0dwk0dwlfv$oo%udqfk6royhg 4xhvwlrq...22 stability & routh hurwitz criterion 295 – 300 23 root...

MathMatics All Branch Solved Question-2005-2017

GATE QUESTION BANK

for

Instrumentation Engineering

By

Mathmatics All Branch Solved 2005-2017

GATE QUESTION BANK Contents

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th

Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page I

Contents Subject Name Topic Name Page No. #1. Mathematics 1-148 1 Linear Algebra 1 – 28

2 Probability & Distribution 29 – 57

3 Numerical Methods 58 – 73

4 Calculus 74 – 112

5 Differential Equations 113 – 131

6 Complex Variables 132 – 143

7 Laplace Transform 144 – 148

#2. Network Theory 149 – 216 8 Network Solution Methodology 149 – 167

9 Transient/Steady State Analysis of RLC Circuits to DC Input

168 – 185

10 Sinusoidal Steady State Analysis 186 – 203


12 Two Port Networks 207 – 214

13 Network Topology 215 – 216

#3. Signals & Systems 217 – 275 14 Introduction to Signals & Systems 217 – 223

15 Linear Time Invariant (LTI) systems 224 – 238

16 Fourier Representation of Signals 239 – 250

17 Z-Transform 251 – 256


19 Frequency response of LTI systems and Diversified Topics

262 – 275

#4. Control Systems 276 – 340 20 Basics of Control System 276 – 282

21 Time Domain Analysis 283 – 294

22 Stability & Routh Hurwitz Criterion 295 – 300

23 Root Locus Technique 301 – 308

24 Frequency Response Analysis using Nyquist plot 309 – 316

25 Frequency Response Analysis using Bode Plot 317 – 322

26 Compensators & Controllers 323 – 329

27 State Variable Analysis 330 – 340

#5. Analog Circuits 341 – 421 28 Diode Circuits - Analysis and Application 341 – 353

29 AC & DC Biasing-BJT and FET 354 – 363

30 Small Signal Modeling Of BJT and FET 364 – 372

31 BJT and JFET Frequency Response 373 – 375

32 Feedback and Oscillator Circuits 376 – 381

33 Operational Amplifiers and Its Applications 382 – 420

34 Power Amplifiers 421

GATE QUESTION BANK Mathematics

THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th

Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 1

Linear Algebra ME – 2005

1. Which one of the following is an

Eigenvector of the matrix[

]?

(A) [

]

(B) [

]

(C) [

]

(D) [

]

2. 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

(B) 2

(C) 3

(D) 4

ME – 2006

3. Multiplication of matrices E and F is G.

Matrices E and G are

E [ os sin sin os

] and

G [

]. What is the matrix F?

(A) [ os sin sin os

]

(B) [sin os os sin

]

(C) [ os sin sin os

]

(D) [sin os os sin

]

4. Eigen values of a matrix

S 0

1are 5 and 1. What are the

Eigenvalues of the matrix = SS?

(A) 1 and 25

(B) 6 and 4

(C) 5 and 1

(D) 2 and 10

5. Match the items in columns I and II.

Column I Column II

P. Singular matrix

1. Determinant is not defined

Q. Non-square matrix

2. Determinant is always one

R. Real symmetric matrix

3. Determinant is zero

S. Orthogonal matrix

4. Eigen values are 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

ME – 2007

6. The number of linearly independent

Eigenvectors of 0

1 is

(A) 0

(B) 1

(C) 2

(D) Infinite

7. 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

(C) 1

(D) 2

9. The matrix [ p

] has one Eigenvalue

equal to 3. The sum of the other two

Eigenvalues is

(A) p

(B) p – 1

(C) p – 2

(D) p – 3



Cross, 10th

Main, Jayanagar 4th


20. One of the Eigenvectors of the matrix

0

1 is

(A) {– }

(B) {– }

(C) 2 3

(D) 2 3

21. Consider a 3×3 real symmetric matrix S

such that two of its Eigenvalues are

with respective Eigenvectors

[

x x x ] [

y y y ] If then x y + x y +x y

equals

(A) a (B) b

(C) ab (D) 0

22. 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 ( )

( ) ( ) where is s l r Let

( ) 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.

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 , - , - and

, -. The order of , ( ) - will be

(A) (2 × 2) (B) (3 × 3

(C) (4 × 3) (D) (3 × 4

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) (B)

(C) (D)

CE – 2007

6. The minimum and the maximum

Eigenvalue of the matrix [

]are 2

and 6, respectively. What is the other

Eigenvalue?

