav 5V Vav

Y (C) Vav

X (A) X = 0 (B) X = 1

1 T/2

Vav

(C) X = 0 or 1 (D) X = Y

IN-2007 2. Two square waves of equal period T, but with a time delay τ are applied to a digital circuit whose truth table is shown in the following figure. X Y Output 0 0 1 0 1 0 1 0 0 1 1 1 X

5V

T/2 (D)

τ

Vav 2.5V T/2

τ

IN-2009 3. The diodes in the circuit shown are ideal. A voltage of 0V represents logic 0 and +5V represents logic1.The function Z realized by the circuit for inputs X and Y is + 5V

t

T

τ

T/2

Y 1

X

τ

τ T/2

Z

Y

t

The high and the low levels of the output of the digital circuit are 5 V and 0 V, respectively. Which one of the following figures shows the correct variation of the

(C) Z =̅̅̅̅̅̅̅̅ (D) Z=̅̅̅̅

(A) Z=X + Y (B) Z=XY

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GATE QUESTION BANK

IN-2010 4. The logic gate circuit shown in the figure realizes the function A X

Z

Y

(A) XOR (B) XNOR

(C) Half adder (D) Full adder

Digital Circuits

EC/EE/IN-2013 5. A bulb in a staircase has two switches, one switch being at the ground floor and the other one at the first floor. The bulb can be turned ON and also can be turned OFF by any one of the switch irrespective of the state of the other switch. The logic of switching of the bulb resembles (A) An AND gate (C) A XOR gate (B) An OR gate (D) A NAND gate

Answer Keys & Explanations ECE 1.

5.

[Ans. D] P = ̅̅̅̅̅̅̅ = ̅ ̅ = 4 Q = ̅̅̅̅ = ̅ + ̅ = 2 R = A ⊕ B = A̅ + ̅B = 3 S = A B = AB + ̅ ̅ = 1

6.

[Ans. B] The output Y expression in the ckt (Majority circuit) So that two or more inputs are ‘1’, always ‘1’.

[Ans. B] A B AB + CD C D

2.

3.

[Ans. D] When P = Q = 1, then OUT = 1 P = Q = 0, then OUT = 0 P = 0, Q = 1, then OUT = 0 P = 1, Q = 0, then OUT = 0 So, it is AND gate

7.

[Ans. A] In NMOS circuit Since & are in parallel so those represent ( ) & is in sense, so it represents ‘dot’ operation and the whole function should be inverted or it is complementary logic. So, ̅̅̅̅̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅ ̅ ̅ ̅ ̅ ( )

8.

[Ans. D] For RAM #1

[Ans. D] X=

;Y=P+Q

Z = XY = =(

. (P + Q) )(

)

⊕ ∴ 4.

⊕

⊕

[Ans. D] For 3 input XNOR for output to be one, two input must be one, and we know that 2- input XOR & XNOR gate are complementary & hence only 1(1’s) will be generated & C=1 is required i.e, When A = 0, B = 0 and C = 1, then F = 1

is

⏟0

0

0

0

⏟1

0 0 0 ⏟ 0

0

⏟ 0 0 0

ower add

⏟ ighest add 0000 ⏟ 1 0 1 1 ⏟ 1⏟1 0 So range of add for RAM #1 0 00 0 which is present only in option D th

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GATE QUESTION BANK

9.

10.

[Ans. A] ( ⊕ )(( ⊕ ) ( ⊕ )(( ⊕ ) ( ⊕ ) ̅) (̅ ̅ ̅

) (

IN 1.

A B

0

[Ans. B] When X = 1, equivalent circuit is

)̅)

Y This circuits act as clock generator.

[Ans. A] C

2. ̅̅̅̅̅̅̅̅

[Ans. C] When τ = 0 X and Y will be same and out-put will be equal to dc of 5V.

1 ̅̅̅̅̅̅̅̅

0

( ( ̅

EE 1.

2.

3.

of each other and output will be equal to dc 0.

AB

̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅

y

When τ = , X and Y will be complement

̅̅̅̅̅̅̅̅

A B

) ̅̅̅̅ ) (̅ ̅

̅)

[Ans. B] ⊕ ⊕ ⊕ from the given diagram. We know that sum of any number of bits is XOR of all bits. So S ⊕ ⊕ ⊕ ⊕ S=Y⊕Y S = either zero or even because LSB is zero (always). [Ans. C] NOR and NAND are designated as universal logic gates, because using any one of them we can implement all the logic gates. [Ans. A] Y= .̅ X 1 0

0 1

̅. ̅ Y

̅

Digital Circuits

When τ increases from 0 to

, O/P will

decrease from 5V to 0V linearly. 3.

[Ans. B] When any of X or Y is zero, Z = 0. For X = Y = 1, Z = 1

4.

[Ans. A] X

y x

Y

y

Z= y. y 5.

Z

= y

y

y =x⊕y

[Ans. C] When both switches in on position, bulb is off When both switches in off position, bulb is off Bulb 0 0 0 0 1 1 1 0 1 1 1 0 It is a XOR gate

1

1 1

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Digital Circuits

Logic Gate Families ECE - 2007 1. The circuit diagram of a standard TTL NOT gate is shown in the figure. When Vi = 2.5V, the modes of operation of the transistors will be

1 4kΩ 4kΩ

Q1

Ignoring the body-effect, the output voltages at P, Q and R are,

100kΩ

R 2

(A) 4 V, 3 V, 2 V (B) 5 V, 5 V, 5 V

Q4 D

4.

Q2

(C) 4 V, 4 V, 4 V (D) 5 V, 4 V, 3 V

The output (Y) of the circuit shown in the figure is

Q3 1kΩ -

̅

(A) Q1: reverse active; Q2: normal active; Q3: saturation; Q4: cut-off (B) Q1: reverse active; Q2: saturation; Q3: saturation; Q4: cut-off (C) Q1: normal active; Q2: cut-off; Q3: cut-off; Q4: saturation (D) Q1: saturation; Q2: saturation; Q3: saturation; Q4: normal active ECE - 2009 2. The full forms of the abbreviations TTL and CMOS in reference of logic families are (A) Triple Transistor Logic and Chip Metal Oxide Semiconductor (B) Tristate Transistor Logic and Chip Metal Oxide Semiconductor (C) Transistor Transistor Logic and Complementary Metal Oxide Semiconductor (D) Tristate Transistor Logic and Complementary Metal Oxide Silicon

̅

(A) ̅ (B)

̅ ̅ ̅

̅

(C) ̅ (D)

̅ ̅

EE - 2010 1. The TTL circuit shown in the figure is fed with the waveform X (also shown). All gates have equal propagation delay of 10ns. The output Y of the circuit is x 100 ns

1

ECE - 2014 3. In the following circuit employing pass transistor logic, all NMOS transistors are identical with a threshold voltage of 1V.

0

th

t

th

th

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GATE QUESTION BANK X

Y

Digital Circuits

IN - 2007 1. A CMOS implementation of a logic gate is shown in the following figure: 5v

(A) Y

X Y

PMOS

1 0

t

NMOS

(B) Y

The Boolean logic function realized by the circuit is. (A) AND (C) NOR (B) NAND (D) OR

1 0

t

(C)

IN - 2014 2. The figure is a logic circuit with inputs A and B and output Y. = + 5 V. The circuit is of type

Y

1

t

0

(D)

Y

1 0

t

(A) NOR (B) AND

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(C) OR (D) NAND

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Digital Circuits

Answer Keys and Explanations ECE 1.

EE 1.

[Ans. B] reverse active,

saturation

saturation, 2.

[Ans. A] X

Y

cut off

[Ans. C] TTL – Transistor Transistor Logic CMOS – Complementary Metal Oxide Semiconductor

A

B

1 X 0

3.

[Ans. C] Suppose all NMOS at saturation

A

B

For 1 &

Y=X

B

4 For (

IN 1.

1)

(4

[Ans. C] NOR Gate

)

4

(4

)

2.

[Ans. D] Given circuit is of the standard 2 input NAND gate.

4 1 (4

4.

)

4 4

[Ans. A] The given circuit is CMOS implementation If the NMOS is connected in series, then the output expression is product of each input with complement to the final product. ̅̅̅̅̅̅̅̅̅ ̅ ̅

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Digital Circuits

Combinational and Sequential Digital Circuits ECE - 2006 1. The point P in the following figure is stuck-at-1. The output f will be A B

1 0 1 1 MSB

A

Q

D

LSB

FULL ADDER

CK f

0 0 1 1 Shift Registers

P

B

Q

D

S

0

Ci

0

Q

C0

CK D

C

(A) ̅̅̅̅̅̅̅ (B) ̅

0

CK

̅ (C) (D) A CLOCK

2.

Two D-flip-flops, as shown below, are to be connected as a synchronous counter that goes through the following Q1Q0 sequence ………… The inputs and respectively should be connected as

̅̅̅̅

(A) S =0, C0= 0 (B) S = 0, C0= 1

ECE - 2007 4. For the circuit shown, the counter state (Q1 Q0) follows the sequence

̅̅̅̅

c

c

(A) (B) (C) (D) 3.

(C) S = 1, C0= 0 (D) S = 1, C0= 1

̅̅̅̅ and ̅̅̅̅ and ̅̅̅̅ and ̅̅̅̅ ̅̅̅̅ ̅̅̅̅and

For the circuit shown in figure below, two 4-bit parallel-in serial-out shift registers loaded with the data shown are used to feed the data to a full adder. Initially, all the flip-flops are in clear state. After applying two clock pulses, the outputs of the full adder should be

(A) (B) (C) (D) 5.

……… ……… ……… ………

The following binary values were applied to the X and Y inputs of the NAND latch shown in the figure in the sequence indicated below: X=0, Y=1; X=0, Y=0; X=1, Y=1 The corresponding stable P, Q outputs will be X P

Q

Y th

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(A) P=1, Q=0; P=1, Q=0; P=1, Q=0 or P=0, Q=1 (B) P=1, Q=0; P=0, Q=1 or P=0, Q=1; P=0, Q=1 (C) P=1, Q=0; P=1, Q=1; P=1, Q=0 or P=0, Q=1 (D) P=1, Q=0; P=1, Q=1; P=1, Q=1

Digital Circuits

1

1

1

1

CLK

1 CLK 0 T

6.

In the following circuit, X is given by 0 1 1 0

I0 I1 I2 I3

4-to-1 MUX Y S1

S0

A

B

0 1 1 0

I0 I1 I2 I3 S1

4-to-1 MUX Y

t

t

Which of the following waveforms correctly represents the output at ?

X

S0

(A)

C

1 0 2T

t 1

(A)

(B)

(B) (C)

0 4T t

(D)

(C)

ECE - 2008 7. For the circuit shown in the following figure, are inputs to the 4:1 multiplexer. R(MSB) and S are control bits.

1 0 2T

t

(D)

1 0 4T t

z

9.

For the circuit shown in the figure, D has a transition from 0 to 1 after CLK changes from 1 to 0. Assume gate delays to be negligible

̅

̅

The output Z can be represented by (A) PQ + P ̅ S + ̅ ̅ ̅ (B) P ̅ +PQ̅ +̅ ̅ ̅ (C) P ̅ ̅ + ̅QR +PQRS + ̅ ̅ ̅ (D) PQ ̅ +PQR̅ +P ̅ ̅S + ̅ ̅ ̅ 8.

Which of the following statements is true? (A) Q goes to 1 at the CLK transition and stays at 1. (B) Q goes to 0 at the CLK transition and stays at 0. (C) Q goes to 1 at the CLK transition and goes to 0 when D goes to 1. (D) Q goes to 0 at the CLK transition and goes to 1 when D goes to 1.

For each of the positive edge – triggered J-K flip flop used in the following figure, the propagation delay is T.

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ECE - 2009 10. What are the counting states (Q1, Q2) for the counter shown in the figure below?

c

(A) (B) (C) (D) 11.

ip

Statement for Linked Answer Question 13 and 14. Two products are sold from a vending machine, which has two push buttons and . When a button is pressed, the price of the corresponding product is displayed in a 7-segment display. If no buttons are pressed ‘ ’ is displayed, signifying ‘ s. ’ If only is pressed, ‘ ’ is disp ayed signifying ‘ s. ’ If only p are pressed, ‘5’ is disp ayed signifying ‘ s.5’ If both a d are pressed, ‘E’ is displayed, Signifying ‘Err r’. The names of the segments in 7-segment display, and the glow of the display for ‘ ’ ‘ ’ ‘5’ a d ‘E’ are sh w be w.

ip

11, 10, 00, 11, 10 . . . . . 01, 10, 11, 00, 01 . . . . . 00, 11, 01, 10, 00, . . . . . 01, 10, 00, 01, 10 . . . . . .

Refer to the NAND and NOR latches shown in the figure. The inputs for both the latches are first made (0, 1) and then, after a few seconds, made (1, 1). The corresponding stable outputs ( ) are P1

Q1 P 1

Digital Circuits

a f Q1

0

g

e

2

5

E

b c

d P2

Q2 P 2

(A) NAND : first (0, 1) then (0, 1) NOR: first (1, 0) then (0, 0) (B) NAND : first (1, 0) then (1, 0) NOR: first (1, 0) then (1, 0) (C) NAND : first (1, 0) then (1, 0) NOR: first (1, 0) then (0, 0) (D) NAND : first (1, 0) then (1, 1) NOR: first (0, 1) then (0, 1) 12.

What are the minimum number of 2 to. 1 multiplexers required to generate a 2-input AND gate and a 2-input Ex-OR gate? (A) 1 and 2 (C) 1 and 1 (B) 1 and 3 (D) 2 and 2

Q2

Consider (i) Push Button pressed/not Pressed in a equivalent to logic 1/0 respectively. (ii) A segment glowing / not glowing in the display is equivalent to logic 1/0 respectively. 13.

If segment a to g are considered as functions of and , then which are of the following is correct? (A) g ̅ d c e (B) g d c e ̅ (C) g e b c (D) g e b c

14.

What are the minimum numbers of NOT gates and 2-input OR gates required to design the logic of the driver for this 7-segment display? (A) a d (B) a d (C) a d (D) a d

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ECE - 2010 15. The Boolean function realized by the logic circuit shown is

Data

Digital Circuits

D

Q

D

Y

Q

Clock

(A) (B) (C) (D)

MUX

19. (A) (B) (C) (D) 16.

∑ ∑ ∑ ∑

5 5 5

cha ged fr “ ”t “ ” cha ged fr “ ”t “ ” changed in either direction not changed

The logic function implemented by the circuit below is (ground implies a logic “ ” MUX

5

5

Assuming that all flips flops are in reset condition initially, the count sequence observed at in the circuit shown is utput

(A) F = AND(P, Q) (B) F = OR(P, Q)

̅

̅

̅

c

(A) (B)

… …

(C) (D)

… …

ECE - 2014 20. Five JK flip-flops are cascaded to form the circuit shown in Figure. Clock pulses at a frequency of 1 MHz are applied as shown. The frequency (in kHz) of the waveform at Q3 is _____.

ECE - 2011 17. Two D flip – flops are connected as a synchronous counter that goes through the following sequence The connections to the inputs are (A)

and

(C) F = XNOR(P, Q) (D) F = XOR(P, Q)

c

c

c

c

c

c

21.

The digital logic shown in the figure satisfies the given state diagram when Q1 is connected to input A of the XOR gate.

(B) (C) (D) ̅̅̅̅

18.

̅̅̅̅

When the output Y in the circuit below is “ ” it i p ies that data has th

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GATE QUESTION BANK

Suppose the XOR gate is replaced by an XNOR gate. Which one of the following options preserves the state diagram? (A) Input A is connected to ̅̅̅̅ (B) Input A is connected to (C) Input A is connected to ̅̅̅̅ complemented (D) Input A is connected to ̅̅̅̅ 22.

23.

Digital Circuits

(A) (B)

(C) (D)

and S is 24.

In a half-subtractor circuit with X and Y as inputs, the Borrow (M) and Difference (N = X Y) are given by (A) (B) ̅ (C) ̅̅̅̅̅̅̅̅ ̅ (D) In the circuit shown, choose the correct timing diagram of the output (y) from the given waveforms a d .

The outputs of the two flip-flops Q1, Q2 in the figure shown are initialized to 0, 0. The sequence generated at Q1 upon application of clock signal is

̅̅̅̅

̅̅̅̅

(A) (B) 25.

… …

(C) (D)

… …

The circuit shown in the figure is a E

̅

atch

̅

E

atch

̅

utput y

(A) Toggle Flip Flop (B) JK Flip Flop (C) SR Latch (D) Master-Slave D Flip Flop ̅

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GATE QUESTION BANK

26.

Digital Circuits

Consider the multiplexer based logic circuit shown in the figure.

Which one of the following Boolean functions is realized by the circuit? ̅ ̅̅̅ (A) (B) (C) (D) 27.

In the circuit shown, W and Y are MSBs of the control inputs. The output F is given by

(A) (B) (C) (D)

28.

̅

29.

̅ ̅ ̅̅ ̅ ̅ ̅ ̅̅ ̅ ̅ ̅ ̅ ̅̅

If X and Y are inputs and the Difference (D = X – Y) and the Borrow (B) are the outputs, which one of the following diagrams implements a half-subractor?

An 8-to-1 multiplexer is used to implement a logical function as shown in the figure. The output Y is given by

30.

th

̅ ̅

̅

(A) (B) (C) (D)

̅ ̅

̅

̅̅

̅

A 16-bit ripple carry adder is realized using 16 identical full adders (FA) as shown in the figure. The carrypropagation delay of each FA is 12 ns and the sum-propagation delay of each FA is 15 ns. The worst case delay (in ns) of this 16-bit adder will be __________

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Digital Circuits

EE - 2011 3. A two-bit counter circuit is shown below.

EE - 2006 1. A 4 1 MUX is used to implement a 3 – input Boolean function as shown in figure. The Boolean Function F(A,B,C) implemented is

B

(A) (B) (C) (D)

F (A,B,C) = F (A,B,C) = F (A,B,C) = F (A,B,C) =

J

Q

K

̅

K

̅

If the state of the counter at the clock time t is “ ” the the state of the counter at t (after three clock cycles) will be (A) 00 (C) 10 (B) 01 (D) 11

F (A, B, C)

‘ ’

Q

CLK

A

‘ ’

J

EE - 2013 4. The clock frequency applied to the digital circuit show in the figure blow is 1 kHz. If the initial state of the output Q of the flip – f p is ‘ ’ the the freque cy f the output wavefrom Q in kHz is

C

(1,2,4,6) (1,2,6) (2,4,5,6) (1,5,6)

X

EE - 2008 2. A 3 line to 8 line decoder, with active low outputs, is used to implement a 3 – variable Boolean function as shown in the figure.

