Engineering maths

1. ENGINEERING MATHEMATICS
Objective & NAT Questions Bank
“Topic wise From GATE 1988 to 2019”
(Containing around 1200 Questions)
(VERSION: 07|12|19)
GATE
Common to all GATE – Engineering branches
Product of,
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place of trust since 2009…

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3. SYLLABUS: ENGG. MATHEMATICS
GATE - 2020
Electronics & Communication (EC)
Linear Algebra: Vector space, basis, linear dependence and independence, matrix algebra, eigen values and
eigen vectors, rank, solution of linear equations – existence and uniqueness.
Calculus: Mean value theorems, theorems of integral calculus, evaluation of definite and improper integrals,
partial derivatives, maxima and minima, multiple integrals, line, surface and volume integrals, Taylor series.
Differential Equations: First order equations (linear and nonlinear), higher order linear differential
equations, Cauchy's and Euler's equations, methods of solution using variation of parameters,
complementary function and particular integral, partial differential equations, variable separable method,
initial and boundary value problems.
Vector Analysis: Vectors in plane and space, vector operations, gradient, divergence and curl, Gauss's,
Green's and Stoke's theorems.
Complex Analysis: Analytic functions, Cauchy's integral theorem, Cauchy's integral formula; Taylor's and
Laurent's series, residue theorem.
Numerical Methods: Solution of nonlinear equations, single and multi-step methods for differential
equations, convergence criteria.
Probability and Statistics: Mean, median, mode and standard deviation; combinatorial probability,
probability distribution functions - binomial, Poisson, exponential and normal; Joint and conditional
probability; Correlation and regression analysis.
Electrical Engineering (EE)
Linear Algebra: Matrix Algebra, Systems of linear equations, Eigenvalues, Eigenvectors.
Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper
integrals, Partial Derivatives, Maxima and minima, Multiple integrals, Fourier series, Vector identities,
Directional derivatives, Line integral, Surface integral, Volume integral, Stokes’s theorem, Gauss’s theorem,
Green’s theorem.
Differential equations: First order equations (linear and nonlinear), Higher order linear differential
equations with constant coefficients, Method of variation of parameters, Cauchy’s equation, Euler’s
equation, Initial and boundary value problems, Partial Differential Equations, Method of separation of
variables.
Complex variables: Analytic functions, Cauchy’s integral theorem, Cauchy’s integral formula, Taylor
series, Laurent series, Residue theorem, Solution integrals.
Probability and Statistics: Sampling theorems, Conditional probability, Mean, Median, Mode, Standard
Deviation, Random variables, Discrete and Continuous distributions, Poisson distribution, Normal
distribution, Binomial distribution, Correlation analysis, Regression analysis.
Numerical Methods: Solutions of nonlinear algebraic equations, Single and Multi‐step methods for
differential equations.
Transform Theory: Fourier Transform, Laplace Transform, z‐Transform.

6. GATE Paper
(This booklet contains the questions from the following
GATE streams)
Code
Aerospace Engineering AE
Agricultural Engineering AG
Biotechnology BT
Civil Engineering CE
Chemical Engineering CH
Computer Science and Information Technology CS
Electronics and Communication Engineering EC
Electrical Engineering EE
Ecology and Evolution EY
Geology and Geophysics GG
Instrumentation Engineering IN
Information & Technology IT
Mathematics MA
Mechanical Engineering ME
Mining Engineering MN
Metallurgical Engineering MT
Petroleum Engineering PE
Physics PH
Production and Industrial Engineering PI
Statistics ST
Textile Engineering and Fiber Science TF
Engineering Sciences XE

