GATE QUESTION BANK Contents
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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
11 Laplace Transform 204 – 206
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
18 Laplace Transform 257 – 261
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
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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
GATE QUESTION BANK Mathematics
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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
GATE QUESTION BANK Mathematics
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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
GATE QUESTION BANK Mathematics
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(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
GATE QUESTION BANK Mathematics
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(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
GATE QUESTION BANK Mathematics
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(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
GATE QUESTION BANK Mathematics
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⟹
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
GATE QUESTION BANK Mathematics
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, | - [
| ]
→
[
| ]
→ [
| ]
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
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( ) 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]
The augmented matrix for given system is
[
| ]
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]
The augmented matrix for given system is
[ k
| ] 6xyz7 [
]
Using Gauss elimination we reduce this to
an upper triangular matrix to find its rank
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[ 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
[
]
|
|
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