CLK

3L x 8L Decoder

z y x

0 1 2 3 4 5 6 7

(A) 0.25 (B) 0.5 F

The simplified form of Boolean function p e e ted i ‘ r duct f u ’ f r wi be (A) (X + Z). (̅ ̅ ̅ ). (Y + Z) (B) (̅ ̅ ). (X + Y + Z). (̅ ̅ ). (C) (̅ ̅+ Z). (̅ + Y + Z). (X + ̅ + Z). (X +Y + ̅ ) ̅ ). (X +̅ + ). (D) (̅ ̅ . (̅ ̅ ̅ ( )

T

Q

>

(C) 1 (D) 2

EE - 2014 5. A state diagram of a logic gate which exhibits a delay in the output is shown in the figure, where X is the d ’t care condition, and Q is the output representing the state. ⁄ ⁄

The logic gate represented by the state diagram is (A) XOR (C) AND (B) OR (D) NAND th

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Q

GATE QUESTION BANK

6.

Digital Circuits

A 3-bit gray counter is used to control the output of the multiplexer as shown in the figure. The initial state of the counter is . The output is pulled high. The output of the circuit follows the sequence -bit gray c u ter

f ip f p

5

̅ ̅ E

8.

Two monoshot multivibrators, one positive edge triggered ( ) and another negative edge triggered ( ), are connected as shown in figure

utput

(A) (B) (C) (D) 7.

5

̅

A JK flip flop can be implemented by T flip-flops. Identify the correct implementation.

̅

The monoshots a d when triggered produce pulses of width a d respectively, where . The steady state output voltage of the circuit is

f ip f p t

̅

f ip f p

̅

f ip f p ̅

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IN - 2006 1. Given that the initial state (Q1Q0) is 00, the counting sequence of the counter shown in the following figure is, Q1Q0 =

Digital Circuits

IN - 2007 Statement for Linked Answer Questions 3 & 4 Consider the circuit shown in the following figure.

‘ ’

J0

Q

J1

Q1

̅

K

̅1

0

1

0

CLK

(A) (B) (C) (D) 2.

K0

00 00 00 00

11 01 11 10

01 11 10 01

10 10 01 11

00 00 00 00

A combinational circuit using a 8-to-1 multiplexer is shown in the following figure. The minimized expression for the output (Z) is 1 1 1 0 0 0 1 0

I0 I1 I2 I3 I4 I5 I6 I7

(MSB)

The correct input-output relationship between Y and ( , ) is (A) Y= + (C) Y= ⨁ (B) Y= (D) Y= ⨁

4.

The

flip-flops are initialized =000. After 1 clock cycle, is equal to (A) 011 (C) 100 (B) 010 (D) 101

5.

A sequential circuit is shown in the figure below. Let the state of the circuit be encoded as . he tati implies that state Y is reachable from state X in a finite number of clock transitions.

Y Z

MUX

A B C Select Inputs

̅ ̅ (A) (B) C (A + B)

3.

(LSB)

̅̅ (C) ̅ (D) ̅ + AB

D

̅A

to

QB

QA

̅

Q

̅

CLK

TA

CLK

Q TB

CLK

Identify the INCORRECT statement. (A) (C) (B) (D)

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GATE QUESTION BANK

6.

A MUX circuit shown in the figure below implements a logic function . The correct expression for is Z

1 MUX out

̅

9.

The output F of the multiplexer circuit shown below expressed in terms of the inputs P, Q and R is

F1

R

I0

̅

I1 4

0 S

X

Digital Circuits

̅

I2

R

I3

1 MUX out

̅

F0

0 Y

(A) (B) (C) (D)

S

(A) (̅̅̅̅̅̅̅̅)⨁ ̅̅̅̅̅̅̅̅) ⨁ (B) (̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅

⨁ ⨁̅ ⨁ +Z

(C) (D)

IN - 2008 Statement for Linked Answer Questions 7 &8 Consider the counter circuit shown below.

10.

F

Y

1 MUX

S1

S0

P

Q

F=P⨁Q⨁R F = PQ + QR + RP F = (P ⨁ Q) R F = (P ⨁ Q) ̅

The inverters in the ring oscillator circuit shown below are identical. If the output waveform has a frequency of 10 MHz, the propagation delay of each inverter is Output

l

J Q

Q0

J Q

Q1

J

Q

Q2

Clock KCLR

KCLR

KCLR

̅̅̅̅̅̅̅

J Q Q3

(A) 5 ns (B) 10 ns

K CLR

̅̅̅̅̅̅̅

IN - 2009 11. The figure below shows a 3-bit ripple counter, with as the MSB. The flipflops are rising-edge triggered. The counting direction is

8.

J

1 Clock

Y

7.

(C) 20 ns (D) 50 ns

In the above figure, Y can be expressed as (A) (C) (B) (D)

Q

1

CLK 1

K

Q

J

Q1

1

1

K

Q

CLK

CLK ̅

J

̅

1

K

̅

(A) always down (B) always up (C) up or down depending on the initial state of only (D) up or down depending on the initial states of and

The above circuit is a (A) Mod – 8 Counter (B) Mod 9 Counter (C) Mod – 10 Counter (D) Mod – 11 Counter

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Q2

GATE QUESTION BANK

12.

In the figure shown, the initial state of Q is 0. The output is observed after the application of each clock pulse. The output sequence at Q is 1

CLK

In this circuit, the race around (A) Does not occur (B) Occurs when CLK = 0 (C) Occurs when CLK = 1 and A =B =1 (D) Occurs when clk = 1 and A = B = 0

̅

K

(A) 0 0 0 0 . . . (B) 1 0 1 0 . . .

(C) 1 1 1 1 . . . (D) 1 0 0 0 . . .

16.

IN - 2011 13. The circuit below shows as up/down counter working with a decoder and a flip-flop. Preset and clear of the flip-flop are asynchronous active-low inputs

̅

Consider the given circuit

Q

J

CLOCK

15.

Digital Circuits

̅ ̅

̅

̅

̅

̅

The state transition diagram for the logic circuit shown is

x ̅

x

e ect

̅

3 to 8 Decoder C ̅̅̅̅̅̅̅̅ reset Q Flip-Flop Clock ̅̅̅̅̅̅̅ c ear ̅

D

Count Down

B

A(L.SB)

Up/down Counter

Count Up Clock

Assuming that the initial value of counter output ( as zero, the counter output in decimal for 12 clock cycles are (A) 0,1,2,3,4,4,3,2,1,1,2,3,4, (B) 0,1,2,3,4,5,0,1,2,3,4,5,0, (C) 0,1,2,3,4,5,5,4,3,2,1,0,1 (D) 0,1,2,3,4,5,4,3,2,1,0,1,2 ECE/EE/IN - 2012 14. The output Y of a 2-bit comparator is logic 1 whenever the 2-bit input A is greater than the 2-bit input B. The number of combinations for which the output is logic 1, is (A) 4 (C) 8 (B) 6 (D) 10 th

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IN - 2013 17. The digital circuit shown below uses two negative edge- triggered D flip- flops. Assuming initial condition of and as zero, the ouput of this circuit is

D1

D0

D-Flip-flop

D-Flip-flop ̅̅̅̅

̅̅̅̅

Digital Circuits

IN - 2014 18. Frequency of an analog periodic signal in the range of 5 kHz - 10 kHz is to be measured with a resolution of 100Hz by measuring its period with a counter. Assuming negligible signal and transition delays the minimum clock frequency and minimum number of bits in the counter needed, respectively, are: (A) 1 MHz, 10-bits (B) 10 MHz, 10-bits (C) 1 MHz, 8-bits (D) 10MHz, 8-bits

Clock

(A) (B) (C) (D)

00, 01, 10, 11, clock 00, 01, 11, 10, 00, 11, 10, 01, 00, 01, 11, 11,

…… …… …… ……

Answer Keys and Explanations ECE 1.

S = 1, [Ans. D] When P is stuck at 1, f = A

4.

[Ans. B] The i/p to first F/F = ̅̅̅̅̅̅̅̅̅̅̅ The i/p to second F/F =

f

B

0

C (

2.

0 0 1 1

3.

)

[Ans. A] Present State

So

,C=1

0 1 1 0

[(

0

0

1 1 0 0

0 1 1 0

1 1 0 0

5.

[Ans. C] When X = 0, Y = 1 then P = 1 and Q = 0 X = 0, Y = 0 then P = 1 and Q = 1 X = 1, Y = 1 then P = 0 and Q = 1 or P = 1 and Q = 0

6.

[Ans. A] Let the output of first MUX is Y Y = ̅B + A̅ = A ⨁ B

and

[Ans. D] Before clock pulse S = C = 0 After first clock pulse A = B = 1 So S = 0, C=1 After second clock pulse A = B = 1 So

(initially at rest)

1st 0 0 1 1 clk 2nd 1 1 0 0 3rd 0 0 0 0 So sequence generated 00, 01, 10, 00

Next State 0 1 1 0

0

)]

th

th

th

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GATE QUESTION BANK

̅ X=̅ ⨁ So X = A ⨁ B ⨁ C ̅ ̅ ̅̅ i.e, ̅ ̅ 7.

So sequence is (00, 11, 10, 00, 11) or …..

[Ans. A] (

ts

11.

[Ans. C] For NAND latch, when ( ) are (0, 1), ( ) will be (1, 0) and when ( ) are (1, 1), ( ) will be (1, 0). For NOR latch, when ( ) are (0, 1), ( ) will be (1, 0) and when ( ) are (1, 1), ( ) will be (0, 0).

12.

[Ans. A]

)

̅̅ ̅̅

̅ ̅̅̅

̅ ̅

̅ ̅

ap PQ

RS

00 00

01

1

11 10 1

1

01

1

1

11

1

1

10

1

Digital Circuits

̅ ̅̅ SP

O Y = AB

PQ

B

̅̅̅

So Now only option (A) has two similar terms, we can say that option A is not most simplified version of Z it can be obtained as

A B Y=A ⨁B 1

PQ RS

11

10

1

1

01

1

1

11

1

1

10

1

00 00

01

1

̅̅ ̅

0 A

P̅S

B

PQ

So option A is correct choice 8.

9.

10.

13.

[Ans. B] At clock will be divide by 4 and will have 2 T delay w.r.t clock. [Ans. C] Initially, when clk is high and D is low, Q = 0 or 1, When clk goes low and D is also low , Q = 1, When clk is low and D goes high, Q = 0.

5

0 1 0 1 1

0 1 0 0 1

0 1 1 1 1

0 1 1 0 1

0 1 0 0 1

14.

a

b

c

d

e

f

g

0

0

1

1

1

1

1

1

0

0

1

1

0

1

1

0

1

1

1

0

1

1

0

1

1

0

1

1

1

1

0

0

1

1

1

1

a b c d e f g g d

[Ans. A]

0 1 0 1 1

[Ans. B]

̅̅̅̅ ̅̅̅̅

…. …. c e ̅̅̅̅ … ̅̅̅̅ … … c

e

[Ans. D] gates gates

th

th

th

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GATE QUESTION BANK

15.

[Ans. D] ̅̅ ̅̅

̅

̅̅

̅

̅̅ ̅

̅

̅̅

̅̅ ̅

21.

[Ans. D] If one input of the gate is kept constant, and A is interchanged with ̅, an XNOR gate acts as XOR gate.

22.

[Ans. C] In half subtractor Difference x y xy x y b rr w x y x y Borrow 0 0 0 0 1 1 1 0 0 1 1 0

̅̅ ̅

̅̅ [ ̅ ̅̅

x

x̅

x.

x ̅ ̅ ̅̅̅

]

̅̅

5 16.

[Ans. D] ⨀

=

Initially 0 After first clk 1 nd After 2 clk 1 After 3rdclk 0 So . . . . . .. 17.

[Ans. D] Q(present) 0 1 0 1

0 1 1 0

Digital Circuits

0 0 1 1

0 0 0 1

23.

Difference 0 1 1 0

[Ans. C]

̅

Q (next) 1 0 1 0

y

1 1 0 0

1 0 1 0

1 1 0 0

̅

⨀ ̅̅̅̅ ̅̅̅̅ ̅ 18.

19.

20.

[Ans. A] Given Y=1, this implies Both the output of D- flip flop should be 1 i.e, input at first flip flop is 1 and for output of 2nd flip flop to be 1, inverted output of first flip should be 1 in previous clock, for which input must be 0 so data is changing from 0 to 1. [Ans. D] From the CKT 0 is connected to & d ‘ ’ is c ected t

So wave form 24.

[Ans. *] Range 62.4 to 62.6 .5

[Ans. D] This is a figure of Johnson counter So 0 1 1 0 0 1 So =

and

is correct

0 0 1 1 0 0

z th

th

th

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GATE QUESTION BANK

25.

[Ans. D]

26.

[Ans. D]

2.

Digital Circuits

[Ans. A] ∑

5

[ 3.

(

[Ans. C] Clock

)

]

Input

Output

̅

̅

utput f first et utput f sec d

Initial state 1 2 3

̅ ̅

28.

[Ans. C] ̅ ̅̅ ̅ ̅

X

̅̅̅ ̅̅

̅ ̅

̅̅ CLK

[Ans. A]

30.

EE 1.

[Ans. C] ̅̅ ̅

1 1 0

̅̅ ̅

1 0 1

1 0 0

[Ans. B]

From above Fig. ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ̅ . ̅̅̅̅̅̅̅̅̅̅̅ ̅ ] [ ̅ ecause ̅̅̅̅̅̅̅̅̅ ̅ a ways

̅ 29.

0 1 0

0

̅ 4.

27.

1 0 1

1

̅

T

Q

Q

>

a ways

̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ̅ . ̅̅̅̅̅̅̅̅̅̅̅ ̅ ̅̅̅̅ ̅ .

[Ans. *] Range 194.9 to 195.1 Worst case propagation delay = carry propagation delay of 15FA stages + max(carry Pd & sum Pd of last FA stage) 5 5 5 s

i put a ways r ‘ ’ f ip flop if input is =1 then output will be complemented at the time of triggering. CLK

T

[Ans. A]

2T

f

F (A, B, C)= ̅̅ ̅̅

̅̅ ̅̅

̅ ̅

̅ ̅ ̅

th

.5 f .5 .5 z

th

th

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GATE QUESTION BANK

5.

[Ans. D]

IN 1.

Digital Circuits

[Ans. A] Clock

x

( 1 1 1 1 1

gate 6.

[Ans. A] Gray code output

o/p 0 0 1 1

0 1 0 1

̅ E

When

0 0 0 0 1 1 1 1

0 0 1 1 1 1 0 0

0 1 1 0 0 1 1 1

2.

1 0 1 0

0 1 0 1 0

0 1 1 0 0

[Ans. C]

(

) )(

)

utput e ect i puts 3.

[Ans. B] Y=

[Ans. A] T-FF to JKFF is given by eq ̅

4.

[Ans. B] a d

This is implemented on options [A] 8.

1 0 1 0

(

utput 7.

1 1 1 1 1

)

[Ans. C] Let is initially high. Since is from will be high for time. The waveform is shown below:

So initially

it means so after one clock cycle will be 010.

5.

[Ans. D] 01 10, both both will toggle.

̅

= 1 so

and

t

6.

[Ans. B]

7.

[Ans. A] ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ Y = ̅̅̅̅̅̅̅ . ̅̅̅̅̅̅̅

trigger t

t

= = t

8.

+

[Ans. C] Whenever Y=1, then clear input of all the s receives ‘ ’ a d utputs f the th

th

th

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GATE QUESTION BANK

counter will be reset. When count = 1010, Y=1 and counter will be reset. Q3 Q2 Q1 Q0 1 0 1 0 1 1 0 0 1 1 1 0 9.

)

(

10.

So sequence is 0,1,2,3,4,5,4,3,2,1,0,1,2. 14.

[Ans. B] Let A1 A0 be the bits of number A and B1B0 be the bits of number B and let Y be the output A1 A0 B1 B0 Y 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 A1 A0 B1 B0 Y 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 From the truth table we see that the u ber f ti es ‘ ’ bec es is

15.

[Ans. A]

[Ans. A] (

(

)

)

[Ans. B] f

t where N – no. of inverters and t - Propagation delay of each, so t sec 11.

[Ans. A] Since triggering in positive edge triggering & Q of pervious flip flop is input to next hence always down

12.

[Ans. C]

Digital Circuits

̅ Also the truth table ̅

J

K ̅

1

1

0

1

0

1

1

0

1 1 ̅

13.

[Ans. D] Initially Q=0 and count up ( ̅ =1) is active so it started counting up and when it reaches to 5 then decoder output at pin 5 becomes 0 and preset will be active and it will set Q and it will make the counter mode down and count becomes 4, then 3 then 2 then 1 then 0, as soon as it reaches 0, decoder output at pin 0 is low and clear is active and Q goes to 0 and ̅ = 1 so up is active a d it c u ts ……

next

= . . = A.CLK + Q

next = A.CLK + If CLK = 1 and A and B = 1

then

} No race around

If CLK = 1 and A = B = 0 } No race around

th

th

th

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GATE QUESTION BANK

Digital Circuits

Thus race around does not occur in the circuit 16.

[Ans. D] State table A

D

0

0

1

1

0

1

0

0

1

0

0

0

1

1

1

1

From State table A=O

Q=O

Q=1 A=O A=1

A=1 17.

[Ans. B] State table Present state

Next state

0 0 0 0 1 1 1 1 1 1 0 0 00, 01, 11, 10, 00 18.

1 1 0 0

0 1 1 0

1 1 0 0

[Ans. C] i c c freque cy z

z

5 i

th

th

th

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GATE QUESTION BANK

Digital Circuits

AD /DA Convertor ECE-2006 1. A 4-bit D/A converter is connected to a free-running 3-bit UP counter, as shown in the following figure. Which of the following waveforms will be observed at V 0?

3.

D3 D2 D1 D0

Q1 Q0 Clock 3-bit Counter

IDEAL Vo

1k

(C) – 3.125 V (D) – 6.250 V

ECE-2008 Statement for linked Answer Questions 4 and 5 In the following circuit, the comparator output is logic “1” if and is logic “0”otherwise. The D/A conversion is done as per the relation = ∑ 2 Volts, where (MSB), and (LSB) are the counter outputs. The counter starts from the clear state.