7. Table of Contents
01. LINEAR ALGEBRA 1
PROPERTY BASED PROBLEM 1
DET. & MULT. 3
ADJOINT - INVERSE 7
EIGEN VALUES & VECTORS 10
RANK 21
HOMOGENOUS & LINEAR EQN 23
HAMILTONS 28
GEOMETRICAL TRANSFORMATION 29
02. CALCULUS 31
2.1 MEAN VALUE THEOREM 32
ROLLE’S MVT 32
LAGRANGES’S MVT 33
CAUCHY’S MVT 34
2.2 MAXIMA AND MINIMA 36
SINGLE VARIABLE 36
DOUBLE VARIABLE 42
2.3 LIMITS 46
LIMIT, CONTINUITY, DIFF. CHECKUP 46
LIMITS 48
Single Variable 48
Double Variable 53
2.4 INTEGRAL & DIFFERENTIAL CALCULAS 56
SINGLE INTEGRATION 56
Simple Improper Integration 59
Laplace form of Integration 60
Beta and Gama Integration 61
AREA & VOLUME CALCULATION 62
Area Calculation 62
Volume Calculation 62
Double and Triple Integration 64
DIFFERENTIAL CALCULUS 65
2.5 SERIES 68
TAYLOR SERIES EXPANSION 68
CONVERGENCE TEST 70
MISCELLANEOUS 71
03. DIFFERENTIAL EQUATIONS 73
LINEARITY/ORDER/DEGREE OF DE 73
FIRST ORDER & DEGREE DE 74
Lebnitz Linear Form 74
Variable Separable Form 76
Exact Differential Equation Form 78
MISCELLANEOUS 78
HIGHER ORDER DE 80
MISCELLANEOUS 88

8. 04. COMPLEX VARIABLE 92
BASIC PROBLEMS 92
ANALYTIC FUNCTION 95
CAUCHY’S INTEGRAL & RESIDUE 97
Cauchy Integral 97
Residue 100
05. PROBABILITY AND STATISTICS 103
PROBABILITY PROBLEMS 103
Combined Problems 103
Problems on Combination 109
Problems from Binomial 113
Problems on Bay’s 114
PROBABILITY DISTRIBUTION 116
Statistics 116
Expectation 118
Normal Distribution 119
Uniform Distribution 121
Combined Continuous Dist. 123
Poisson Distribution 127
Miscellaneous 128
06. NUMERICAL METHODS 132
BASIC PROBLEMS 132
ROOTS FINDING METHODS 134
Newton Raphson Method 134
Other Methods 137
SOLUTION OF DIFFERENTIAL EQUATION 137
Eulers Method 137
Runge Kutta Method 138
NUMERICAL INTEGRATION 138
Trapezoidal Rule 138
Simpsons Rule 140
Mixed 141
MISCELLANEOUS 141

9. www.targate.org Page 1
01
Linear Algebra
Property Based Problem
B [GATE-EE-2011-IITM]
1. The matrix [A] =
2 1
4 1
 
 

 
is decomposed
into a product of lower triangular matrix [L]
and an upper triangular [U]. The property
decomposed [L] and [U] matrices
respectively are
(A)
1 0
4 1
 
 

 
and
1 1
0 2
 
 

 
(B)
2 0
4 1
 
 

 
and
1 1
0 1
 
 

 
(C)
1 0
4 1
 
 
 
and
2 1
0 1
 
 

 
(D)
2 0
4 3
 
 

 
and
1 0.5
0 1
 
 
 
D [GATE-CS-1994-IITKGP]
2. If A and B are real symmetric matrices of
order n then which of the following is true.
(A) A AT
= I (B) A = A-1
(C) AB = BA (D) (AB)T
= BT
AT
B [GATE-CE-1998-IITD]
3. If A is a real square matrix then A+AT
is
(A) Un symmetric
(B) Always symmetric
(C) Skew – symmetric
(D) Sometimes symmetric
C[GATE-EC-2005-IITB]
4. Given an orthogonal matrix A =
0 0 0 1
0 0 1 0
1 0 0 0
0 1 0 0
 
 
 
 
 