(A) (B)

(C) (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 + 2 Y + Z

(A) 2, 7 (B) 3, 8

(C) 8, 3 (D) 7, 2



Cross, 10th

Main, Jayanagar 4th


20. The determinant of matrix [

]

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

1

(A) n (B) n

(C) n (D) 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

4. Let A be a 4x4 matrix with Eigenvalues

–5, –2, 1, 4. Which of the following is an

Eigenvalue of 0 II

1, where I is the 4x4

identity matrix?

(A) (B)

(C) (D)

CS – 2007

5. Consider the set of (column) vectors

defined by X={xR3 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 0

1 n 0

1

(A) One (B) two

(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 (B) x = 5, y = 8

(C) x = 3, y = 9 (D) x = 4, y = 10



Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com

(B) [

⁄

⁄

⁄

⁄

]

(C) [

]

(D) [

⁄

⁄

⁄

⁄

]

2. Let, A=0

1 and = 0 ⁄

1.

Then (a + b)=

(A) ⁄ (B) ⁄

(C) ⁄ (D) ⁄

3. Given the matrix 0

1 the

Eigenvector is

(A) 0 1

(B) 0 1

(C) 0 1

(D) 0 1

ECE – 2006

4. For the matrix 0

1 , the Eigenvalue

corresponding to the Eigenvector

0

1 is

(A) 2 (B) 4

(C) 6 (D) 8

5. The Eigenvalues and the corresponding

Eigenvectors of a 2 2 matrix are given

by

Eigenvalue Eigenvector

= 8 v = 0 1

= 4 v = 0 1

The matrix is

(A) 0

1

(B) 0

1

(C) 0

1

(D) 0

1

6. The rank of the matrix [

]

(A) 0 (B) 1

(C) 2 (D) 3

ECE – 2007

7. It is given that X1 , X2 …… M are M non-

zero, 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.

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

9. All the four entries of the 2 x 2 matrix

P = 0p p p 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

ECE – 2009

10. The Eigen values of the following matrix

are

[

]

(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



Cross, 10th

Main, Jayanagar 4th


(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. If R = [

] , then top row of is

(A) , - (B) , -

(C) , - (D) , -

2. For the matrix p = [

] , one of

the Eigenvalues is equal to 2 . Which of

the following is an Eigenvector?

(A) [ ]

(B) [ ]

(C) [ ]

(D) [ ]

3. 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

EE – 2006

Statement for Linked Answer Questions 4

and 5.

P = [ ]

, Q = [ ]

, R = [ ]

are

three vectors

4. An orthogonal set of vectors having a

span that contains P,Q, R is

(A) [ ] [

]

(B) [ ] [

] [

]

(C) [ ] [ ] [

]

(D) [ ] [

] [

]

5. The following vector is linearly

dependent upon the solution to the

previous problem

(A) [ ]

(B) [ ]

(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 (B) Has rank 1

(C) Is orthogonal (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

L(x) = M [

x x x ]

Then the Eigenvalues of M are

(A) 0, +1, 1 (B) 1, 1, 1

(C) i, i, 1 (D) i, i, 0

8. Let x and y be two vectors in a 3

dimensional space and <x, y> denote

their dot product. Then the determinant

det 0 x x x y y x y y 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



Cross, 10th

Main, Jayanagar 4th


(B) 0

1 and 0

1

(C) 0

1 and 0

1

(D) 0

1 and 0

1

EE – 2013

19. The equation 0

1 0x x 1 0

1 has

(A) No solution

(B) Only one solution 0x x 1 0

1.

(C) Non – zero unique solution

(D) Multiple solution

20. A matrix has Eigenvalues – 1 and – 2. The

corresponding Eigenvectors are 0 1 and

0 1 respectively. The matrix is

(A) 0

1

(B) 0

1

(C) 0

1

(D) 0

1

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. 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.

23. Two matrices A and B are given below:

0p qr s

1 [p q pr qs

pr qs r s ]

If the rank of matrix A is N, then the rank

of matrix B is

(A) N (B) N

(C) N (D) N

IN – 2005

1. Identify which one of the following is an

Eigenvector of the matrix A = 0

1?

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

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

2. 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

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 (B) 2

(C) 3 (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 matrix A, it is observed

that

0 1 0

1 n 0

1 0

1

Then matrix A is



Cross, 10th

Main, Jayanagar 4th


⟹

Hence real Eigen value.

8. [Ans. B]

Let 0

1 eigenv lues re n

Eigen vector corresponding to

is ( I)

.

/ .xy/ .

/

By simplifying

.K / .

/ y t king K

Eigen vector corresponding to =2

is ( I)

.

/ .xy/ .