1k Q2

The voltage Vo is (A) – 0.781 V (B) – 1.562 V

D/A Convertor

In the figure shown above, the ground has been shown by the symbol (A)

4 bit D/A converter +5V

Binary to BCD

̅̅̅̅

(B) ̅̅̅̅̅

2 Digit LED Display

4bit Upcounter

= 6.2

Clock

(C) 4.

The stable reading of the LED displays is (A) 06 (C) 12 (B) 07 (D) 13

5.

The magnitude of the error between and at steady state in volts is (A) 0.2 (C) 0.5 (B) 0.3 (D) 1.0

(D)

ECE-2007 Statement for Linked Answer Questions 2 and 3 In the Digital-to-Analog converter circuit shown in the figure below, = 10 V and R = 10kΩ. R

R

R

I

2R

V R

2R

2R

2R

2R

R +

2.

The current I is (A) 31.25 μA (B) 62.5 μA

V0

ECE-2011 6. The output of a 3 – stage Johnson (twisted – ring) counter is fed to a digital – to – analog (D/A) converter as shown in the figure below. Assume all states of the counter to be unset initially. The waveform which represents the D/A converter output is

(C) 125 μA (D) 250 μA

th

th

th

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GATE QUESTION BANK

(B) droop rate decreases and acquisition time increases (C) droop rate increases and acquisition time decreases (D) droop rate increases and acquisition time increases

D/A Converter D

Clock

D

D

Johnson Counter

(A)

Digital Circuits

EE-2006 1. A student has made a 3 – bit binary down counter and connected to the R – 2R ladder type DAC [Gain = ( 1 KΩ/2R)] as shown in figure to generate a staircase waveform. The output achieved is different as shown in figure. What could be the possible cause of this error?

(B)

2R

1kHz clock

R

R

R

2R

2R

2R

1kΩ +12V Vo 12V

Counter 10kΩ

7 6 (C)

5 4 3 2 1 0 ms

0 1 2 3 4 5 6 7

(A) The resistance values are incorrect (B) The counter is not working properly (C) The connection from the counter to DAC is not proper (D) The R and 2R resistances are interchanged

(D)

ECE-2014 7. For a given sample-and-hold circuit, if the value of the hold capacitor is increased, then (A) droop rate decreases and acquisition time decreases

IN-2006 Common Data for Questions 1 and 2 An R-2R ladder type DAC is shown below. If a switch status is ‘0’ 0 is applied and if a switch status is ‘1’ 5 is applied to the corresponding terminal of the DAC.

th

th

th

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GATE QUESTION BANK R

R 2R

2R

2R

1.5V will produce a digital output of (A) 90H (C) 9BH (B) 96H (D) A0H

+

2

+ 5V

1.

What is the output voltage (V0) for the switch status S0=0,S1=1,S2=1? (A)

V

(B) 2.

V

(C) (D)

. .

Digital Circuits

V

IN - 2010 5. A 4-bit successive approximation type ADC has a full scale value of 15V. The sequence of the states, the SAR will traverse, for the conversion of an input of 8.15V is (A) ( )

V

What is the step size of the DAC? (A) 0.125 V (C) 0.625 V (B) 0.525 V (D) 0.75 V

IN-2007 3. The circuit shown in the figure below works as a 2-bit analog to digital converter for 0 ≤ ≤3 . 3V 0.5kΩ

(C)

tart conversion

tart conversion

1 0 0 0

1 0 0 0

1 1 0 0

0 1 0 0

1 0 1 0

0 0 1 0

1 0 0 1

0 0 0 1

1 0 0 0

0 0 0 0

End conversion

End conversion

(D)

tart conversion

tart conversion

1.0kΩ 1.0kΩ

Digital

1 0 0 0

1 0 0 0

Circuit

0 1 0 0

1 1 0 0

0 1 1 0

1 1 1 0

0 1 1 1

1 1 1 1

1 0 0 0

1 1 1 1

End conversion

End conversion

0.5kΩ

The MSB of the output , expressed as a Boolean function of the inputs , , , is given by (A) (C) (B) (D) + IN-2009 4. An 8- it ADC with 2’s complement output, has a nominal input range of 2V to +2V. It generates a digital code of 00H for an analog input in the range – 7.8125mV to +7.8125mV. An input of

IN-2014 6. A thermopile is constructed using 10 junctions of Chromel-Constantan (sensitivity 60μ /°C for each junction) connected in series. The output is fed to an amplifier having an infinite input impedance and a gain of 10. The output from the amplifier is acquired using a

th

th

th

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GATE QUESTION BANK

10-bit ADC, with reference voltage of 5 V. The resolution of this system in units of °C is _________ 7.

DAC output and input to be positive, the maximum error in conversion of the analog sample value is: Input Comparator ample

An N-bit ADC has an analog reference voltage V. Assuming zero mean and uniform distribution of the quantization error, the quantization noise power will be: (A) (B)

(

)

(

)

(C) (D)

(

Digital Circuits

tart of conversion ( C) Control logic

l l

R 2R adder DAC

)

Clock

l

l

p counter l

utpt uffer

l

√

l

8.

Reset l

l

The circuit in the figure represents a counter-based unipolar ADC. When SOC is asserted the counter is reset and clock is enabled so that the counter counts up and the DAC output grows. When the DAC output exceeds the input sample value, the comparator switches from logic 0 to logic 1, disabling the clock and enabling the output buffer by asserting EOC. Assuming all components to be ideal,

(A) (B) (C) (D)

th

th

Ena le End of Conversion l (E C)

directly proportional to inversely proportional to l independent of directly proportional to frequency

th

clock

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GATE QUESTION BANK

Digital Circuits

Answer Keys & Explanations ECE 1.

2.

7.

[Ans. B]

[Ans. B] The i/p to D/A will be 0000, 0001, 0010, 0011, 1000, 1001, 1010, 1011. So O/P will be in B waveform

In a capacitor drop rate in given an we know i = C

dv 1 dt C From the above relation it is clear that if capacitor value increases then the drop rate decreases because of inversion relation. C Also = C = it ⇒ t = ⇒t c i From the above relation, it is clear that if capacitor value increases acquisition time also increases as of proportionality relation ⇒

[Ans. B] I 2

I V

R

2R

R

I 4

R

2R

I 8 R

I 16

2R

2R

2R

I 16

I 4

R V0

A

Due to virtual ground, node A can be considered as ground so = 10K So I = So I = 3.

EE 1.

[Ans. C] =

4.

= 1mA = 62.5 μA

I R[ 4

I ]= 16

[Ans. C] Sequence generated is 7→111 0→000 3→011 1→001 5→101 2→010 1→001 3→011 6→110 Instead of 4→100 2→010 5→101 4→100 6→110 0→000 7→111 Simply just by observation we can say that error is because of lack of proper connections

3.125

[Ans. D] =

2

4

volts

Counter will stop when VDAC> 6.2 volt So counter will stop when = 1101 o ED will display 1101 i.e CD ⇒ 13 5.

6.

[Ans. B] In steady state = 0.5 6.5V So error = 6.5 6.2 = 0.3V

0

2

dv dt

4= IN 1.

[Ans. B] At switch status =0 = =1

[Ans. A] For the Johnson counter sequence D D D 0 0 0 0 1 0 0 4 1 1 0 6 1 1 1 7 0 1 1 3 0 0 1 1 0 0 0 0

R

2R

2R

R

2R

2R

Thevenin s equivalent

th

th

th

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Digital Circuits

i.e 2V → 128 2R

2R

1.5V →

2R

2R ≡

5 2

2R 5

5.

[Ans. A] By characteristics of SAR ADC.

6.

[Ans. * ] Range 0.800 to 0.833 For 10 junctions the thermopile voltage will be 600 μ = 0.6 m When it measured through amplifier of gain 10, its voltage will be 6 mV. So, the thermopile sensitivity after amplification is 0.006 V/° C Which will be input to ADC of resolution = /2 1 ADC resolution = 5/1023=0.00488 In terms of temperature, the resolution is 0.00488/0.006=0.814

7.

[Ans. A]

8.

[Ans. A] Maximum error is equal to step size which is directly proportional to .

R ≡

∴ 2.

15 2

R

=

[Ans. C] Step size =

3.

4.

= = 0.625

[Ans. B] Truth table of ADC is 0 0 0 1 ⇒

0 0 1 1 =

0 1 1 1

0 0 1 1

0 1 0 1

128 = 96

It is 96 for 1.5 analog input, while is given an 96 = 2 complement of +96 = 2’s complement of 01100000 =10100000 = A0H

5

5

.

[Ans. D] 8 – it ADC o/p is in 2’s complement form i.e it represents 127 128 i/p voltage Range = 2 to 2

aximum Error =

th

th

th

2

1

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Digital Circuits

Semiconductor Memory ECE-2014 1. If WL is the Word Line and BL is the Bit Line, an SRAM cell is shown in

̅̅̅̅

̅̅̅̅

̅̅̅̅

̅̅̅̅

EE-2009 1. The increasing order of speed of data access for the following devices is i. Cache Memory ii. CDROM iii. Dynamic RAM iv. Processor Registers v. Magnetic Tape (A) (v), (ii), (iii), (iv), (i ) (B) (v), (ii), (iii), (i), (iv) (C) (ii), (i), (iii), (iv), (v) (D) (v), ( ii), (i) , (iii), (iv) IN-2011 1. An bit RAM is interfaced to an 8085 microprocessor. In a fully decoded Scheme if the address of the last memory location of this RAM is 4FFFH, the address of the first memory location of the RAM will be, (A) 1000 H (C) 3000 H (B) 2000 H (D) 4000 H

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GATE QUESTION BANK

Digital Circuits

Answer Keys & Explanations ECE 1.

EE 1.

IN 1.

[Ans. B] For an SRAM construction four MOSFETs are required (2-PMOS and 2-NMOS) with interchanged outputs connected to each CMOS inverter. So option (B) is correct.

[Ans. B] Processor registers has highest speed. Followed by cache memory then dynamic ram (slower than static ram because of refreshing required)

[Ans. C] Capacity of chip = last memory address – First memory address+1 = last memory address – Capacity of chip+1 = 4FFFH – 2000H+1 =3000H

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th

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GATE QUESTION BANK

Digital Circuits

Microprocessors ECE-2006 1. An I/O peripheral device shown in Figure (b) below is to be interfaced to an 8085 microprocessor. To select the I/O device in the I/O address range D4 H - D7 H, its chip-select (̅̅̅) should be connected to the output of the decoder shown in Figure (a) below. A2

LSB

A3

3 8 Decoder

A4

MSB

3.

In the circuit shown, the device connected to Y5 can have address in the range

o de ice hip e ec o decode

0 1 2 3 4 5 6 7

̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅

(A) 2000 20FF (B) 2D00 2DFF ̅̅̅̅

A7 A6 ̅̅̅

Data ̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅

Fig. (a) I/O Peripheral

A1 A0

̅̅̅ Fig. (b)

(A) output 7 (B) output 5

(C) 2E00 2EFF (D) FD00 FDFF

(C) output 2 (D) output 0

ECE-2010 2. For the 8085 assembly language program given below, the content of the accumulator after the execution of the program is 3000 MVI A, 45H 3002 MOV B, A 3003 STC 3004 CMC 3005 RAR 3006 XRA B (A) 00H (C) 67H (B) 45H (D) E7H

ECE-2011 4. An 8085 assembly language program is given below. Assume that the carry flag is initially unset. The content of the accumulator after the execution of the program is MVI A, 07H RLC MOV B, A RLC RLC ADD B RRC (A) 8CH (B) 64H

(C) 23H (D) 15H

ECE-2014 5. For the 8085 microprocessor, the interfacing circuit to input 8-bit digital data ( ) from an external device is shown in the figure. The instruction for correct data transfer is

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GATE QUESTION BANK o ecode

⁄

̅̅̅̅̅ ̅̅̅̅̅̅

̅̅̅̅̅

̅̅̅̅̅

(A) MVI A, F8H (B) IN F8H 6.

(C) OUT F8H (D) LDA F8F8H

n

mic op oce o execu e “ H” wi h a ing add e oca ion 1FFEH (STA copies the contents of the accumulator to the 16-bit address location). While the instruction is fetched and executed, the sequence of values written at the address pins is (A) 1FH, 1FH, 20H, 12H (B) 1FH, FEH, 1FH, FFH, 12H (C) 1FH, 1FH, 12H, 12H (D) 1FH, 1FH, 12H, 20H, 12H

EE-2006 1. In an 8085 microprocessor based system, it is desired to increment the contents of memory location whose address is available in (D, E) register pair and store the result in same location. The sequence of instructions is (A) XCH G (C) INXD INRM XCH G (B) XCH G (D) INRM INXH XCH G 2.

How many times DCR L instruction will be executed? (A) 255 (C) 65025 (B) 510 (D) 65279

e ice

aa u

igi a inpu

Digital Circuits

A software delay subroutine is written as given below: DELAY: MVI H, 255 D MVI L, 255 D LOOP: DCR L JNZ LOOP DCR H JNZ LOOP

EE-2008 3. The contents (in Hexadecimal) of some of the memory location in an 8085A based system are given below Address Contents .. .. 26FE 00 26FF 01 2700 02 2701 03 2702 04 .. .. The contents of stack pointer (SP), Program counter (PC) and (H, L) are 2700H, 2100H and 0000H respectively, when the following sequence of instruction are executed, 2100 H: DAD SP 2101 H: PCHL The contents of (SP) and (PC) at the end of execution will be (A) PC = 2102 H, SP = 2700 H (B) PC = 2700 H, SP = 2700 H (C) PC = 2800 H, SP = 26 FE H (D) PC = 2A02 H, SP= 2702 H EE-2009 4. In an 8085 microprocessor, the contents of the Accumulator, after the following instructions are executed will become XRA A MVI B, F0H SUB B (A) 01 H (C) F0 H (B) 0F H (D) 10 H EE-2010 5. hen a “ dd ” in uc ion i executed, the CPU carries out the following sequential operations internally: th

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Note: (R) means content of register R ((R)) means content of memory location pointed to by R PC means Program Counter SP means Stack Pointer (A) (SP) incremented (PC) Addr ((SP)) (PC) (B) (PC) Addr ((SP)) (PC) (SP) incremented (C) (PC) Addr (SP) incremented ((SP)) (PC) (D) ((SP)) (PC) (SP) incremented (PC) Addr EE-2011 6. A portion of the main program to call a subroutine SUB in an 8085 environment is given below. : : LXI D, DISP LP: CALL SUB : : It is desired that control be returned to LP + DISP + 3 when the RET instruction is executed in the subroutine. The set of instructions that precede the RET instruction in the subroutine are (A) POP D (C) POP H DAD H DAD D PUSH D PUSH H (B) POP H DAD D INX H INX H INX H PUSH H

(D) XTHL INXD INX D INX D XTHL

Digital Circuits

EE-2014 7. An output device is interfaced with 8-bit microprocessor 8085A. The interfacing circuit is shown in figure

ecode u pu

̅ ̅

u pu

o

e ice

̅̅̅̅̅

̅

The interfacing circuit makes use of 3 Line to 8 Line decoder having 3 enable ̅ ̅ The address of the device lines is (A) (C) (B) (D) 8.

In an 8085 microprocessor, the following program is executed Address location - Instruction 2000H XRA A 2001H MVI B, 04H 2003H MVI A, 03H 2005H RAR 2006H DCR B 2007H JNZ 2005 200AH HLT At the end of program, register A contains (A) 60H (C) 06H (B) 30H (D) 03H

9.

In 8085 microprocessor, the operation performed by the instruction LHLD is (A) H (B) H (C) H (D) H

IN-2006 1. An 8085 assembly language program is given as follows. The execution time of

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GATE QUESTION BANK

2.

Digital Circuits

each instruction is given against the instruction in terms of T-state. Instruction T-states MVI B, 0AH 7T LOOP: MVIC, 05H 7T DCR C 4T DCR B 4T JNZ LOOP 10T/7T The execution time of the program in terms of T-states is (A) 247 T (C) 254 T (B) 250 T (D) 257 T

4.

A memory mapped I/O device has an address of 00F0H. Which of the following 8085 instructions outputs the content of the accumulator to the I/O device? (A) LXI H, 00F0H MOV M, A (B) LXI H, 00F0H OUT M (C) LXI H, 00F0H OUT F0H (D) LXI H, 00F0H MOV A, M

IN-2008 5. A part of a program written for an 8085 microprocessor is shown below. When the program execution reaches LOOP2, the value of register C will be SUB A MOV C, A LOOP1: INR A DAA JC LOOP2 JNC LOOP1 LOOP2: NOP (A) 63H (C) 99H (B) 64H (D) 100H

IN-2007 3. 8-bi igned in ege in ’ comp emen form are read into the accumulator of an 8085 microprocessor from an I/O port using the following assembly language program segment with symbolic addresses. BEGIN: INPORT RAL JNC BEGIN RAR END: HLT This program (A) Halts upon reading a negative number (B) Halts upon reading a positive number (C) Halts upon reading a zero (D) Never halts

A snapshot of the address, data and control buses of an 8085 microprocessor executing program is given below: Address 2020H Data 24H Logic high IO/ ̅ ̅̅̅̅ Logic high ̅̅̅̅̅ Logic Low The assembly language instruction being executed is (A) IN 24H (C) OUT 24H (B) IN 20H (D) OUT 20H

6.

A 2k×8 bit RAM is interfaced to an 8-bit microprocessor. If the address of the first memory location in the RAM is 0800H, the address of the last memory location will be (A) 1000H (C) 4800H (B) 0FFFH (D) 47FFH

IN-2009 7. The following is an assembly language program for 8085 microprocessors Address Instruction Mnemonic Code 1000H 3E06 MVI A, 06H 1002H C6 70 ADI 70H 1004H 32 07 10 STA 1007H 1007H AF XRA A 1008H 76 HLT th

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GATE QUESTION BANK

When this program halts, the accumulator contains (A) 00H (C) 70H (B) 06H (D) 76H 8.

Consider a system consisting of a microprocessor, memory, and peripheral devices connected by a common bus. During DMA data transfer, the microprocessor (A) only reads from the bus (B) only writes to the bus (C) both reads from and writes to the bus (D) neither reads from nor writes to the bus

IN-2010 9. In an 8085 processor, the main program calls the subroutine SUB1 given below. When the program returns to the main program after executing SUB1, the value in the accumulator is Address Opcode Mnemonic 2000 3E 00 SUB1: MVIA,00h 2002 CD 05 20 CALL SUB2 2005 3C SUB2: INR A 2006 C9 RET (A) 00 (C) 02 (B) 01 (D) 03 10.