 
1
( )
T
AA 
is ______
(A) 4
1
4
I (B) 4
1
2
I
(C) 4
I (D) 4
1
3
I
A [GATE-CS-2001-IITK]
5. Consider the following statements
S1: The sum of two singular matrices may
be singular.
S2 : The sum of two non-singulars may be
non-singular.
This of the following statements is true.
(A) S1 & S2 are both true
(B) S1 & S2 are both false
(C) S1 is true and S2 is false
(D) S1 is false and S2 is true
D [GATE-CS-2011-IITM]
6. [A] is a square matrix which is neither
symmetric nor skew-symmetric and [A]T
is
its transpose. The sum and differences of
these matrices are defined as [S] = [A] +
[A]T
and [D] = [A] – [A]T
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
5tAD [GATE-EC-2014-IITKGP]
7. For matrices of same dimension M, N and
scalar c, which one of these properties
DOES NOT ALWAYS hold?
(A)  
T
T
M M

(B)    
T T
cM c M

(C)  
T T T
M N M N
  
(D) MN NM


10. ENGINEERING MATHEMATICS
Page 2 TARGATE EDUCATION GATE-(EE/EC)
Statement for Linked Answer Questions for next
two problems
Given that three vector as
T T T
10 2 2
P 1 ,Q 5 ,R 7
3 9 12
 
     
     
    
     
     

     
AA [GATE-EE-2006-IITKGP]
8. An orthogonal set of vectors having a span
that contains P, Q, R is
(A)
6 4
3 2
6 3

  
  
 
  
  
  
(B)
4 5 8
2 7 2
4 11 3

   
   
   
   
  
   
(C)
6 3 3
7 2 9
1 2 4

   
   
   
   
  
   
(D)
4 1 5
3 31 3
11 3 4
   
   
   
   
   
AB [GATE-EE-2006-IITKGP]
9. The following vector is linearly dependent
upon the solution to the previous problem
(A)
8
9
3
 
 
 
 
 
(B)
2
17
30

 
 

 
 
 
(C)
4
4
5
 
 
 
 
 
(D)
13
2
3
 
 
 
 

 
AB [GATE-EE-1997-IITM]
10. A square matrix is called singular if its
(A) Determinant is unity
(B) Determinant is zero
(C) Determinant is infinity
(D) Rank is unity
AA [GATE-ME-2004-IITD]
11. For which value of x will be the matrix
given below become singular?
8 x 0
4 0 2
12 6 0
 
 
 
 
 
(A) 4 (B) 6
(C) 8 (D) 12
AC [GATE-IN-2010-IITG]
12. X and Y are non-zero square matrices of size
n n
 . If XY= n n
0  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

AA [GATE-CE-2009-IITR]
13. A square matrix B is skew symmetric if
(A)
T
B B
  (B)
T
B B

(C)
1
B B

 (D)
1 T
B B


AC [GATE-CS-2004-IITD]
14. The number of differential n n
 symmetric
matrices with each element being either 0 or
1 is: (Note: power(2, x) is same as
x
2 ).
(A)
n
Power 2
 
 
(B)
2
n
Power 2
 
 
(C)
 
2
n n
2
Power 2

 
 
 
 
(D)
 
2
n n
2
Power 2

 
 
 
 
AA [GATE-CS-2000-IITKGP]
15. An n n
 array V is defined as follows
 
V i, j ,i j
  for all i, j, 1 i, j n
  then the
sum of the elements of the array V is
(A) 0 (B) n – 1
(C) 2
n 3n 2
  (D) n(n + 1)
AB [GATE-CH-2013-IITB]
16. Which of the following statements are
TRUE?
P. The eigen values of a symmetric matrix
are real.
Q. The value of the determinant of an
orthogonal matrix can only be +1.
R. The transpose of a square matrix A has
the same eigen values as those of A
S. The inverse of an 'n n'
 matrix exists
if and only if the rank is less than ‘n’