/

By simplifying ( K K ) 4

⁄5 by

taking K

⁄

⁄

9. [Ans. C]

Sum of the diagonal elements = Sum of

the Eigenvalues

⟹ 1 + 0 + p = 3+S

⟹ S= p 2

10. [Ans. B]

( ⁄ ) [

]

[

]

→ →

[

]

→ [

]

If system will h ve solution

11. [Ans. A]

iven M M → MM I

[

x

] [

x

] 0

1

Equating the elements x ⁄

12. [Ans. A]

0

1 → Eigenv lues re

Eigenve tor is x x verify the options

13. [Ans. C]

[

] [

]

→ [

]

→ [

]

( ) infinite m ny solutions

14. [Ans. B]

Eigenvalues of a real symmetric matrix

are always real

15. [Ans. B]

0

1 eigenv lues v lue

Eigen vector will be . /

Norm lize ve tor

[

√( ) ( )

√( ) ( ) ]

*

√ ⁄

√ ⁄

+

16. [Ans. C]

The given system is

x y z

x y z

x y z

Use Gauss elimination method as follows

Augmented matrix is



Cross, 10th

Main, Jayanagar 4th


, | - [

| ]

→

[

| ]

→ [

| ]

nk ( )

nk ( | )

So, Rank (A) = Rank (A|B) = 2 < n (no. of

variables)

So, we have infinite number of solutions

17. [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 x.

So Ax = x … ①

and x x……②

king tr nspose of equ tion ②

x x … ③

[( ) n is s l r ]

x x x x

x x x x … , -

x x x x

(x x) (x x) ( re s l r )

( x x re Eigenve tors they nnot e zero )

i i

i 0

Hence Eigenvalue of a symmetric matrix

are real

18. [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.

19. [Ans. A]

|

|

So, |

|

|

|

(Taking 2 common from each row)

( )

20. [Ans. D]

0

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 x x ] [

y y y ] x y x y x y

22. [Ans. D]

( )

In case of matrix PQ QP (generally)

CE

1. [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



Cross, 10th

Main, Jayanagar 4th


( ) 3×3

P 2×3

3×2

P( ) (2×3) (3×3) (3×2)

2 × 2

( ( ) ) 2×2

4. [Ans. D]

The augmented matrix for given system is

[

| ] → [

| ]

Then by Gauss elimination procedure

[

| ]

→ [

| ]

→ [

| ]

( ⁄ )

( )

( ) ( ⁄ )

∴ olution is non – existent for above

system.

5. [Ans. B]

∑ = Trace (A)

+ + = Trace (A)

= 2 + ( 1) + 0 = 1

Now = 3

∴ 3 + + = 1

Only choice (B) satisfies this condition.

6. [Ans. B]

∑ = Trace (A)

+ + = 1 + 5 + 1 = 7

Now = 2, = 6

∴ 2 + 6 + = 7

= 3

7. [Ans. A]


[

| ]

Using Gauss elimination we reduce this to

an upper triangular matrix to find its

rank

[

| ] →

[

|

]

→ [

|

]

Now for infinite solution last row must be

completely zero

i e – 2 = 0 n – 7 = 0

n

8. [Ans. A]

Inverse of 0

1 is

0

1

( )0

1

∴ 0

1

( )0

1

0

1

9. [Ans. B]

( ) P = ( ) P

( ) ( )

= ( ) (I) =

10. [Ans. B]

A = 0

1

Characteristic equation of A is

|

| = 0

(4 ) ( 5 ) 2 × 5 =0

+ 30 = 0

6, 5

11. [Ans. D]


[ k

| ] 6xyz7 [

]

Using Gauss elimination we reduce this to

an upper triangular matrix to find its rank



Cross, 10th

Main, Jayanagar 4th


[ 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 skew-

symmetric if and only if = B

13. [Ans. D]

By definition A + is always symmetric

is symmetri

is lw ys skew symmetri

is skew symmetri

14. [Ans. B]

0

1

=

( )0

1

∴ 0 i i i i

1

,( i)( i) i -0 i ii i

1

=

0 i ii i

1

15. [Ans. B]

0

1

Sum of the Eigenvalues = 17

Product of the Eigenvalues =

From options, 3.48 + 13.53 = 17

(3.48)(13.53) = 47

16. [Ans. 0.5]

0.5

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 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

[

]

→

[

]

→

[

]

→

[

]

Interchanging Column 1& Column 2 and

taking transpose

[

]

|

|

0dwk0dwlfv$oo%udqfk6royhg 4xhvwlrq...22 stability & routh hurwitz criterion 295 – 300 23 root...

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