(A) 3000H (B) 4FFFH 11.

d

a a ̅̅̅

b b b b

The subroutine SBX given below is executed by an 8085 processor. The value in the accumulator immediately after the execution of the subroutine will be: SBX: MVI A, 99H ADI 11H MOV C, A RET (A) 00H (C) 99H (B) 11H (D) AAH

dd e

u

x og ammab e

x

n e up con o e

aa u

x

Assuming vectored interrupt, a correct sequence of operations when a single external interrupt (Ext INT1) is received will be : (A) x → → a a ead→ (B) x → → → a a ead (C) x → → → dd e Write (D) x → → aa ead→ dd e ie

d

̅̅̅

(C) AFFFH (D) C000H

IN-2014 12. A microprocessor accepts external interrupts (Ext INT) through a Programmable Interrupt Controller as shown in the figure.

A 8-bit DAC is interfaced with a microprocessor having 16 address lines (A0...A15) as shown in the adjoining figure. A possible valid address for this DAC is ine o ine ecode

Digital Circuits

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GATE QUESTION BANK

Digital Circuits

Answer Keys & Explanations ECE 1.

2.

0100 [Ans. B] I/O address range D4 – D7 H ⇒ xx So, and should be. So, decoder output should be 5.

3.

5.

⏟

0111

RLC

1110

MOV B, A

1110

RLC

1100

RLC

1000

ADD B A 0000 1110 + B 0011 1000

⏟

⏟

⏟

That is F8 F8 memory address is selected So LDA F8F8 [Load accumulator from the content of memory location F8F8H] should be the instruction for data transfer

[Ans. B] Address bits should be as follows:

[Ans. C] MVI A, 07 H

H

[Ans. D] In this figure the chip select and will be selected if the memory address will be

Content of register A = 45 B = 45 CY =1 CY =0 A = 22 ⊕

6.

(and for hey can be he o 2D00 – 2DFF (Range) 4.

→

[Ans. C] MVI A, 45 MOV B, A STC CMC RAR XRA B

0110

[Ans. A] STA 1234H is stored as follows 1FFE Opcode of STA 1FFF 34 H 2000 12H After this 1234H will be loaded in the address bus. So the correct sequence of values at are 1F, 1F, 20, 12

)

he content of ‘ ’ he content of ‘ ’ he content of ‘ ’ he content of ‘ ’ he content of ‘ ’

EE 1.

[Ans. A] XCHG (HL) (DE) INR M (HL) = (HL) + 1 So this sequence stores the address into HL pair and then increment the content of memory location specified by HL pair.

2.

[Ans. D] DCR L executed = 255 × 255 + 255 – 1 = 65279 times

3.

[Ans. B] Given, (SP) = 2700H (PC) = 2100 H (HL) = 0000H th

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2100 H : DAD SP (SP) + (HL) → (HL) 2700 H + 0000 H = 2700 H stored in HL pair 2101 H : PCHL : The content of HL are transferred to (PC) So now (PC) = 2700H and (SP) also unchanged (PC) = 2700 H (SP) = 2700 H 4.

5.

6.

[Ans. D] XRA A MVI B, F0H SUB B

Output line 2 is selected So ⇒

⏟

[Ans. A] H XRA A H MVI B, H H MVI A, H H RAR

⇒ ⇒ ⇒ – B will go to accumulator 00 – F0 = 10

SUB

⇒ ⏟

⏟

[Ans. D] First of all content of PC is loaded into stack. i.e. address of next instruction to be executed is loaded onto stack. i.e. SP is decremented then PC is loaded by address given in call instruction.

CALL

̅

⏟

8.

[Ans. C] Call take 3 address locations. RET always returns to LP + 3 location, this stored in SP. So to return to LP + DISP + 3 we have to add DISP to SP. POP H DAD D PUSH H Normal call operation shown.

Digital Circuits

→ clear accumulator → B= H → A=

H

→ Rotate accumulation to right A = 10000001 → Decrement B, B= → ump o when z

H H

DCR B INZ 2005 200 AH HLT After 3 rotation H 9.

[Ans. C] LHLD instruction loads the value at memory location specified by the immediate Value in H and L pair register Value at will be stored in L register value of will be stored in H register

IN 1.

[Ans. C]

RET

2. 7.

[Ans. A] Since I/O device is memory mapped I/O Memory related instructions will be used for data transfer

[Ans. B] To enable decoder

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H H → load the address of memory in HL pair → load the memory by content of A 3.

4.

5.

Digital Circuits

6.

[Ans. B] Starting address 0800H, so last address = 0800 + 7FF = 0FFFH.

7.

[Ans. A] Last instruction is XRA A, so accumulator contents will be 00

8.

[Ans. C] In DMA transfer of data between source and destination takes place without any involvement of microprocessor. During both in read/write mode.

9.

[Ans. B] SUB 1: MVI A, 00H → H CALL SUB 2 → program will shift to SUB 2 address location SUB 2: INR A →A = 00H + 01H=001H RET → returned to main program The contents of Accumulation after execution of the above SUB 2 is 01H

[Ans. D] Address 2020H: The contents of lower and higher address lines are same i.e. 20H. So, this indicates that the technique is I/O mapped I/O. Data 24H: This indicates that the content of accumulator is 24 H, which have to sent to I/O port. ̅ (Logic High): ̅ high indicates that this is input/output operation. ̅̅̅̅ (Logic High): ̅̅̅̅ logic high indicates that read operation is inactive. ̅̅̅̅̅ (Logic Low): ̅̅̅̅̅ logic low indicates that write operation is active. So, by observing all the operations the appropriate instruction is OUT20H.

10.

[Ans. A] To select 2-4 line decoder, A15 = 0 To select the DAC, b2 should be active, i.e. A14 = 0 and A13 = 1. Reset all address lines can be either 0 or 1. So, address can be 001x xxxx xxxx xxxx Out of four choices this is satisfied by 3000 H only.

11.

[Ans. D] Instructions MVI A, 99H ADI 11H MOV C,A

[Ans. A] Execution will go to Loop 2 when A contain 64H = , at that time C contains 63 H.

12.

[Ans. A] e.g. 01H positive number → After RAL CY Accumulator 0 0000 0010 or 00000011 JNC is TRUE and again IN instruction will load accumulator with same port address, and microprocessor will go into infinite loop. Negative number → after RAL CY Accumulator 1 0000 0001 or 00000000 JNC is false and microprocessor will come out from the loop. So, this program halts upon reading a negative number.

th

Content of register A = 99 A = 99 + 11 = AAH C = AAH

[Ans. B] The correct sequence is external interrupts occurs at PIC, then it is transferred to microprocessor, then interrupt is acknowledge and finally data is read. th

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GATE QUESTION BANK

Communication

Amplitude Modulation (AM) ECE - 2006 1. The diagonal clipping in Amplitude demodulation (using envelope detector) can be avoided if RC time-constant of the envelope detector satisfies the following condition, (here W is message bandwidth and is carrier frequency both in )

2.

3.

(A)

(C)

(B)

(D)

Common Data for Questions 2 & 3 Consider the following Amplitude Modulated (AM) signal, where () ( ) The average side-band power for the AM signal given above is (A) 25 (C) 6.25 (B) 12.5 (D) 3.125 The AM signal gets added to a noise with Power Spectral Density ( )given in the figure below. The ratio of average sideband power to mean noise power would be ()

N0/2

-fc-B

fc

fc+B

fc-B

(A)

(C)

(B)

(D)

fc fc+B

f

ECE - 2008 4. Consider the amplitude modulated (AM) signal . For demodulating the signal using envelope detector , the minimum value of should be (A) 2 (C) 0.5 (B) 1 (D) 0

ECE - 2009 5. A message signal given by m(t) = . /

. /

is amplitude- modulated with a carrier of frequency to generate s(t) = [1+m(t)] . What is the power efficiency achieved by this modulation scheme? (A) 8.33% (C) 20% (B) 11.11% (D) 25% ECE - 2010 6. Suppose that the modulating signal is () ( ) and the carrier signal is ( ) ( ). Which one of the following is a conventional AM signal without over modulation? (A) ( ) () ( ) ( )(B) ( ) , ( ) (C)

()

(

(D)

()

( )

(

)

() (

) (

( )

)

ECE - 2011 7. A message signal () modulates the carrier () =1MHz to produce an AM signal. For demodulating the generated AM signal using an envelope detector, the time constant RC of the detector circuit should satisfy. (A) 0.5 ms< RC < 1 ms (B) μ (C) μ (D) RC >> 0.5 ms ECE - 2014 8. Consider sinusoidal modulation in an AM system. Assuming no overmodulation, the modulation index (𝜇) when the maximum and minimum values of the envelope, respectively, are 3 V and 1 V, is ________. th

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)

GATE QUESTION BANK

9.

Communication

In the figure, M(f) is the Fourier transform of the message signal m(t) where A = 100 Hz and B = 40 Hz. Given () ( ) and w(t)=cos( ( +A)t), where >A. The cutoff frequencies of both the filters are .

()

() ()

()

The bandwidth of the signal at the output of the modulator (in Hz) is _____. 10.

() ( ) Let ( ) be sampled at 20 Hz and reconstructed using an ideal low-pass filter with cut-off frequency of 20 Hz. The frequency/frequencies present in the reconstructed signal is/are (A) 5 Hz and 15 Hz only (B) 10 Hz and 15 Hz only (C) 5 Hz, 10 Hz and 15 Hz only (D) 5 Hz only

11.

Consider an FM signal , f(t) The maximum deviation of the instantaneous frequency from the carrier frequency is (A) (C) (B) (D)

IN - 2008 1. Consider the AM signal s(t) = , ( )( ). It is given that the bandwidth of the real, low – pass message signal m(t) is 2 kHz. If the bandwidth of the bandpass signal s(t) will be (A) 2.004 MHz (C) 4 kHz (B) 2 MHz (D) 2 kHz

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GATE QUESTION BANK

Communication

Answer Keys and Explanations ECE 1.

2.

()

, ( in option (B),

[Ans. A] Diagonal clipping can be avoided if

Average SB power =

( 3.

μ μ

)

)

(

μ

(

) (

()

)

So, it is conventional AM signal but with over-modulation. In option (C),

[Ans. C]

*(

( ))

()

(

)

()

(

) +

(

Therefore, it is convectional AM signal without over-modulation.

)

[Ans. B]

7.

[Ans. B]

μ 4.

[Ans. A] Modulated signal () , Condition for envelope detection of an AM signal is

8.

[Ans. *] Range 0.45 to 0.55 μ

9.

[Ans. *] Range 59.9 to 60.1 ()

Therefore minimum value of AC should be 2. 5.

()

() ()

[Ans. C]

()

√ Power efficiency

√ =

After HPF with cutoff

⁄ ⁄

/2

6.

[Ans. C] Convectional AM signal is th

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)

GATE QUESTION BANK

After multiplied by ( ) and passing through LPF with cutoff

IN 1.

Communication

[Ans. C] In case of AM modulation B.W becomes double of that of message signal B.W.

1/4

So bandwidth of the modulator is (A – B) Hz = 10.

[Ans. A] ()

After sampling (

)

After passing through LPF with only 5 Hz and 15 Hz will be retained 11.

[Ans. A] (

|

)|

Maximum phase deviation | |

| |

(

)

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GATE QUESTION BANK

Communication

DSBSC, SSB and VSB Modulation ECE - 2006 1. A message signal with bandwidth 10 kHz is Lower-Side Band SSB modulated with carrier frequency The resulting signal is then passed through a Narrow-Band Frequency Modulator with carrier frequency The bandwidth of the output would be (A) (C) (B) (D) ECE - 2009 2. For a message signal m(t) = cos (2 ) and carrier of frequency which of the following represents a single side- band (SSB) signal? (A) () () (B) () (C) , ( )(D) , ( )( ) ECE - 2012 3. The signal m(t) as shown is applied both to a phase modulator (with kP as the phase constant) and a frequency modulator (with kf as the frequency constant) having the same carrier frequency.

ECE - 2013 4. The bit rate of a digital communication system is R kbits/s. The modulation used is 32-QAM. The minimum bandwidth required for ISI free transmission is (A) R/10 Hz (C) R/5 Hz (B) R/10 kHz (D) R/5 kHz ECE - 2014 5. A real band-limited random process X(t) has two-sided power spectral density () ( | |) || { where f is the frequency expressed in Hz. The signal X(t) modulates a carrier cos16000 t and the resultant signal is passed through an ideal band-pass filter of unity gain with centre frequency of 8 kHz and band-width of 2 kHz. The output power (in Watts) is________

6.

In a double side-band (DSB) full carrier AM transmission system, if the modulation index is doubled, then the ratio of total sideband power to the carrier power increases by a factor of __________

2

–2

0

2

4

6

8

t(seconds)

–2

The ratio kp/kf (in rad/Hz) for the same maximum phase deviation is (A) 8 (C) (B) (D)

IN - 2008 1. Ten real, band pass message signals each of bandwidth 3 kHz, are to be frequency division multiplexed on a band pass channel with bandwidth B kHz. If the guard band is between any two adjacent signals should be of 500 Hz width and there is no need to provide any guard band at the edge of the band pass channel the value of B should be at least (A) 30 (C) 35 (B) 34.5 (D) 35.5

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GATE QUESTION BANK

IN - 2013 2. A band – limited signal with a maximum frequency of 5 kHz is to be sampled. According to the sampling theorem, the sampling frequency in kHz which is not valid is (A) 5 (C) 15 (B) 12 (D) 20 3.

Communication

digital samples are converted by a parallel to serial converter to generate a serial PCM signal. This PCM signal is frequency modulated with FSK modulator with 1200Hz as 1 and 960Hz as 0. The minimum band allocation required for faithful reproduction of the signal by the FSK receiver without considering noise is. (A) 840 Hz to 1320 Hz (B) 960 Hz to 1200 Hz (C) 1080 Hz to 1320 Hz (D) 720 Hz to 1440 Hz

Signals from fifteen thermocouples are multiplexed and each one is sampled once per second with a 16 – bit ADC. The

Answer Keys and Explanations ECE 1.

[Ans. B]

(

)

*∫

()

(

)

*∫

+

+

()

( ) Given ( ) 8

So required frequency B.W ( ( ( ) ( )

)

[Ans. C] Option C is for SSB with upper side band used

3.

[Ans. B] For phase modulator () () Maximum phase deviation is ( ) , ( )................(1) For frequency modulator

(

)

∫ *∫

()

(

)

[Ans. B] Number of bits per symbol =

)

2.

()

4.

8

⁄ = ⁄ kilo symbols per second Minimum Bandwidth required for ISI transmission ⁄ (symbol rate) ⁄ 5.

[Ans. *] Range 2.4 to 2.6 ()

()

(

After modulation with

+ th

th

th

(

)

)

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GATE QUESTION BANK

()

, (

)

(

B = 10 × 3 kHz + 9 = 34.5 kHz

)-

This is obtain the power spectral density random process y(t), we shift the given power spectral density process y(t)

2

[Ans. A] so 5 kHz is not a valid one

3

8

(

8

Communication

[Ans. D] T

‘ ’ for FSK

)

After passing through a unity gain bandpass filter having 8

T

T

Transmission of 0

( )

T

8

(

8

After FSK modulation

1

, 6.

-

[Ans. *] Range 3.95 to 4.05 μ ( )

Given So data rate from ADC is = 240 bits/sec

μ

[

Ratio of total side band power to carrier power

T

bits/sec

]

T

μ

T Minimum Band allocation = 720 Hz to 1440 Hz For cross verification of obtained answer As we know Bandwidth of FSK is ( ) 8

. /

So if μ is doubled then, the ratio will be 4 times IN 1

T

)

The output power = Total area of shaded regular , )0

NRZ coding

[Ans. B] If these is ten signals then there will be needed of 9 guard band So, the required bandwidth of the channel is

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th

th

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Communication

Angle Modulation ECE - 2008 1. Consider the frequency modulated signal ( ) , ( )with carrier frequency of . The modulation index is (A) 12.5 (C) 7.5 (B) 10 (D) 5 2.

The signal (A) FM only (B) AM only (C) both AM and FM (D) neither AM nor FM

Column – 1 P. Power efficient transmission signals Q. Most bandwidth efficient transmission of voice signals R. Simplest receiver structure S. Bandwidth efficient transmission of Signals with significant dc component Codes: (A) P-IV, Q-II, R-I, S-III (B) P-II, Q-IV, R-I, S-III (C) P-III, Q-II, R-I, S-IV (D) P-II, Q-IV, R-III, S-1

is

ECE - 2010 3. Consider an angle modulated signal () (8 ) , (8 )- V. The average power of ( ) is (A) 10 W (C) 20 W (B) 18 W (D) 28 W

Column – 2 I. Conventio nal AM II. FM

III. VSB IV. SSB-SC

ECE - 2011 4. The column-1 lists the attributes and the column-2 lists the modulation system. Match the attribute to the modulation system that best meets it.

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th

th

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Communication

Answer Keys and Explanations ECE 1.

contains no intelligible information below 300Hz. AM-VSB – SC is same as AM, expects a vestige of LSB is transmitted. This requires less band width.

[Ans. B] Modulation index,

Where maximum frequency deviation fm = maximum frequency component Given that fm = 1500 Hz Deviation, ( ) ( )

(

2.

)

(

)

)

[Ans. C] () ()

() ()

() √

() (

() ,

( ))

, ( Hence x(t) is both AM and FM 3.

(

)-

[Ans. B] Signal x(t) is Here, Power =

4.

8

[Ans. B] In AM, the power of the carrier after modulation is more than before Modulation. But, in FM, it remains the same. AM-SSB-SC is more suitable for Voice signal Transmission, since the signal th

th

th

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Communication

Receivers ECE-2007 1. In a GSM system, 8 channels can co-exist in 200kHz bandwidth using TDMA. A GSM based cellular operator is allocated 5 MHz bandwidth. Assuming a frequency reuse factor of , i.e. a five-cell repeat pattern,

ECE - 2013 Common Data Questions 4 and 5 Bits 1 and 0 are transmitted with equal probability. At the receiver, the PDF of the respective received signals for both bits are as shown below.

the maximum number of simultaneous channels that can exist in one cell is (A) 200 (C) 25 (B) 40 (D) 5 2.