11. TOPIC 1 - LINEAR ALGEBRA
www.targate.org Page 3
(A) P and Q only (B) P and R only
(C) Q and R only (D) P and S only
AD [GATE-AG-2017-IITR]
17. Matrix
0 0.5 1.5
0.5 0 2.5
1.5 2.5 0
 
 

 
 
 
 
is a
(A) Diagonal matrix
(B) Symmetric matrix
(C) Orthogonal matrix
(D) Skew-symmetric matrix
AC [GATE-CE-2017-IITR]
18. The matrix P is a inverse of a matrix Q. If I
denotes the identity matrix, which one of the
following options is correct?
(A) PQ I but QP I
 
(B) QP I but PQ I
 
(C) PQ I and QP I
 
(D) PQ QP I
 
AD [GATE-ME-2017-IITR]
19. Consider the matrix
1 1
0
2 2
0 1 0
1 1
0
2 2
P
 
 
 
  
 

 
 
 
.
Which one of the following statements about
P is INCORRECT ?
(A) Determinant of P is equal to 1.
(B) P is orthogonal.
(C) Inverse of P is equal to its transpose.
(D) All eigenvalues of P are real numbers.
A2.8 to 3.0 [GATE-GG-2018-IITG]
20. The highest singular value of the matrix
1 2 1
1 2 0
G
 
  

 
is ______.
A–6 T1.1 [GATE-BT-2019-IITM]
21. Matrix
0 6
A=
0
p
 
 
 
will be skew-symmetric
when p = _____.
AD T1.1 [GATE-MN-2019-IITM]
22. Matrix
0 2
A
 
 
 
   
 
 
  
 
is orthogonal. The
values of  ,  and  respectively are
(A) 1 1 1
, ,
3 2 6
  
(B) 1 1 1
, ,
3 6 2
  
(C) 1 1 1
, ,
6 2
3
  
(D) 1 1 1
, ,
2 6 3
  
C [GATE-IN-2014-IITKGP]
23. A scalar valued function is defined as
( ) T T
f x x Ax b x c
   , 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 f(x)
will occur when x equals
(A)  
1
T
A A b

(B)  
1
T
A A b


(C)
1
2
A b

 
  
 
(D)
1
2
A b

**********
Det. & Mult.
199to201 [GATE-EC-2014-IITKGP]
24. The determinant of matrix A is 5 and the
determinant of matrix B is 40. The
determinant of matrix AB is ------. .
10 [GATE-BT-2018-IITG]
25. The determinant of the matrix
4 6
3 2

 
 

 
is
_________.
A160 [GATE-BT-2016-IISc]
26. The value of determinant A given below is
__________
5 16 81
0 2 2
0 0 16
A
 
 
  
 
 
D [GATE-PI-1994-IITKGP]
27. The value of the following determinant
1 4 9
4 9 16
9 16 25
is :
(A) 8 (B) 12
(C) – 12 (D) – 8
B [GATE-CE-2001-IITK]
28. The determinant of the following matrix
5 3 2
1 2 6
3 5 10
 
 
 
 
 

12. ENGINEERING MATHEMATICS
Page 4 TARGATE EDUCATION GATE-(EE/EC)
(A) – 76 (B) – 28
(C) 28 (D) 72
B [GATE-PI-2009-IITR]
29. The value of the determinant
1 3 2
4 1 1
2 1 3
is :
(A) – 28 (B) – 24
(C) 32 (D) 36
A [GATE-CE-1997-IITM]
30. If the determinant of the matrix
1 3 2
0 5 6
2 7 8
 
 

 
 
 
is 26 then the determinant of the matrix
2 7 8
0 5 6
1 3 2
 
 

 
 
 
is :
(A) – 26 (B) 26
(C) 0 (D) 52
B [GATE-CS-1998-IITD]
31. If  =
1
1
1
a bc
b ca
c ab
then which of the
following is a factor of  .
(A) a + b (B) a - b
(C) abc (D) a + b + c
B [GATE-CE-1999-IITB]
32. The equation
2
2 1 1
1 1 1 0
y x x
  represents a
parabola passing through the points.
(A) (0,1), (0,2),(0,-1)
(B) (0,0), (-1,1),(1,2)
(C) (1,1), (0,0), (2,2)
(D) (1,2), (2,1), (0,0)
C [GATE-EE-2002-IISc]
33. The determinant of the matrix
1 0 0 0
100 1 0 0
100 200 1 0
100 200 300 1
 
 
 
 
 
 
is
(A) 100 (B) 200
(C) 1 (D) 300
A [GATE-EC-2005-IITB]
34. The determinant of the matrix given below is
0 1 0 2
1 1 1 3
0 0 0 1
1 2 0 1
 
 

 
 
 

 
(A) -1 (B) 0
(C) 1 (D) 2
C [GATE-CE-1999-IITB]
35. If A is any n n
 matrix and k is a scalar then
| | | |
kA α A
 where  is
(A) kn (B)
k
n
(C)
n
k (D)
k
n
A [GATE-CS-1996-IISc]
36. The matrices
cos sin
sin cos
θ θ
θ θ

 
 
 
and
0
0
a
b
 
 
 
commute under multiplication.
(A) If a = b (or) ,
θ nπ
 n is an integer
(B) Always
(C) never
(D) If a cos sin
θ b θ

AA [GATE-ME-2015-IITK]
37. If any two columns of a determinant
4 7 8
P 3 1 5
9 6 2
 
 
  
 
 
are interchanged, which one
of the following statements regarding the
value of the determinant is CORRECT ?
(A) Absolute value remains unchanged but
sign will change.
(B) Both absolute value and sign will
change.
(C) Absolute value will change but sign
will not change .
(D) Both absolute value and sign will
remain unchanged.
A1 [GATE-EC-2014-IITKGP]
38. Consider the matrix:
6
0 0 0 0 0 1
0 0 0 0 0
1
0 0 0 0 0
1
J
0 0 0 0 0
1
0 0 0 0 0
1
0 0 0 0 0
1


13. TOPIC 1 - LINEAR ALGEBRA
www.targate.org Page 5
Which is obtained by reversing the order of
the columns of the identity matrix 6
I .
Let 6 6
P I J
   , where  is a non-negative
real number. The value of  for which
det(P)=0 is ___________.
AB [GATE-EC-2013-IITB]
39. Let A be m n
 matrix and B an
n m
 matrix. It is given that determinant
( )
m
I AB
  determinant ( )
n
I BA
 , where
k
I is the k k
 identity matrix. Using the
above property, the determinant of the
matrix given below is
2 1 1 1
1 2 1 1
1 1 2 1
1 1 1 2
 
 
 
 
 
 
(A) 2 (B) 5
(B) 8 (C) 16
AB [GATE-EE-2007-IITK]
40. Let x and y be two vectors in a 3
dimensional space and < x, y > denote their
dot product. Then the determinant
x,x x, y
det
y, x y, y
   
 
 
   
 
(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
A88 [GATE-CE-2012-IITD]
41. The determinant of the matrix
0 1 2 3
1 0 3 0
2 3 0 1
3 0 1 2
 
 
 
 
 
 
is______.
A23 [GATE-CE-2014-IITKGP]
42. Given the matrices
3 2 1
J 2 4 2
1 2 6
 
 
  
 
 
and
1
K 2
1
 
 
  
 

 
, the product of 1
K JK

is
_________.
AA [GATE-ME-2014-IITKGP]
43. Given that the determinant of the matrix
1 3 0
2 6 4
1 0 2
 
 
 
 

 
is -12, the determinant of the
matrix
2 6 0
4 12 8
1 0 2
 
 
 
 

 
is :
(A) -96 (B) -24
(C) 24 (D) 96
AA [GATE-BT-2013-IITB]
44. If P =
1 1
2 2
 
 
 