In a direct sequence CDMA system the chip rate is 88 chips per second. If the processing gain is desired to be AT LEAST 100, the data rate (A) must be less than or equal to 88 bits per sec (B) must be greater than 88 bits per sec (C) must be exactly equal to 88 bits per sec (D) can take any value less than 88 bits per sec

ECE - 2012 3. A binary symmetric channel (BSC) has a transition probability of 1/8. If the binary transmit symbol X is such that P(X = 0) = 9/10, then the probability of error for an optimum receiver will be (A) 7/80 (C) 9/10 (B) 63/80 (D) 1/10

Pdf of received 1 signal for bit 0 0.5

1

0

1

2

Pdf of received signal for bit 1

4

4.

If the deflection threshold is 1, the BER will be (A) ⁄ (C) ⁄8 ⁄ (B) (D) ⁄

5.

The optimum threshold to achieve minimum bit error rate (BER) is (A) ⁄ (C) (B) ⁄ (D) ⁄

ECE - 2014 6. The phase response of a passband waveform at the receiver is given by () ( ) where is the centre frequency, and 𝛼 and 𝛽 are positive constants. The actual signal propagation delay from the transmitter to receiver is (C) (A) (D)

(B)

th

th

th

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Communication

Answer Keys and Explanations ECE 1.

4.

[Ans. D]

[Ans. B] Allocated Band width = 5 MHz Frequency reuse factor = 1/5 Bandwidth allocated for 1 cell Maximum number of channel =

8 = 40 ( ) ( )

2.

I

[Ans. A]

‘ ’

‘ ’ ‘ ’

defined so Gain

100

[ ]

[ ]

8

So Bit error rate ( )

( ) [ ]

88

8

88 3.

bit/sec

[Ans. D]

5.

T

[Ans. B] Optimum threshold = the point of ’ ( ) | | | |

8 ( )

( )

(

)

8 (

. /

)

Point of intersection which decides Optimum threshold

For optimum receiver ( )

( ) 8 8

( ) [ ]

( ) ( )

( ) ( ) 8

6.

[Ans. C] Given phase response () ( ) So, propagation delay, ( )

8

0 [

th

th

.

/

1

]

th

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Communication

Noise in Analog Modulation ECE - 2006 1. A zero- mean white Gaussian noise is passed through an ideal low pass filter of bandwidth 10kHz. The output is then uniformly sampled with sampling period

constellations. Assume that the four symbols in both the constellations are equiprobable. Let

spectral density of white Gaussian noise. √ a

The samples so obtained would be (A) Co-related (B) statistically independent (C) uncorrelated (D) Orthogonal

2.

0

√ a

√ a constellation 1

by driving a

Linear-Time-Invariant system by zero mean white noise (as voltage process) with power spectral density being constant equal to 1. The system which can perform the desired task could be (A) first order low-pass R-L filter (B) first order high-pass R-C filter (C) tuned L-C filter (D) series R-L-C filter 3.

√ a

Common Data for questions 2 and 3 The following two questions refer to wide sense stationary stochastic processes It is desired to generate a stochastic process (as voltage process) with power spectral density ( )

denote the power

The parameters of the system obtained in Q.2 would be (A) first order R-L low-pass filter would have 1H (B) first order R-C high-pass filter would have (C) tuned L-C filter would have

a

a

0

a

-a constellation 2

4.

The ratio of the average energy of constellation 1 to the average energy of constellation 2 is (A) 4a2 (C) 2 (B) 4 (D) 8

5.

If these constellations are used for digital communication over an AWGN channel, then which of the following statement is true? (A) probability of symbol error for constellation 1 is lower (B) probability of symbol error for constellation 1 is higher (C) probability of symbol error is equal for both the constellations (D) The value of N0 will determine which of the two constellations has a lower probability of symbol error.

(D) series R-L-C low-pass filter would have . ECE - 2007 Common data for questions 4 and 5 Two 4-ary signal constellations are shown. It is given that constitute an orthonormal basis for the two th

th

th

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6.

During the transmission over a certain binary communication channel, bit errors occur independently with probability p. The probability of AT MOST one bit in error in a block of n bits given by (A) (B) (C) ( ) ( ) (D) ( )

7.

A Hilbert transformer is a (A) non-linear system (B) non casual-system (C) time-varying system (D) low-pass system

ECE - 2009 11. A communication channel with AWGN operating at a signal to noise ratio SNR >> 1 and bandwidth B has capacity . If the SNR is doubled keeping B constant, the resulting capacity is given by (A) (C) +2B (B) (D) +0.3B ECE - 2010 12. ( ) is a stationary process with the power spectral density ( ) for all . The process is passed through a system shown below.

ECE - 2008 8. Consider a Binary Symmetric Channel (BSC) with probability of error being p. To transmit a bit, say 1, we transmit a sequence of three 1s. The receiver will interpret the received sequence to represent 1 if at least two bits are 1. The probability that the transmitted bit will be received in error is ( ) (A) (B) ) (C) ( (D) ( ) 9.

Noise with double – sided power spectral density of K over all frequencies is passed through a RC low pass filter with 3 dB cut –off frequency of . The noise power at the filter output is (A) K (C) K (B) K (D)

10.

A memory less source emits n symbols each with a probability p. The entropy of the source as a function of n (A) Increases as log n (B) decreases as log (1/n) (C) Increases as n (D) increases as n log n

Communication

()

()

Let ( ) be the power spectral density of ( ). Which one of the following statements is correct? () (A) for all () (B) for | | kHz () (C) for kHz, is any integer () (D) for ( ) , kHz, is any integer ECE - 2011 13. X(t) is a stationary random process with autocorrelation function ( ) ( ) This process passed through the system shown below. The power spectral density of the output process Y(t) is H(f) = j

+

X(t)

Y(t) –

(A) (B) (C) (D) th

( ( ( ( th

) ) ) )

( ( ( ( th

) ) ) )

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ECE - 2012 14. The power spectral density of a real process X(t) for positive frequencies is shown below. The values of , ( )- and , ( )-, respectively, are Sx ( )

( - 104)

(C) (D)

( (

Communication

( (

)) ))

IN – 2006 1. Which of the following integrals provides a measure of the rapidity of change in a random variable f(t)? T

(A) lim

6

T



f(d) dt

T T

(B) lim T

0

(A) (B) (C) (D)

9

10

11

(

3

T



T

T

rad/s)



(C) [ lim T

() ]

T T

(D) lim

( √ )

T

( √ )

ECE - 2014 15. Let X be a real-valued random variable with E[X] and , - denoting the mean values of X and , respectively. The relation which always holds true is , (A) ( , -) (B) ( , -) ( , -) (C) , - ( , -) (D) , - ( , -) 16.



T

()

Consider a random process () √ ( ) where the random phase is uniformly distributed in the interval [0,2 ]. The autocorrelation , ( ) ( )- is (A) ( ( )) (B) ( ( ))



() (

)

T

IN - 2011 2. A square wave (amplitude ± 10 mV, frequency 5 kHz, duty cycle 50%) is passed through an ideal low pass filter with pass band gain and cut-off frequency of 0dB and 10kHz, respectively. The “ ” additively into a zero mean noise process of one sided power spectral density (PSD) of 25pWH up to a frequency of 2 MHz. the PSD of the noise is assumed to be zero beyond 2 MHz. the signal to noise ratio of the output is (A) 0 dB (C) 1.0 dB (B) 0.1 dB (D) 3 dB

th

th

th

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Communication

Answer Keys and Explanations ECE 1.

() () Average

[Ans. A]

Energy

of

constellation

The required Ratio is 4 So signal can be recovered easily So samples will be co-related in time 2.

[Ans. A] ( )

5.

[Ans. A] The distance between the two closet points in constellation -1 is d1 = 2a. The same in constellation-2, d2 = √ Since d1> d2, Probability of symbol error in constellation 1 is lower

6.

[Ans. C] Probability of no error in n bits = (1 – p)n Probability of one error in n bits = n.p(1-p)n-1 = ( – ) + nP ( – )

7.

[Ans. C] The o/p y(t) of an Hilbert Transformer for

( ) | ( )| | ( )|

( )

j

Low pass RL filter 3.

[Ans. A]

L

an i/p of ( )

()

( )

∫

This is not a Time Invariant system

R

8.

(j )

j Comparing with (i) & (ii) R = 4, L = 1 4.

[Ans. B] Constellation 1 () () ()

= 9.

+3

√

(

)

[Ans. C] ()

j

j

√ | ( )|

√

() √ Energy of ( ) Average

[Ans. A] Probability of Error = 3 error + 3 (2 error and 1 correct) = ( )

8 Energy

√

| ( )| of

constellation ∫(

)

Constellation 2 () () th

th

th

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GATE QUESTION BANK

13.

Communication

[Ans. A]

∫ ( 10.

x(t)

y(t) –

[Ans. A] ( )

∑

()

Where Pi = Probability of individual symbol. Since probability of each symbol is same, therefore,

T ()

( ) 11.

+

)

()

() j () ,j - () ( ) |j | () () ( ) () () T, ( )()

∑

()

, ()

[Ans. B] SNR >> 1

14. (

()

→ ,

-

[Ans. B] Sx ( )

)

-

( - 104)

( ) When SNR is doubled (

)

( )

⏟

⏟

10

9

0

11

(

3rad/s)

As we know P.S.D is even function so part can be expected, ( )

12.

(

[Ans. D] ()

, ()

( ) ( As ( | ( | (

(

)

(

)

T)-

( ) j ( ) ( ) ) j j ( ) ( ) ( ) | ( )| we know ) )| ( ) )| [ ] j

Either T ( T ( ( )

⁄ )

(

for , ( )∫

( )

[(

)

]

) , ( )-

)

As there in no component on So No DC component , ( )th

th

th

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15.

[Ans. B] We know that standard deviation of any real valued random variable x is greater than or equal to zero

2.

Communication

[Ans. *] Square wave is as V 10 mV t

, , 16.

10 mV

( , -) ( , -)

It has half wave symmetry so it will contain odd harmonics only as it is also odd signal therefore only odd harmonics of sine term will present. Cut off frequency of L.P.F is 10 kHz so it will allow only to pass fundamental

[Ans. D]

() √ ( , ( ) ( ), (

) (

) ,

)-

(

( )

) -

component at 5 kHz of amplitude

The first term on the right hand side contains no Real value ) , ( Now, ,

∫ ( IN 1.

)

(

) ,

(

(y out can calculate) So power of message (

-

8 watt Power in noise signal

)-

[Ans. C] Standard deviation is a measure of rapidity of change in a random variable. It can be expressed as *

T

∫

)

(S/N) in dB = 10log.

/

= 2.098 dB

() +

th

th

th

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Communication

Digital Communications ECE - 2006 Common Data for Question 1 and 2 Let g(t) = p( ) * p(t), where * denotes convolution and p(t)=u(t) u(t 1) () 1. The impulse response of filter matched to the signal s(t) = g(t) (t 2)* g(t) is given as (A) s(1 t) (C) s(t) (B) s(1 t) (D) s(t)

5.

The minimum step-size required for a Delta-Modulator operating at 32K samples/Sec to track the signal (here u(t) is the unit-step function) ( ), () X(t) ( ), ( ) ( )so that slope-overload is avoided, would be (A) (C) (D) (B)

2.

6.

The minimum sampling frequency(in samples/sec) required to reconstruct the following signal from its samples without distortion

3.

An Amplitude Modulated signal is given () ( )) cos as ( () in the interval 0 . One set of possible values of the modulating signal and modulation index would be (A) t, 0.5 (C) t, 2.0 (B) t, 1.0 (D) , 0.5 In the following figure the minimum value of the constant 'C', which is to be added to ( ) are (t) such that ( )and different, is Quantizer Q with L

Same Quantizer Q

()

allowable signal dynamic range ( V,V)

()

X(t) with range [

(A) (B)

]

⁄

(C) (D)

⁄ ⁄

()

(

)

would be (A) (B)

.

/

(C) (D) 8

ECE - 2007 Statement for Linked Answer Questions 7 &8 An input to a 6-level quantizer has the probability density function f(x) as shown in the figure. Decision boundaries of the quantizer are chosen so as to maximize the entropy of the quantizer output. It is given that 3 consecutive decision ‘ ’ ’ ’ ‘ ’ f(x) a

4.

A source generates three symbols with probabilities 0.25, 0.25, 0.50 at a rate of 3000 symbols per second. Assuming independent generation of symbols, the most efficient source encoder would have average bit rate as (A) 6000 bits/sec (C) 3000 bits/sec (B) 4500 bits/sec (D) 1500 bits/sec

b 5

7.

0

1

1

5

x

The values of a and b are (A) a = 1/6 and b = 1/12 (B) a = 1/5 and b = 3/40 (C) a = 1/4 and b = 1/16 (D) a = 1/3 and b = 1/24

th

th

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8.

9.

Assuming that the reconstruction levels of the quantizer are the mid-points of the decision boundaries, the ratio of signal power to quantization noise power is ⁄ ⁄ (A) (C) ⁄ (B) (D) 28

(

at

)

0

B

2B

(C)

+2B

f

Y(f)

f

0

B

. The value of p(t)

+B

+B Y(f)

(D)

is

(A) 0.5 (B) 0

11.

Y(f)

(B)

The raised cosine pulse p(t)is used for zero ISI in digital communications. The expression for p(t) with unity roll-off factor is given by ()

10.

Communication

(C) 0.5 (D)

In delta modulation, the slope overload distortion can be reduced by (A) Decreasing the step size (B) decreasing the granular noise (C) decreasing the sampling rate (D) increasing the step size

12.

+B

B

2B

+2B

f

Three functions ( ) ( ) ( ), which are zero outside the interval [0,T], are shown in the figure. Which of the following statements is correct? ()

In the following scheme, if the spectrum M(f) of m(t) is as shown, then the spectrum Y(f) of y(t) will be

T

T

T

M(f) ()

0

B

f

+B

(

)

()

m(t) + +

Hilbert transformer (

y(t)

Y(f)

(A)

B

0

+B

( ( ( (

(A) (B) (C) (D)

)

) ) ) )

() () () ()

f

th

th

th

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ECE - 2008 Common Data for Questions 13, 14 and 15 A speech signal, band limited to 4kHz and peak voltage varying between +5 V and 5V, is sampled at the Nyquist rate. Each sample is quantized and represented by 8bits. 13. If the bits 0 and 1 are transmitted using bipolar pulses, the minimum band width required for distortion free transmission is (A) 64kHz (C) 8kHz (B) 32kHz (D) 4kHz 14.

15.

16.

Assuming the signal to be uniformly distributed between its peak to peak value, the signal to noise ratio at the quantizer output is (A) 16dB (C) 48dB (B) 32dB (D) 64dB

18.

If the positive values of the signal are uniformly quantized with a step size of 0.05 V, and the negative values are uniformly quantized with a step size of 0.1V, the resulting signal to quantization noise ratio is approximately (A) 46 dB (C) 42 dB (B) 43.8 dB (D) 40 dB

ECE - 2010 Statement for Linked Answer Questions 19 and 20 Consider a baseband binary PAM receiver shown below. The additive channel noise ( ) is white with power spectral density ()

W/Hz. The low pass

filter is ideal with unity gain and cut-off frequency 1 MHz. Let represent the random variable ( ). if transmitted bit if transmitted bit where represents the noise sample value. The noise sample has a probability | | ( ) density function,

The number of quantization levels required to reduce the quantization noise by a factor of 4 would be (A) 1024 (C) 256 (B) 512 (D) 64.

(This has mean zero and variance 2/ ). Assume transmitted bits to be equiprobable and threshold z is set to a/2 = V.

Four messages band limited to W, W, 2W, 3W respectively are to be multiplexed using Time Division Multiplexing (TDM). The minimum bandwidth required for transmission of this TDM signal is (A) W (C) 6W (B) 3W (D) 7W

ECE - 2009 Common Data for Questions 17 & 18 The amplitude of a random signal is uniformly distributed between – 5V and 5V. 17. If the signal to quantization noise ratio required in uniformly quantizing the signals is 43.5dB, the step size of the quantization is approximately (A) 0.0333V (C) 0.0667V (B) 0.05V (D) 0.10V

Communication

() ()

()

()

( ) ( ) ( )

{ T

19.

The value of the parameter ( (A) (C) 1.414× (B) (D) 2 ×

20.

The probability of bit error is (C) 0.5 × (A) 0.5 × (D) 0.5 × (B) 0.5 ×

) is

ECE - 2011 21. An analog signal is band-limited to 4 KHz. Sampled at the Nyquist rate and the samples are quantized into 4 levels. The th

th

th

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quantized levels are assumed to be independent and equally probable. If we transmit two quantized samples per second, the information rate is (A) 1 bit/sec (C) 3 bits/sec (B) 2 bits/sec (D) 4 bits/sec Statement for Linked question 22 and 23. A four-phase and an eight-phase signal constellation are show in the figure below.

22.

23.

For the constraint that the minimum distance between pairs of signal points be d for both constellations, the radii r1 and r2 of the circles are (A) 8 (B) (C) (D) Assuming high SNR and that all signals are equally probable, the additional average transmitted signal energy required by the 8-PSK signal to achieve the same error probability as the 4-PSK signal is (A) 11.90 dB (C) 6.79 dB (B) 8.73 dB (D) 5.33 dB

ECE - 2012 24. A BPSK scheme operating over an AWGN channel with noise power spectral density of N0/2, uses equiprobable signals () ()

(

√ √

received signal, the probability of error in the resulting system is

) (

)

over the symbol interval, (0, T). If the local oscillator in a coherent receiver is ahead in phase by with respect to the

(A)

(√ )

(C)

(√

)

(B)

(√ )

(D)

(√

)

25.

A source alphabet consists of N symbols with the probability of the first two symbols being the same. A source encoder increases the probability of the ε ε encoding, the entropy of the source (A) Increases (B) Remains the same (C) Increases only if N = 2 (D) Decreases

26.

In a baseband communications link, frequencies upto 3500 Hz are used for signaling. Using a raised cosine pulse with 75% excess bandwidth and for no intersymbol interference, the maximum possible signaling rate in symbols per second is (A) 1750 (C) 4000 (B) 2625 (D) 5250

I

I

Communication

ECE/IN - 2013 27. Let U and V be two independent and identically distributed random variables such that P (U=+1) = P(U= 1)

⁄ .

The entropy H(U+V) in bits is (A) ⁄ (B) 1

(C) (D)

⁄

ECE - 2013 28. Consider two identically distributed zeromean random variables U and V. Let the cumulative distribution functions of U and 2V be F(x) and G(x) respectively. Then, for all values of x (A) ( ) ( ) (B) ( ) ( ) ( )) (C) ( ( ) (D) ( ( ) th

th

( )) th

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GATE QUESTION BANK

ECE - 2014 29. In a code-division multiple access (CDMA) system with N = 8 chips, the maximum number of users who can be assigned mutually orthogonal signature sequences is _____. 30.

31.

The capacity of a Binary Symmetric Channel (BSC) with cross-over probability 0.5 is ______.

(A) 1.44 (B) 1.08

The input to a 1-bit quantizer is a random variable X with pdf ( ) and ( ) . For outputs to be of equal probability, the quantizer threshold should be _____.

35.

Coherent orthogonal binary FSK modulation is used to transmit two equiprobable symbol waveforms () and () , where 𝛼=4 mV. Assume an AWGN channel with two-sided noise power

Let Q (√ ) be the BER of a BPSK system over an AWGN channel with two-sided noise power spectral density . The parameter is a function of bit energy and noise power spectral density. A system with two independent and identical AWGN channels with noise power spectral density is shown in the figure. The BPSK demodulator receives the sum of outputs of both the channels.

spectral density ( )

√

A fair coin is tossed repeatedly until a ‘ ’ the number of tosses to get this first ‘ ’ T ( ) ts is_____.

given by

.

The power spectral density of a real stationary random process 𝑋(𝑡) is given by

For a fixed capacity (in bandwidth (𝑊

The 0

with

||

value () .

of

the

expectation

/1 is ________

37.

An analog voltage in the range 0 to 8 V is divided in 16 equal intervals for conversion to 4-bit digital output. The maximum quantization error (in V) is __________

38.

Let X(t) be a wide sense stationary (WSS) random process with power spectral density (f). If Y(t) is the process defined as Y(t) = X(2t 1), the power spectral density (f) is

1000, the channel kbps) )

{

||

/ bits per

second (bps), where W is the channel bandwidth, 𝑃 is the average power received and is the one-sided power spectral density of the AWGN.

, the bit error

∫

probability for a data rate of 500 kbps is (A) ( ) (C) ( ) (B) ( √ ) (D) ( √ )

()

The capacity of a band-limited additive white Gaussian noise (AWGN) channel is

W/Hz.

Using an optimal receiver and the relation

If the BER of this system is Q( √ ), then the value of b is _____

33.

(C) 0.72 (D) 0.36

34.

36.

32.

Communication

infinite

th

th

th

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GATE QUESTION BANK

39.

40.

(A)

()

. /

(B)

()

. /

(C)

()

. /

(D)

()

. /

In a PCM system, the signal ) * ( m(t) ( )+V is sampled at the Nyquist rate. The samples are processed by a uniform quantizer with step size 0.75 V. The minimum data rate of the PCM system in bits per second is _________. A binary random variable X takes the value of 1 with probability 1/3. X is input to a cascade of 2 independent identical binary symmetric channels (BSCs) each with crossover probability 1/2. The output of BSCs are the random variables 𝑌 and 𝑌 as shown in the figure.

The values of __________.

(𝑌 )

If calls arrive at a telephone exchange such that the time of arrival of any call is independent of the time of arrival of earlier or future calls, the probability distribution function of the total number of calls in a fixed time interval will be (A) Poisson (C) Exponential (B) Gaussian (D) Gamma

42.

Consider a communication scheme where the binary valued signal X satisfies P{X = +1} = 0.75 and * + = 0.25. The received signal Y = X + Z, where Z is a Gaussian random variable with zero mean and variance . The received signal Y is fed to the threshold detector. The output of the threshold detector ̂ is: ̂

To achieve a minimum probability of error P{̂ }, the threshold should be (A) strictly positive (B) zero (C) strictly negative (D) strictly positive, zero, or strictly negative depending on the nonzero value of 43.

Consider the Z-channel given in the figure. The input is 0 or 1 with equal probability. I

If the output is 0, the probability that the input is also 0 equals __________ 44.

An M-level PSK modulation scheme is used to transmit independent binary digits over a band-pass channel with bandwidth 100 kHz. The bit rate is 200 kbps and the system characteristic is a raised- cosine spectrum with 100% excess bandwidth. The minimum value of M is______________

45.

Consider

(𝑌 ) in bits is

41.

Communication

a discrete-time channel , where the additive noise Z is signal-dependent. In particular, given the * + at any transmitted symbol instant, the noise sample Z is chosen independently from a Gaussian distribution with mean X and unit variance. Assume a threshold detector with zero threshold at the receiver. When = 0, the BER was found to be ( ) ( ( )

√

∫ ( )

When 𝛽 (A) (B)

2 th

th

)

0.3, the BER is closest to (C) (D) th

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GATE QUESTION BANK

Communication

IN - 2009 1. A 50% duty cycles square wave with zero mean used as a baseband signal in an ideal frequency modulator with sinusoidal carrier of frequency . the modulated signal is given as and input to an ideal phase demodulation circuit (Circuit that produces an output proportional to the difference in phase of the modulates signal from that of the carrier). The output of the circuit is (A) a square wave (B) a train of impulses with alternating signs (C) a triangular wave (D) a sinusoidal wave IN - 2010 2. In a pulse code modulated (PCM) signal sampled at and encoded into a n – bit code, the minimum bandwidth required for faithful reconstruction is ⁄ (A) 2 (C) (B) (D) 3.

A signal with frequency components 50Hz, 100Hz and 200 Hz only is sampled at 150 samples/sec. The ideally reconstructed signal will have frequency components of (A) 50 Hz only (B) 75 Hz only (C) 50 Hz and 75 Hz (D) 50 Hz, 70 Hz and 100 Hz

IN - 2014 4. A full duplex binary FSK transmission is made through a channel of bandwidth 10 kHz. In each direction of transmission the two carriers used for the two states are separated by 2 kHz. The maximum baud rate for this transmission is: (A) 2000 bps (C) 5000 bps (B) 3000 bps (D) 10000 bps

th

th

th

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GATE QUESTION BANK

Communication

Answer Keys and Explanations ECE 1.

()

[Ans. C] P(t) = u(t)

u(t 1)

P(t)

h(t)= s(t)

0

2.

t

1

[Ans. A] ()

, ()

( )-

()

p(t)

g(t) = P(t) * P(t) g(t)

0 0

1

S(t)=g(t)

2

t

1

t g(t)

(t 2)*g(t)=g(t) g(t 2)

g(t – 2)

t

S(t) = g(t)

g(t

t

4

3

2

0

s(t)=10(1+m(t)) 𝑡 (𝑡) m(t) =0.5g(t), comparing AM equation with (i)=0.5t Modulating signal = t(form equation of line) Modulation index = 0.5

2)

s(t)

0

1

2

3

2

4

Impulse response of match filter h(t) = s(T t)

th

th

th

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GATE QUESTION BANK

3.

Communication

[Ans. B] ∫ ( )

∫

∫

For (𝑡) and (𝑡) to be different minimum step size of is needed else they will be same. 4.

8.

[Ans. D]

[Ans. B]

∫

= [0.25 (0.25) + 0.25 0.5 0.5] = 1.5 3000 = 4500 bits/sec

∫

[Ans. B] x(t) = 125t [u(t) – u(t – 1)] + (250 125t)[u(t 1) u(t 2)] To avoid slope overload () | T

9.

[Ans. C] ()

(

7.

)

)

.

/

As it comes in the form of 0/0 so applying LH rule

x(t)

6.

) 8

(

1 2 5 0

∫

(0.25) +

( 5.

( )

∫

( 1

2

)

(

)

t

[Ans. C] Sampling rate = Nyquist Rate For first term NR = 6k and For second term NR = 4k Total NR = 6k Hz [Ans. A] To maximize the entropy, all the decision boundaries should be equiprobable.

8

() 10.

[Ans. D] By increasing step size, slope overload distortion can be reduced.

11.

[Ans. A] y(t) is equal to SSB wave having LSB and carrier frequency equal to B.

th

th

th

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GATE QUESTION BANK

Highest frequency component in message is also of B Hz.

Where n = number of bits per sample quantized

M(f)

(

15. 0

B

Y(f)

12.

B

8

8

[Ans. B] ,where =

quantization Noise

SSB having LSB

0

B

)

Quantization Noise =

f

B

Communication

To Reduce Noise by a factor 4. L should be increased by a factor of 2. So required levels 2 256 = 512 levels.

f

16.

[Ans. D]

[Ans. A] Two functions f(x) & g(x) are said to be orthogonal if

T If all have same B.W then TDM B.W

∫ ( ) ( )

T () ()

∫ T

T

T

∫ () ()

T

17.

[Ans. C] We know that

13.

[Ans. B] While using the bipolar pulses to transmit bits 0 and 1, the minimum bandwidth requires for distortion free transmission is four times the theoretical bandwidth (Nyquist Bandwidth).

8

(

Nyquist bandwidth, 8 Minimum bandwidth in bipolar signaling is 8 14.

Hence closest answer is 0.0667 V so correct option is C 18.

[Ans. C] Signal to noise ratio (

)

[Ans. B] For +ve values

) th

th

th

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GATE QUESTION BANK

(

)

(

)

(– r1, 0). The distance between them is ⁄√ . Hence, √ In M-ary PSK, the points in the constellation diagram lie on a circle, with √ Where E = {log2(M)}.Eb, where Eb is the energy in one bit duration. The points in the Constellation diagram are given by *√

)( )- √ ,( ) ( )I The given points in the constellation diagram of 8-PSK are with co-ordinates (r2, 0) and (r2 √ 2 √ ) √ 2 The distance between the points is 0.76r2=d. Hence r2 = 1.307d

Best SNR 8 19.

20.

[Ans. B]

[Ans. D] Symbol error Probability in M-ary PSK is erfc[√( η) ( )b in 4-PSK and 3.Eb in 8-PSK. Considering in general as erfc(x), the ratio between x in QPSK to 8-PSk is 1.508. This is the requirement in increase in the signal energy per bit. For, 3 bits, the increase is (1.508)3. This in decibels is 10log (1.508)3, i.e.5.35dB

24.

[Ans. B] Probability of error in coherent BPSK

T

( )

∫

,(

23.

[Ans. D] When a 1 transmitted:

( )

*√ ∫ (

Communication

+

) Phase difference 450 decreases the signal

21.

22.

[Ans. D] Since all the 4 levels are equi-probable, the entropy of the source will be maximum. Since, two quantized samples are transmitted per sec, message rate r = 2. Thus, the information rate R = r.H=4bps.

energy by a factor of Cos2 *√

25.

[Ans. D] The points in the Constellation diagram of 4-PSk are with co-ordinates (0, r1) and th

+

[Ans. D] If all the symbols are equi-probable, the entropy of the source will be maximum. Since , only the first two are of same probability, the entropy will not be maximum, but will be less than the th

th

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GATE QUESTION BANK

maximum. Because of the source encoder, the first two are also of different probability. Hence, the entropy will be further reduced. 26.

Chip rate = 8 symbol rate 8 ( )

[Ans. C]

[Ans. C] u v +1 +1 +1 +1

8

Spread factor (or) process gain and determine to a certain extent the upper limit of the total number of uses supported simultaneously by a station. 30.

Symbol rate = 4000 (OR) T “ ” T “ S” “ ” “ S” 27.

Communication

31.

[Ans. *] Range Channel capacity of BSC is ( ) (

)

[Ans. *] Range 1.4 to 1.42

2 0 0

,

-

( ) ( )

*

+ So signal energy increases 4 times as

-

𝑃,

(

)

(

) Noise energy increase 2 times

-

𝑃,

( ) ( ( 28.

29.

)

(√ )

)

(√ √ )

[Ans. D] F (x) = P{X x} G (x) = P {2X x} = P {X x/2} For positive value of x, F(x) G(x) is always greater than zero For negative value of x. F(x) G(x) is ve but .[F (x) G(x)].x 0

32.

)

√ )

[Ans. *] Range 1.99 to 2.01 The number of losses read and the probability of its occurrence is given below L=1

(

L=2

(

L=3

(

L=4

(

[Ans. *] Range 7.99 to 8.01 (

(

(√

If single symbol is represented by a code of 8 chips

) ) ) )

∑

th

th

th

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)

GATE QUESTION BANK

36. ∑

∑

Communication

[Ans. *] Range 3.9 to 4.1 We know auto correlation

function

↔ 33.

[Ans. A]

() (

)

(

|| ||

) *

( (

)

()

∫

)

+

∫

,

-

j [

34.

{

[Ans. *] Range 0.34 to 0.36 For output to be of equal probability

0

1 ]

j

[ () (

∫

)]

() (

[

)]

( ){ 37. [

]

,

[Ans. *] Range 0.24 to 0.26 T

8 T

T

35.

38.

[Ans. C] For BFSK, we know the probability of symbol error is

average

[Ans. C] Shifting operation does not change PSD () ( ) () ( ) () ( ) Scaling operating change PSD (

(√

)

(

)

( )

) () (

T

39.

)

(√

( )

) j

(√

()

)

[Ans. *] Range 199 to 201 Nyquist frequency The peak to peak amplitude = 2V Step size = 0.75V

( ) ,

-

⁄ th

th

th

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GATE QUESTION BANK

Communication

( ) ( ) ( ) ( ) ( )

Thus, minimum data rate 40.

[Ans. *] Range 1.9 to 2.1

43.

[Ans. *] Range 0.79 to 0.81 I

(

)

(

)

To get ( ) we should find probability of (

)

(

)

)

(

)

(

)

(

)

(

|

42.

)

)

( )

(

[

)

(

)

) (

]

(

) ) ( )

| (

41.

(

8 (

( )

(

[Ans. A] It is properly of Poisson distribution event occur with known average rate and independent of time

)

8

8

44.

[Ans. *] Range 15.9 to 16.1 For M-PSK we know

45.

[Ans. C] ( ) ( ( ( ( ( )

[Ans. C] (

)

( )

(

)

√ 2 (

(

)

√ (

(

)

)

)

√ At optimum threshold probability of error ( ⁄ ) ( ) | ( ⁄ ) ( )

for minimum

th

th

)

|

)

| ) )

| |

th

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GATE QUESTION BANK

Communication

( ) ( )

( )

( (

)) ( ) ( )

√

IN 1.

( ) (

)

√

[Ans. A] We will get square wave at the output

2.

[Ans. C] According to Nyquist sampling theorem for the faithful reconstruction of signal

3.

[Ans. A] Given samples/sec We know that,

samples/sec In given signals only 50 Hz signal is satisfying Nyquist criteria, and there are no signals like 70 Hz and 75 Hz. Hence correct option is (a) 4.

[Ans. B] Bandwidth for each direction = baud rate +separation of carriers (10Hz/2) = baud rate +2kH Then, baud rate is = 5000 2000=3000bps th

th

th

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EDC

Semiconductor Theory ECE - 2006 1. A heavily doped n-type semiconductor has the following data Hole-electron mobility ratio: 0.4 Doping concentration: 4.2 108atoms/m3 intrinsic concentration:1.5 104atoms/m3 The ratio of conductance of the n-type semiconductor to that of the intrinsic semiconductor of same material and at the same temperature is given by (A) 0.00005 (C) 10,000 (B) 2,000 (D) 20,000 2.

3.

4.

ECE - 2007 5. The electron and hole concentration in an intrinsic semiconductor are per cm3 at 300 K. Now, if acceptor impurities are introduced with a concentration of NA per cm3 (where NA ni), the electron 3 concentration per cm at 300 K will be (A) (C) NA (B) + NA ⁄ (D) Common Data Question 6, 7 and 8: The figure shows the high-frequency capacitance-voltage (C-V) characteristics of a Metal/ /silicon (MOS) capacitor having an area of Assume ) of silicon that the permittivities ( ⁄ and are and ⁄ respectively C

The majority carriers in an n-type semiconductor have an average drift velocity V in a direction perpendicular to a uniform magnetic field B. The electric field E induced due to Hall effect acts in the direction (A) (C) (B) (D) Under low level injection assumption, the injected minority carrier current for an extrinsic semiconductor is essentially the (A) Diffusion current (B) Drift current (C) Recombination current (D) Induced current The concentration of minority carriers in an extrinsic semiconductor under equilibrium is: (A) directly proportional to the doping concentration (B) inversely proportional to the doping concentration (C) directly proportional to the intrinsic concentration (D) inversely proportional to the intrinsic concentration

7pF

1pF

V

0 6.

The gate oxide thickness in the MOS capacitor is (A) 50nm (C) 350nm (B) 143nm (D) μ

7.

The maximum depletion layer width in silicon is (A) 4 μ (C) μ (B) 8 7μ (D) 4 μ

8.

Consider the following statements about the C-V characteristics plot: S1: The MOS capacitor has as n-type substrate. S2: If positive charges are introduced in the oxide, the C-V plot will shift to the left. th

th

th

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Then which of the following is true? (A) Both S1 and S2 are true (B) S1 is true and S2 is false (C) S1 is false and S2 is true (D) Both S1 and S2 are false

12.

ECE - 2008 9. Silicon is doped with boron to a concentration of 4 atoms /cm3. Assume the intrinsic carrier concentration of silicon to be /cm3

and the value of

13.

to be

25mV at 300K. Compared to undoped silicon, the Fermi level of doped silicon (A) goes down by 0.13 eV (B) goes up by 0.13 eV (C) goes down by 0.427 eV (D) goes up by 0.427 eV ECE - 2009 10. The ratio of the mobility to the diffusion coefficient in a semiconductor has the units (A) (C) V. (D) (B) cm 11.