,
2 1
Q
2 2
 
  
 
and
3 0
R
1 3
 
  
 
which one of the following
statements is TRUE?
(A) PQ = PR (B) QR = RP
(C) QP = RP (D) PQ = QR
AD [GATE-ME-2014-IITKGP]
45. Which one of the following equations is a
identity for arbitrary 3 3
 real matrices P, Q
and R?
(A)  
P Q R PQ RP
  
(B)  
2 2 2
P Q P 2PQ Q
   
(C)  
det P Q detP detQ
  
(D)  
2 2 2
P Q P PQ QP Q
    
A16 [GATE-CE-2013-IITB]
46. There are three matrixes  4 2
P 
,  2 4
Q 
and
 4 1
R 
. The minimum number of
multiplication required to compute the
matrix PQR is
AA [GATE-CE-2004-IITD]
47. Real matrices          
3 1 3 3 3 5 5 3 5 5
A B C D E
    
and  5 1
F 
are given. Matrices [B] and [C] are
symmetric. Following statements are made
with respect to these matrices.
(1) Matrix product        
T T
F C B C F is a
scalar.
(2) Matrix product     
T
D F D is always
symmetric.
With reference to above statements, which
of the following applies?
(A) Statement 1 is true but 2 is false
(B) Statement 1 is false but 2 is true

14. ENGINEERING MATHEMATICS
Page 6 TARGATE EDUCATION GATE-(EE/EC)
(C) Both the statements are true
(D) Both the statements are false
AB [GATE-CE-1999-IITB]
48. The number of terms in the expansion of
general determinant of the order n is
(A)
2
n (B) n!
(C) n (D)  
2
n 1

AC [GATE-IN-2006-IITKGP]
49. For a given 2 2
 matrix A, it is observed
that
1 1
A
1 1
   
 
   
 
   
and
1 1
A 2
2 2
   
 
   
 
   
.
Then the matrix A is :
(A)
2 2 1 0 1 1
A
1 1 0 2 1 1

     
      
    
     
(B)
1 1 1 0 2 1
A
1 2 0 2 1 1
     
      
   
     
(C)
1 1 1 0 2 1
A
1 2 0 2 1 1

     
      
    
     
(D)
0 2
A
1 3

 
  

 
AA [GATE-PI-2007-IITK]
50. The determinant
1 b b 1
b 1 b 1
1 2b 1

 evaluates to
(A) 0 (B) 2b(b - 1)
(C) 2(1 - b)(1 + 2b) (D) 3b(1 + b)
A0 [GATE-CS-2014-IITKGP]
51. If the matrix A is such that
 
2
A 4 1 9 5
7
 
 
 
 
 
 
Then the determinant of A is equal to
_______.
AD [GATE-CS-2013-IITB]
52. Which one of the following determinant
does NOT equal to
2
2
2
1 x x
1 y y
1 z z
 
 
 
 
 
?
(A)
 
 
 
1 x x 1 x
1 y y 1 y
1 z z 1 z
  
 

 
 

 
(B)
2
2
2
1 x 1 x 1
1 y 1 y 1
1 z 1 z 1
 
 
 
 
 
 
 
 
(C)
2 2
2 2
2
0 x y x y
0 y z y z
1 z z
 
 
 
 
 
 
 
(D)
2 2
2 2
2
2 x y x y
2 y z y z
1 z z
 
 
 
 
 
 
 
AA [GATE-CS-2000-IITKGP]
53. The determinant of the matrix
2 0 0 0
8 1 7 2
2 0 2 0
9 0 6 1
 
 
 
 
 
 
is :
(A) 4 (B) 0
(C) 15 (D) 20
AC [GATE-CS-1997-IITM]
54. Let n n
A  be matrix of order n and 12
I be the
matrix obtained by interchanging the first
and second rows of n
I . Then 12
AI is such
that its first
(A) Row is the same as its second row
(B) Row is the same as second row of A
(C) Column is same as the second column
of A
(D) Row is a zero row
[GATE-CS-1996-IISc]
55. Let 11 12
21 22
a a
A
a a
 