In an n-type silicon crystal at room temperature, which of the following can have a concentration of 4 × ? (A) Silicon atoms (B) Holes (C) Dopant atoms (D) Valence electrons Common Data for Question 12 and 13 Consider a silicon p-n junction at room temperature having the following parameters: Doping on the n-side=1× Depletion width on the n- d = μ Depletion width on the p- d = μ Intrinsic carrier concentration = 1.4 × Thermal voltage = 26V

EDC

Permittivity of free space = 8.85 × Dielectric constant of silicon = 12 The built-in potential of the juncition (A) is 0.70 V (B) is 0.76 V (C) is 0.82 V (D) Cannot be estimated from the data given The peak electric field in the device is (A) 0.15 MV. , directed from region to n-region (B) 0.15 MV. , directed from region to p-region (C) 1.80 MV. , directed from region to n-region (D) 1.80 MV. , directed from region to p-region

pnpn-

ECE - 2010 14. At room temperature, a possible value for the mobility of electrons in the inversion layer of a silicon n channel MOSFET is (A) 450 /V s (C) 1800 /V s (B) 1350 /V s (D) 3600 /V s ECE - 2011 15. Drift current in semiconductors depends upon (A) Only the electric field (B) Only the carrier concentration gradient (C) Both the electric field and the carrier concentration (D) Both the electric field and the carrier concentration gradient ECE - 2014 16. The doping concentrations on the p-side and n-side of a silicon diode are and , respectively. A forward bias of 0.3 V is applied to the diode. At T = 300 K, the intrinsic carrier concentration of silicon = and = 26 mV. th

th

th

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GATE QUESTION BANK

The electron concentration at the edge of the depletion region on the p-side is (A) (B) (C) (D) 17.

20.

Assume electronic charge ⁄ = = and electron mobility μ = 1000 /V s. If the concentration gradient of electrons injected into a P-type silicon sample is , the magnitude of electron diffusion current density (in A/ ) is _________.

19.

When a silicon diode having a doping concentration of =9 on p-side and = on n-side is reverse biased, the total depletion width is found to be 3 μm. Given that the permittivity of silicon is 4 , the depletion width on the p-side and the maximum electric field in the depletion region, respectively, are ⁄ (A) 7μ d ⁄ (B) μ d4 ⁄ (C) μ d 4 ⁄ (D) μ d 4

At T = 300 K, the hole mobility of a semiconductor μ = 500 /V-s and = 26 mV. The hole diffusion constant in

21.

A silicon bar is doped with donor impurities = atoms/ . Given the intrinsic carrier concentration of silicon at T = 300 K is = . Assuming complete impurity ionization, the equilibrium electron and hole concentrations are (A) = = (B) = = (C) = = (D) = =

18.

EDC

/s is ______

The donor and accepter impurities in an abrupt junction silicon diode are and , respectively. Assume that the intrinsic carrier concentration in silicon =

at 300K.

= 26 mV

and the permittivity of silicon = 4 F/cm. The built-in potential and the depletion width of the diode under thermal equilibrium conditions, respectively, are (A) 7 d (B) 8 d (C) 7 d (D) 8 d 22.

An ideal MOS capacitor has boron dopingconcentration of in the substrate. When a gate voltage is applied, a depletion region of width 0.5µm is formed with a surface (channel) potential of 0.2 V. Given that = 88 4 F/cm and the relative permittivities of silicon and silicon dioxide are 12 and 4, respectively, the peak electric field (in V/µm) in the oxide region is__________

23.

The cut-off wavelength (in μ ) of light that can be used for intrinsic excitation of a semiconductor material of bandgap = 1.1 eV is__________

24.

Consider a silicon sample doped with = donor atoms. Assume that the intrinsic carrier concentration = . If the sample is additionally doped with

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GATE QUESTION BANK

25.

= acceptor atoms, the approximate number of electrons/ in the sample, at T=300 K, will be________

( )

An N-type semiconductor having uniform doping is biased as shown in the figure.

( )

EDC

( ) N type semiconductor If is the lowest energy level of the conduction band, is the highest energy level of the valance band and is the Fermi level, which one of the following represents the energy band diagram for the biased N-type semiconductor?

( )

Answer Keys and Explanations 1.

[Ans. D] = =

2.

(

)

=

(

= Using mass action law

)

So,

= 20, 000

[Ans. B] The electric field will be perpendicular to both V and B, and in B V direction.

6.

=

[Ans. A] The injected minority carrier current is diffusion current and it is because of concentration gradient.

4.

[Ans. B] Minority concentration

= 7

= = = 7.

[Ans. B] =

=

=7 7 =

= i,e, 5.

=

[Ans. A]

= 3.

=

d(

[Ans. D] By the law of electrical neutrality = as =

)

= = =

(

)

(

) (

)

=

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GATE QUESTION BANK

= =

15.

[Ans. C]

7 8 7

=

= =μ = μ Hence, = μ So, it depends upon carrier concentration and electric field

8 7μ

Put,

8.

[Ans. D] It is p-type substrate ,and also add positive charge, C V plot moves right side

9.

[Ans. C] For p–type material, ( ) = 4 = 1.5

16.

(

[Ans. A] =

)

= 17.1 25 mv = 0.427eV It is p-type so Fermi level goes down by 0.427eV.

μ

11.

=

[Ans. B] N side is heavily doped =( =

=

=

( )

=

18.

[Ans. *] Range 3990 to 4010 d d d = d d ⁄ = d = = =4

19.

[Ans. B] = =

) 7 volts

[Ans. *] = = = =

14.

)

=

⁄

13.

(

[Ans. C] =

12.

)

[Ans. D] = = =

= μ

=( =

d = 17.

[Ans. A]

d

= The electron concentration at the edge of the depletion region on the

= 17.1

10.

EDC

μ 88 4 9 d

(

) d

[Ans. B] The mobility of electrons in Si is 1350 / V-sec.

9 =

=

μ

= = =4 th

th

th

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GATE QUESTION BANK

20.

EDC

[Ans. *] Range 12.9 to 13.1

=

(

= =

= = 21.

(

)

=

(

=

)

=*

(

)

8 d

=*

(

= 22.

(

)+ )+

23.

=

)

= = =

[Ans. *] Range 2.3 to 2.5 24.

8 μ

[Ans. *] Range 224.9 to 225.1 The approximate number of electrons/ in sample is =

d

(

)

[Ans. *] Range 1.12 to 1.14

4

25.

d

(

= 8 μ Since there is a dielectric- dielectric boundary at x = 0, we have to apply boundary condition at the oxide – semiconductor interface, since the field lines are everywhere perpendicular to the interface, we get = ( ) 4 = 8 = 4 μ (

[Ans. D] =

)

(

)

=

[Ans. D]

)

N type

Given = = μ d d = = An ideal MOS capacitor does not contain any stored charge in the oxide region. So the field in the oxide region is constant. Also since the substrate is uniformly doped the field distribution within the depletion region is similar to that of a pn junction, it varies linearly with distance, being zero in the bulk p-region (substrate) ( ) = Area under the Now, = triangular field distribution

Point B is at higher potential than point A. so the potential variation across the semiconductor will be like

Since, electron energy varies in opposite direction with respect to potential since =

the energy band diagram will

be like option D

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th

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GATE QUESTION BANK

EDC

P - N Junction Theory and Characteristics ECE - 2006 1. Find the correct match between Group 1 and Group 2 Group 1 Group 2 E. Varactor 1. Voltage reference diode F. PIN dode 2. High-frequency switch G. Zener 3. Tuned circuits diode H. Schottky 4. Current controlled diode attenuator (A) (B) (C) (D) 2.

1 2 5V

3.

5.

In a p+ n junction diode under reverse bias, the magnitude of electric field is maximum at (A) the edge of the depletion region on the p-side (B) the edge of the depletion region on the n-side (C) the p+ n junction (D) The centre of the depletion region on the n-side

E - 4, F - 2, G -1, H – 3 E- 2, F - 4, G - 1, H – 3 E- 3, F - 4, G- 1, H – 2 E - 1, F - 3, G - 2, H-4

In the circuit shown below, the switch was connected to position 1 at t < 0 and at t = 0, it is changed to position 2. Assume that y the diode has zero voltage drop and a storage time For VR is given by (all in Volts)

(A) (B)

ECE - 2007 4. A p+ n junction has built-in potential of 0.8 V. The depletion layer width at a reverse bias of 1.2 V of 2 µm. For a reverse bias of 7.2 V, the depletion layer width will be (A) 4 µm (C) 8 µm (B) 4.9 µm (D) 12 µm

1K VR

5V

= =

(C) (D)

The values of voltage (VD) across a tunnel-diode corresponding to peak and valley currents are VP and VV respectively. The range of tunnel-diode voltage VD for which the slope of its I-VD characteristics is negative would be (A) (C) (B) 0 VD < VP (D)

ECE - 2008 6. Consider the following assertions. S1: For Zener effect to occur, a very abrupt junction is required S2: For quantum tunneling to occur, a very narrow energy barrier is required. Which of the following is correct? (A) Only S2 is true (B) S1 and S2 are both true but S2 is not a reason for S1 (C) S1 and S2 are both true and S2 is a reason for S1 (D) Both S1 and S2 are false 7.

Which of the following is NOT associated with a p-n junction? (A) Junction Capacitance (B) Charge Storage Capacitance (C) Depletion Capacitance (D) Channel Length Modulation

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th

th

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GATE QUESTION BANK

ECE - 2010 Statement for Linked Answer Questions 8 and 9 The silicon sample with unit cross sectional area shown below is in thermal equilibrium. The following information is given: T = 300 K, electronic charge = 1.6 × , thermal voltage = 26 mV and electron mobility = 1350

=

=

= μ

8.

The magnitude of the electric field at = is (A) 1 kV/cm (C) 10 kV/cm (B) 5 kV/cm (D) 26 kV/cm

9.

The magnitude of the electron drift current density at = is (A) 2.16 × A/ (B) 1.08 × A/ (C) 4.32 × A/ (D) 6.48 × A/

10.

Compared to a p n junction with = = , which one of the following statements is TRUE for a p n junction with = = ? (A) Reverse breakdown voltage is lower and depletion capacitance is lower (B) Reverse breakdown voltage is higher and depletion capacitance is lower (C) Reverse breakdown voltage is lower and depletion capacitance is higher (D) Reverse breakdown voltage is higher and depletion capacitance is higher

EDC

ECE - 2011 11. A silicon PN junction is forward biased with a constant current at room temperature. When the temperature is d ℃ d voltage across the PN junction (A) increases by 60 mV (B) decreases by 60 mV (C) increases by 25 mV (D) decreases by 25 mV ECE - 2013 12. In a forward biased pn junction diode, the sequence of events that best describes the mechanism of current flow is (A) Injection, and subsequent diffusion and recombination of minority carriers (B) Injection, and subsequent drift and generation of minority carriers (C) Extraction, and subsequent diffusion and generation of minority carriers (D) Extraction, and subsequent drift and recombination of minority carriers ECE - 2014 13. Consider an abrupt PN junction (at T = 300 K) shown in the figure. The depletion region width on the N-side of the junction is 0.2 and the permittivity of silicon ( ) is 44 F/cm. At the junction, the approximate value of the peak electric field (in kV/cm) is _________.

=

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th

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GATE QUESTION BANK

EDC

Answer Keys and Explanations 1.

[Ans. C] Varactor diodes are used in tuned circuits and zener diodes is used as voltage reference when it is in reverse bias. Schottky diode is used as high frequency switch.

2.

[Ans. A] During storage time, diode will be ON, so = 5V during 0 a (B) 0 for r < a and 1/r for r > a (C) r for r < a and 1/r for r > a (D) 0 for r < a and 1/r2 for r > a

10.

A hole of radius b (b < a) is now drilled along the length of the wire at a distance d from the center of the wire as shown below.

ECE - 2014 12. In spherical co-ordinates, let â â denote unit vectors along the , directions. r

si

cos

t

r â

m

And .

si cos t r â m r represent the electric and magnetic field components of the EM wave at large distances r from a dipole antenna, in free space. The average power (W) crossing the hemispherical shell located at r m is _______. 13.

th

xâ yâ zâ a |r| r iv r ∇ r = ____________

If r th

th

th

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GATE QUESTION BANK

14.

Given the vector cos x si y â si x cos y â where â â denote unit vectors along x,y directions, respectively. The magnitude of curl of A is __________

15.

The electric field (assumed to be onedimensional) between two points A and B is shown. Let a be the electrostatic potentials at A and B, respectively. The value of in Volts is_________

EE - 2007 Statement for Linked Answer Questions 2 &3 An inductor designed with 400 turns coil wound on an iron core of 16 cm2 cross sectional area and with a cut of an air gap length of 1mm. The coil is connected to a 230 V, 50 Hz ac supply. Neglect coil resistance, core loss, iron reluctance and a ag i ucta c . μo 4 x -7 H/m) 2. The current in the inductor is (A) 18.08 A (C) 4.56 A (B) 9.04 A (D) 2.28 A 3.

The average force on the core to reduce the air gap will be (A) 832.29N (C) 3332.47N (B) 1666.22N (D) 6664.84 N

4.

The total reactance and total susceptance of a lossless overhead EHV line, operating at 50 Hz, are given by 0.045 pu and 1.2 pu respectively. If the velocity of wave propagation is × km/s, then the approximate length of the line is (A) 122 km (C) 222 Km (B) 172 km (D) 272 km

5.

A solid sphere made of insulating material has a radius R and has a total charge Q distributed uniformly in its volume. What is the magnitude of the electric field intensity, E, at a distance r (0 < r < R) inside the sphere?

⁄cm

4 ⁄cm ⁄cm

μm

B

16.

Given ⃑ zâ xâ yâ . If S represents the portion of the sphere x y z =1 for z the ∫ ∇ × ⃑ . ⃑⃑⃑⃑s is ___________

17.

If ⃑

y

yz x̂

xz ŷ xyz ẑ is the electric field in a source free region, a valid expression for the electrostatic potential is (A) xy yz (C) y xyz (B) xy xyz (D) xy xyz

xy

EE - 2006 1. Which of the following statements holds for the divergence of electric and magnetic flux densities? (A) Both are zero (B) These are zero for static densities but non zero for time varying densities (C) It is zero for the electric flux density (D) It is zero for the magnetic flux density

EMT

6.

(A)

(C)

(B)

(D)

Divergence of the vector field V( x, y, z) = (x cos xy + y) i + (y cos xy) j + (sin z2 + x2 + y2) k is (A) 2z cos z2 (B) sin xy + 2z cos z2 (C) x sin xy – cos z (D) None of these.

th

th

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GATE QUESTION BANK

EE - 2008 7. A coil of 300 turns is wound on a nonmagnetic core having a mean circumference of 300 mm and a crosssectional area of 300 mm2. The inductance of the coil corresponding to a magnetizing current of 3A will be (Given that μ = 4 x 10-7 H/m) (A) 37.68 μH (C) 37. 68 μH (B) 113.04 μH (D) 113. 04 μH 8.

9.

10.

A capacitor consists of two metal plates each 500 × 500 mm2 and spaced 6 mm apart. The space between the metal plates is filled with a glass plate of 4mm thickness and a layer of paper of 2 mm thickness. The relative permittivities of the glass and paper are 8 and 2 respectively. Neglecting the fringing effect, the capacitance will be (Given that = 8.85 x F/m) (A) 983.33 pF (C) 637.5 pF (B) 1475 pF (D) 9956.25 pF Two point charges = 10 μC and = 20μC are placed at coordinates (1,1,0) and ( 1, 1,0) respectively, the total electric flux passing through a plane z = 20 will be (A) 7.5 μC (C) 15.0 μC (B) 13.5 μC (D) 22.5 μC An extra high voltage transmission line of the length 300 km can be approximated by a lossless line having propagation constant . radians per km. Then the percentage ratio of the length to wavelength will given by (A) 24.24% (C) 19.05% (B) 12.12% (D) 6.06%

EMT

EE - 2013 12. The curl of the gradient of the scalar field defined by x y y z 4z x is (A) 4xya yza zxa (B) 4a a a (C) 4xy 4z a x yz a

13.

14.

y zx a (D) The flux density at a point in space given by B 4xa ya a m . The value of constant k must be equal to (A) 2 (C) +0.5 (B) 0.5 (D) +2 A dielectric slab with 500 mm x 500 mm cross-section is 0.4 m long. The slab is subjected to a uniform electric field of a a mm. The relative permittivity of the dielectric material is equal to 2. The value of constant is . × m . The energy stored in the dielectric in Joules is (C) .5 (A) . × (D) (B) . ×

EE - 2014 15. is the capacitance of a parallel plate capacitor with air as dielectric (as in figure (a)). If, half of the entire gap as shown in figure (b) is filled with a dielectric of permittivity , the expression for the modified capacitance is a

⁄

(A) (B) 16.

EE - 2012 11. The direction of vector A is radially outward from the origin, with | | r where r x y z and k is a constant. The value of n for which ∇ A=0 is (A) 2 (C) 1 (B) 2 (D) 0 th

(C) (D)

⁄

The following four vector fields are given in Cartesian co-ordinate system. The vector field which does not satisfy the property of magnetic flux density is (A) y a z a x a (B) z a x a y a (C) x a y a z a (D) y z a x z a x y a

th

th

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GATE QUESTION BANK

17.

A parallel plate capacitor consisting two dielectric materials is shown in the figure. The middle dielectric slab is placed symmetrically with respect to the plates.

19.

A hollow metallic sphere of radius r is kept at potential of 1 Volt. The total electric flux coming out of the concentric spherical surface of radius R r is (A) 4 r (C) 4 (B) 4 r (D) 4

20.

A perfectly conducting metal plate is placed in x-y plane in a right handed coordinate system. A charge of √ columbs is placed at coordinate (0, 0, 2). is the permittivity of free space. Assume ̂ ̂ ̂ to be unit vectors along x, y and z axes respectively. At the coordinate √ √ ) the electric field vector ⃑

ot

If the potential difference between one of the plates and the nearest surface of dielectric interface is 2 Volts, then the ratio is (A) 1:4 (C) 3:2 (B) 2:3 (D) 4:1 18.

EMT

(Newtons/Columb) will be ⁄√ o um s y

The magnitude of magnetic flux density ⃑B at a point having normal distance d meters from an infinitely extended wire carrying current of

is

i

ucti g

√ √

u its .

x

An infinitely extended wire is laid along the x-axis and is carrying current of 4 A in the +ve x direction. Another infinitely extended wire is laid along the y-axis and is carrying 2 A current in the +ve y direction. μ is permeability of free space. Assume ̂ ̂ ̂ to be unit vectors along x, y and z axes respectively.

am s

rf ct y co m ta at

(C) ̂ (D) √ ̂

(A) √ ̂ ̂ (B)

μ

B

4

z

Assuming

right

coordinate system, magnetic field intensity, ⃑⃑ at coordinate (2,1,0) will be ̂w B

4

handed

r m

̂

m

̂

m m th

th

th

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GATE QUESTION BANK

EMT

Answer Keys and Explanations ECE 1.

[Ans. D]

μ

*

B x

B +a y

∇× From stokes theorem .

∮ 2.

∬(

5.