  
 
and 11 12
21 22
b b
B
b b
 
  
 
be
two matrices such that AB = I. Let
1 0
C A
1 1
 
  
 
and CD = I. Express the
elements of D in terms of the elements of B.
11 12
11 21 12 22
b b
[D]
b b b b
 
  
   
 
ANS :
AA [GATE-CE-2017-IITR]
56. If
1 5
A
6 2
 
  
 
and
3 7
B
8 4
 
  
 
, T
AB is
equal to
(A)
38 28
32 56
 
 
 
(B)
3 40
42 8
 
 
 
(C)
43 27
34 50
 
 
 
(D)
38 32
28 56
 
 
 
AD [GATE-MT-2017-IITR]
57. For the matrix,
1 1 2
2 1 1 ,
1 1 2
T
A AA
 
 
  
 
 
is

15. TOPIC 1 - LINEAR ALGEBRA
www.targate.org Page 7
(A)
6 5 6
5 6 6
6 5 6
 
 
 
 
 
(B)
6 5 6
5 6 6
5 5 6
 
 
 
 
 
(C)
6 5 6
5 6 5
6 6 6
 
 
 
 
 
(D)
6 5 6
5 6 5
6 5 6
 
 
 
 
 
AA [GATE-PE-2017-IITR]
58. For the two matrices
1 2 3 7 0
,
4 5 6 8 1
X Y
   
 
   

   
, the product
YX will be :
(A)
7 14 21
4 11 18
YX
 
  
 
(B)
4 11 18
7 14 21
YX
 
  
 
(C)
7 14 18
14 11 21
YX
 
  
 
(D)
7 14 21
18 5 6
YX
 
  
 
AD [GATE-TF-2018-IITG]
59. Let
2
a b
A
b
 
  

 
and
1
1
X

 
  
 
. If
3
1
AX

 
  
 
, then | |
A is equal to
(A) 2 (B) –2
(C) –6 (D) 6
AC [GATE-MN-2018-IITG]
60. If
cos sin
sin cos
X
 
 
  
  
 
, then T
XX is
(A)
0 1
1 0
 
 
 
(B)
1 0
0 1

 
 

 
(C)
1 0
0 1
 
 
 
(D)
0 1
1 0

 
 

 
AD [GATE-EE-2016-IISc]
61. Let
3 1
1 3
P
 
 
 
. Consider the set S of all
vectors
x
y
 
 
 
such that 2 2
1
a b
  where
a x
P
b y
   

   
   
. Then S is :
(A) a circle of radius 10
(B) a circle of radius
1
10
(C) an ellipse with major axis along
1
1
 
 
 
(D) an ellipse with minor axis along
1
1
 
 
 
AD [GATE-MN-2018-IITG]
62. The values of x satisfying the following
condition are :
4 3
0
3 6
x
x



(A) 6, 4 (B) 4, 9
(C) 5, 6 (D) 3,7
A0A5.5 [GATE-EE-2018-IITG]
63. Consider a non-singular 2 2
 square matrix
A . If (A) 4
trace  and 2
(A ) 5
trace  , the
determinant of the matrix A is
_________(up to 1 decimal place).
AC T1.2 [GATE-AG-2019-IITM]
64. The determinant of the matrix
2 1 1
2 3 2
1 2 1
A
 
 
  
 
 
is
(A) 1 (B) 0
(C) -1 (D) 2
AB A2 T1.2 [GATE-PE-2019-IITM]
65. Let
1 2 1
,
2 1 0
a
A X
b
   
 
   
   
and
3 1
3 2
Y
 
  
 
. If AX Y
 , then a b
 equals
______.
**********
Adjoint - Inverse
AC [GATE-MN-2016-IISc]
66. If     
A B I
 then
(A)    
T
B A
 (B)    
T
A B