). s

[Ans. B] rom sto

’s th or m

∮

[Ans. A]

∮∇×

s

z axis

iv

∇×

∮ ⃑ ⃑⃑⃑ x axis

∬ ⃑ ⃑⃑⃑⃑s

x axis

6.

[Ans. D] z

z axis

cos

√

(

√

) y

cos ⃑ ⃑ 3.

.

.

.

√

.

[Ans. D] rom Maxw ∇.B

â

x

â × â â â × â â Non-zero field components x and z

â

’s quatio 7.

[Ans. C] ⃑ xyâ

∇×

xâ

For static magnetic field,

yâ xâ

∇× 4.

⃑.

[Ans. C] ∇×B μ ∇×B μ a μ

*

yâ

xy x

x

y

y

3

a

||

a

1

S

R

P

Q

|| B

μ

x â

B a z

X

0

B

⁄ √

B a z B ( x

P-Q:

y=1 dy=0

∫ ⃑ . ⃑⃑⃑

B )a + y

⁄ √

∫

⁄ √

x x

⁄ √

x

|

⁄ √ ⁄ √

Q-R: th

th

th

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GATE QUESTION BANK

∫ ⃑ . ⃑⃑⃑

11.

x

√

∫ (

√

)

y

4

[Ans. D] By curl stroke theorem ∮⃑ . c ∬ ∇ × ⃑ . s ov r o

R-S: Y=3 ,dy=0 ∫ ⃑ . ⃑⃑⃑

∫

⁄√

⁄ √

x x

12.

x | ⁄√

⁄ √

= (

∫ ⃑ . ⃑⃑⃑

â

√

)

y

∫ ⃑.

=1

9.

[Ans. D] Apply the divergence theorem

10.

a

r

a

r

a

r

a

r

a

si ∫

si

∫

.

∭ ∇. r

r

. .

and r is

13.

[Ans. C] r

â

â r si

∫

( ∇. r ∭ v the position vector)

si

r

. si â r Average power crossing

∫ ⃑ . ⃑⃑⃑

∫ ⃑ . ⃑⃑⃑

∯ r. ⃑ x

.

si

r

∫⃑

8.

m

si â m r So, average poynting vector is ⃑ (⃑ × ⃑⃑ )

∫ (

So, ∮ ⃑ .

si

r .

⃑⃑

x

√

∫ ⃑ . ⃑⃑⃑

)

surfac

[Ans. *] Range 55.4 to 55.6 In phasor notation ⃑

S-P: x=

EMT

r .ra r a

[Ans. *] Range 2.9 to 3.1 r̅ xâ yâ zâ r x y z r x r y r z x r y r z r r

x x

r

∇

r

r .∇

r

x

r

iv r . ∇

r

r

r . â r x

r

r r r [ . â x r

r ) x x r (r. ) z z r ( ) x (r.

[Ans. C] Magnetic held inside hole depends on radius of hole i.e b and also on the location of hole from center of the conductor ie d. Hole has uniform cross section magnetic field is uniform

x

(r.

r. [ th

r

th

r ) x

r.

x r ] r x

r r â â y r z r r â â ] y z

r x

y

(r.

r ) y

x ( ) r th

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GATE QUESTION BANK

x x ) ( ) r r r x x ( ) r r r (r. ) x x

∫si

r. (

14.

15.

[Ans. *] Range 0.01 to 0.01 si x × cos y cos x × si y cur x y cos x × cos y cos x × cos y Magnitude of curl of A = 0

17.

×

x ⁄

×

x

(4 ×

x x)|

×

×

×

× × ×

16.

×

EE 1.

[Ans. *] Range 3.13 to 3.15 ̅ zâ xâ yâ â â â ∇× ̅

|

x z

â x y sphere

y x â

z

z y â

| 2.

[cos

cos

. ]

[Ans. D] ̅ y

. 4

yz x̂

xy xyz ẑ

xz ŷ

[Ans. D] Inductance ×

L=

represents a hemi-

×

×

×

= 0.32115H

r si cos r si si r cos

∫ ∇× ̅ . s

.

∫si

[Ans. D] ∇.B ∇. ot a ways Z ro

a x y z

si

si

∇v Hence tio ∇v [ y â xy z â yz â ] tio B [ y ∇v yz x̂ xy xz ŷ ] xyz ẑ tio [yz x̂ ∇v y xz ŷ xyz âz] tio [ y ∇v yz x̂] xy xz ŷ xyz ẑ Hence D option is correct

×

4×

∫

∫si

̅

.

∫

cos

∫ ∇×̅ . s

[Ans. *] Range . . ⁄cm ⁄cm ; A= ⁄cm 4 ⁄cm B= × 4 x × 4× x ∫

∫

EMT

I= x

y

z

=

×

r s ∫(â

â

× .

.

si â ) . s th

th

th

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GATE QUESTION BANK

3.

[Ans. A]

9.

[Ans. C] Net flux will be half of the total charge,

Force = Energy = F= 4.

=

= 0.835J

.

. 4

10.

[Ans. D] m wav

√ ×

f

.4 .

.

u

√

(

×

u

Length = 3×

× .

×

×

×

× ×

.

11.

.

=185 km 5.

= 15µC

[Ans. B] v

EMT

[Ans. A] ⃑ r ̂ ∇. ⃑

×

m 4 4 .

. gth ) gth

.

4 4 .

r . r .

’i radius r =

ra ia s

gth

i wav

[Ans. A] harg

.

r

r

. r

r or ∇. ⃑ r r

6.

r

r

[Ans. A] iv rg c

12.

[Ans. D] cur [∇ sca ar]

13.

[Ans. A]

∇. =

∇. B

cos xy yx si xy cos xy y si xy cos z . z zcos z 7.

[Ans. B] L= =

×

×

x

4x

y

y

z

4 14.

[Ans. B]

15.

[Ans. A] Case (i)

× ×

= 113.09µH 8.

[Ans. B]

h

C=

. .

×

× ×

×

×

× × ×

×

×

= 4.4nF

×

. C = 1.467nF th

th

th

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GATE QUESTION BANK

.h h

EMT

gio 4

.

Case(ii) h

(

i-

In any system the charge Q = CV is constant

ctric

18.

.h

)

[Ans. C]

h [ 16.

m t r

]

am s

[Ans. C] ̅ magnetic flux density [w If B ̅ ∇. B ̅ B â i. . if B B â B â ̅ ∇. B B B B Must x y z ̅ Option (A) : ∇. B ̅ Option (B) : ∇. B ̅ Option (C) : ∇. B x y ̅ Option (D) : ∇. B

17.

sourc m t r

m ]

4 am s sourc

Z

qua to z ro ̅ B

̅ ̅ B B μ 4 m s ̂

̅ B z

μ

̅ B

μ

̅ B

[Ans. C] ̅ B

am s

4μ

μ

( ̂) ̂w

̂w

m ̂w

m

m

μ ̂ am

⃑⃑

m

̅ ̅ = cross product of Note: Direction of B current direction and radial drections respectively [Biot – savart’s aw] 4

gio

4

4 th

th

th

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GATE QUESTION BANK

19.

[Ans. A]

EMT

× √ â 4 |r | |r̅ | √ (√ √ )

⃑ |r̅ | â

√ â |r | 4

√

√ √ √ √

The net flux leaving any closed surface is equal to charge enclosed in it Total electric flux leaving the co c tric s h r of ra ius ‘ ’ charg co tai y th m ta s h r of ra ius ‘r’ ot tia at a oi t o m ta ic s h r 4 4

(√ √

â

) √ √ √ √

√

⃑

*

√ √

√ √

√

̅

√ [ 4̂]

̅

̂

√

+

r wto ⁄ o um

r

iv

ot 4 r (flux ) leaving the sphere of radius ‘ ’ 4 r 20.

[Ans. B] Method of image: A perfect conducting plate acts like a mirror for the existing charge by the introduction of virtual charge opposite to the existing charge and equivalent distant from it z axis

X

√ ou om s

r̅ √ √

r̅

√

X

ou om s

⃑ ⃑

⃑

⃑ 4

imag

√ â |r |

√ â |r |

th

th

th

THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30 Cross, 10 Main, Jayanagar 4 Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 536

GATE QUESTION BANK

EMT

EM Wave Propagation ECE - 2006 1. The electric field of an electromagnetic wave propagating in the positive z-direction is given by E= âx sin (t – βz) + ây sin (t – βz + ) The wave is (A) linearly polarized in the z-direction (B) elliptically polarized (C) left-hand circularly polarized (D) right-hand circularly polarized 2.

3.

When a plane wave travelling in freespace is incident normally on a medium having the fraction of power transmitted into the medium is given by (A) ⁄ (C) ⁄ (B) ⁄ (D) ⁄

meter from the interface. Due to the total internal reflection, the transmitted beam has a circular cross-section over the interface. The area of the beam crosssection at the interface is given by (A) (C) ⁄ (B) (D) ECE - 2007 ⁄ ) of a plane wave The ⃗⃗ field propagating in free space given by

5.

⃗⃗

√

βz) + ̂

(

βz + )

The time average power flow density in Watts is ⁄ (A) (C) ⁄ ⁄ (B) (D) 6.

A right circularly polarized (RCP) plane wave is incident at an angle of to the normal, on an air-dielectric interface. If the reflected wave is linearly polarized, the relative dielectric constant is

A medium is divided into regions I and II about plane, as shown in the figure below. An electromagnetic wave with electric field ̂ + ̂ + ̂ is incident normally on the interface from region-I, The electric field in region-II at the interface is Region 1

̂

Linearly

Polarized

RCP

Region 2 Air

σ1

μ1 μ0 r1 = 3

σ2

E1

μ2 r2 = 4

μ0

Dielectric

E2

(C) 2 (D) 3

(A) √ (B) √

x>0

x 0 . The angle of transmission in the dielectric slab is _________ degrees.

th

th

th

THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30 Cross, 10 Main, Jayanagar 4 Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 546

GATE QUESTION BANK

EMT

Answer Keys and Explanations 1.

[Ans. D] When at t= 0 switch is closed voltage which will travel towards load is

=

V+ = V

=

Where R is battary resistance V+ = 30×

=

4.

[Ans. B] Scattering Parameters

= 30V Reflected wave voltage V V At t = 400 µs total voltage at load V V V V V (1 ) 4 3 (1 )

[ ]

= =

5

3Z 2

15 Z

1

5.

Ω

[Ans. D] Z

2.

3.

[Ans. D] V Z 4 1 .4 Amp

Z [

a Z

6.

]

and

and l Z

[Ans. D] For

][ ]

Where a and a are incident wave and b b are reflected wave at port 1 and port 2 respectively. For matched network, Reflection coefficient is zero. So, S S And complete power transfer occurs so S S 1, so scattering matrix will be, 1 [ ] 1

1

Z 2Z

[

2 π

. 1m l 2 π . 1 j Z an l = j 0.73 Z . So Inductive

[Ans. D]

⁄4

ec ion Z ⁄4

Z

Z Z

36

5

Z

Z

Z

5 Z

Z

(5 ) 1

Z

(

Z Z

)

Z

=

Z

=

. .

2 =

2

So Z

12.5

Z .

∴Z

25

Z

= 8.34

25 Ω

3 th

th

th

THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30 Cross, 10 Main, Jayanagar 4 Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 547

GATE QUESTION BANK

7.

[Ans. D] Z 5 R = 0.1 for distortion less line  RC = LG  = √R √

Z Z



Phase velocity, v 2π 1 ha e

√

=

10.

= √R

1.6

1

m

[Ans. A] Voltage maximum in the line is observed exactly at

. (

1 rad ec hif per uni leng h rad m

v

√

G=

EMT

Therefore ‘z ’ hould be real

)

= √ .1

. (

. )

VSWR

=0.002

z

1 Ω (∵ Vol age

minimum at load) 8.

[Ans. B] For transmission line, Z (

Z

√ )

11.

[Ans. *] 6 Z ln(b a) √ 6 ln(2.4) √1 .89 15.5 Ω

12.

[Ans. B] Concepts: Matching section should be have length l odd multiple of ( ⁄4) opera ing waveleng h Solution:

= 30 × 2 = 60

For transmission line, Z

Z [

Z

=Z [

]

but Z

Z

jZ an

]

jZ = j30 Z and Zin2 are in series Z Z Z Therefore, Reflection co-efficient,

l 5 Ω

=

1

=

Ω

2

Ω

ma ching ec ion

=

or 429

z l

4

4 1

.175m

√

VSWR = 9.

| |

√ √

=

. .

or 1

. 75 m 4 4 2 Length l should be integral multiple of both l and l ∴ l mul iple of of l l mul iple of .525m Hence 1.05 is the appropriate solution

1.64

[Ans. C] The line is matched as Z Z 5 Ω and hence reflected wave a absent. For the travelling wave, given: Phase difference for a length of 2 mm π 4 rad Frequency of excitation = 10 GHz

th

th

z l

th

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GATE QUESTION BANK

13.

[Ans. A] Given- Return loss of a device =20dB To find: Voltage standing wave ratio (VSWR) and the magnitude reflection coefficient Solution overview: Return Loss in dB= 2 log Where i he magni ude of reflec ion coefficien .

15.

[Ans. *] Range 39 to 41 Z

5 Ω

Z

R

jx

8

Z jZ which i real Z jZ R 3 j 5 j Z 5 jR 3 j R 8 j Z 2 Rj R 8 j 2 Rj Z 2 Rj 1 Rj Since Z i purely real Z 16 j R j R 4 ohm Z (

1 1 Solution: Return Loss in dB=2 2 log ∴ 1 log .1 . VSWR 1.22 . Conclusion: Option (A) is correct. VSWR

14.

EMT

8)

[Ans. B] 16.

[Ans. *] Range 70 to 72 ⁄4 ran former

W Z

Z

5 Ω

1

Ω

h

The

characteristic

transformer is √1 7 .71Ω h W W h Z V

impedance

of

5

17.

[Ans. B] TEM wave travelling in the positive x direction Electric field and magnetic field goes in – y and z direction respectively

18.

[Ans. *] Range 32.99 to 34.01

√ √

Since the separation between the plates reduced by a factor of 2 1. Phase velocity not effected 2. Characteristic impedance becomes half 3. Characteristic impedance becomes half of the original

A Z 1

.5

5 Ω

Z

Z

Z

4

Z th

5 Ω

Ω

Z th

(1

Z Z Ω)

5 ohm

th

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GATE QUESTION BANK

Z Z 19.

EMT

(1 )(5 ) (15 ) 33.33 ohm

[Ans. *] Range 29 to 31 ⃗ 1 co ( 3x √3z)â vol ⁄m (3 ̂ (Z

√3 ̂ ) ree pace

)

Z (Z

)

3

ielec ric

Incident angle (3â √3â ). (â ) co √12 1 co ( ) 6 2 in in in in

√

in

2

nell

aw

√

√3 1 2 √3 3

1 2

th

th

th

THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30 Cross, 10 Main, Jayanagar 4 Block, Bangalore-11 : 080-65700750,  [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 550

GATE QUESTION BANK

EMT

Guided E.M Waves ECE - 2006 1. A rectangular waveguide having mode as dominant mode is having a cutoff frequency of GHz for the mode. The inner broad-wall dimension of the rectangular waveguide is (A) ⁄ (C) ⁄ (B) (D)

2.

In a microwave test bench, why is the microwave signal amplitude modulated at the 1 kHz ? (A) To increase the sensitivity of measurement (B) To transmit the signal to a far-off place (C) To study amplitude modulation (D) Because crystal detector fails at microwave frequencies

6.

In the design of a single mode step index optical fiber close to upper cut-off, the single –mode operation is NOT preserved if, (A) Radius as well as operating wavelength are halved (B) Radius as well as operating wavelength are doubled (C) Radius is halved and operating wavelength is doubled (D) Radius is doubled and operating wavelength is halved

ECE - 2009 7. Which of the following statements is true regarding the fundamental mode of the metallic waveguides shown?

ECE - 2007 3. An air filled rectangular wave guide has inner dimensions of 3cm 2cm. The wave impedance of the mode of propagation in the waveguide at a frequency of 30GHz is (free space impedance ) (A) 308 (C) (B) (D) 4.

The ⃗ field in rectangular waveguide of inner dimension given by ⃗

( )

(

)

(

)̂

Where is a constant ,and a and b are the dimensions along the x-axis and the y-axis respectively. The mode of propagation in the waveguide is (A) (C) (B) (D) ECE - 2008 5. A rectangular waveguide of internal dimensions (a = 4 cm and b = 3 cm) is to be operated in mode. The minimum operating frequency is (A) 6.25 (C) 5.0 (B) 6.0 (D) 3.75

P: Coaxial

Q: cylindrical

R: Rectangular

(A) (B) (C) (D)

Only P has no cut-off –frequency Only Q has no cut-off -frequency Only R has no cut-off -frequency All three have cut-off –frequencies

ECE - 2011 8. The modes in a rectangular waveguide are denoted by where m and n are the Eigen numbers along the larger and smaller dimensions of the waveguide respectively. Which one of the following statements is TRUE? (A) The mode of the waveguide does not exist (B) The mode of the waveguide does not exist (C) The and the modes both exist and have the same cut – off frequencies (D) The and the modes both exist and have the same cut – off frequencies th

th

th

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GATE QUESTION BANK

ECE - 2012 9. A plane wave propagating in air with ) ⃗ ( ̂ ̂ ̂ ) ( is incident on perfectly conducting slab positioned at x 0. The ⃗ field of the

10.

reflected wave is (A) ( ̂ ̂

̂ )

(

)

(B) (

̂

̂

̂ )

(

)

(C) (

̂

̂

̂ )

(

)

(D) (

̂

̂

̂ )

(

)

The magnetic field along the propagation direction inside a rectangular waveguide with the cross-section shown in the figure is ( ) ( ) ( )

EMT

Which one of the following statements is TRUE about the distortion of the received signal due to impedance mismatch? (A) The signal gets distorted if , irrespective of the value of (B) The signal gets distorted if , irrespective of the value of (C) Signal distortion implies impedance mismatch at both ends: (D) Impedance mismatches do NOT result in signal distortion but reduce power transfer efficiency 12.

For a rectangular waveguide of internal dimensions ( ) the cut-off frequency for the mode is the arithmetic mean of the cut-off frequencies for mode and mode. If 𝑎=√ cm, the value of 𝑏 (in cm) is _____.

y 1.2 cm

13.

3 cm

x

The phase velocity vp of the wave inside the waveguide satisfies (A) vp> c (C) 0