(C)    
1
B A

 (D)    
B A


16. ENGINEERING MATHEMATICS
Page 8 TARGATE EDUCATION GATE-(EE/EC)
A [GATE-EE-1999-IITB]
67. If A =
1 2 1
2 3 1
0 5 2
 
 
 
 
 

 
and adj (A) =
11 9 1
4 2 3
10 7
k
 
 
 
 
 
 
 
Then k =
(A) – 5 (B) 3
(C) – 3 (D) 5
AA [GATE-EE-2005-IITB]
68. If A =
2 0.1
0 3

 
 
 
and 1 1 / 2
0
a
A
b
  
  
 
then __________
a b
 
(A)
7
20
(B)
3
20
(C)
19
60
(D)
11
20
A [GATE-ME-2009-IITR]
69. For a matrix [M] =
3 / 5 4 / 5
3 / 5
x
 
 
 
. The
transpose of the matrix is equal to the
inverse of the matrix,
1
[ ] [ ] .
T
M M 
 The
value of x is given by
(A)
4
5
 (B)
3
5

(C)
3
5
(D)
4
5
B [GATE-CE-2010-IITG]
70. The inverse of the matrix
3 2
3 2
i i
i i

 
 
 
 
is
(A)
3 2
1
3 2
2
i i
i i
 
 
 

 
(B)
3 2
1
3 2
12
i i
i i
 
 
 

 
(C)
3 2
1
3 2
14
i i
i i
 
 
 

 
(D)
3 2
1
3 2
14
i i
i i
 
 
 

 
A [GATE-CE-2007-IITK]
71. The inverse of 2 2
 matrix
1 2
5 7
 
 
 
is :
(A)
7 2
1
5 1
3

 
 

 
(B)
7 2
1
5 1
3
 
 
 
(C)
7 2
1
5 1
3

 
 

 
(D)
7 2
1
5 1
3
 
 
 
 
 
D [GATE-EE-1995-IITK]
72. The inverse of the matrix S =
1 1 0
1 1 1
0 0 1

 
 
 
 
 
is
(A)
1 0 1
0 0 0
0 1 1
 
 
 
 
 
(B)
0 1 1
1 1 1
1 0 1
 
 
 
 
 
 
(C)
2 2 2
2 2 2
0 2 2
 
 
 
 
 
 
(D)
1/ 2 1/ 2 1/ 2
1/ 2 1/ 2 1/ 2
0 0 1

 
 
 
 
 
 
A0.25 [GATE-ME-2018-IITG]
73. If
1 2 3
0 4 5
0 0 1
A
 
 
  
 
 
then 1
det( )
A
is ______
(correct to two decimal places).
AB [GATE-TF-2016-IISc]
74. Let
1
1
2
A
1
1
2
 
 
  
 
 
 
. The determinant of 1
A
is
equal to
(A)
1
2
(B)
4
3
(C)
3
4
(D) 2
A [GATE-EE-1998-IITD]
75. If A =
5 0 2
0 3 0
2 0 1
 
 
 
 
 
then 1
A
=
(A)
1 0 2
0 1/ 3 0
2 0 5

 
 
 
 

 
(B)
5 0 2
0 1/ 3 0
2 0 1
 
 

 
 
 

17. TOPIC 1 - LINEAR ALGEBRA
www.targate.org Page 9
(C)
1/5 0 1/ 2
0 1/3 0
1/ 2 0 1
 
 
 
 
 
(D)
1/ 5 0 1/ 2
0 1/ 3 0
1/ 2 0 1

 
 
 
 

 
B [GATE-CE-2000-IITKGP]
76. If A, B, C are square matrices of the same
order then
1
( )
ABC 
is equal be
(A)
1 1 1
C A B
  
(B)
1 1 1
C B A
  
(C)
1 1 1
A B C
  
(D)
1 1 1
A C B
  
AA [GATE-ME-2015-IITK]
77. For a given matrix
4 3
4 3
 
 
  

 
i i
P
i i
,
where 1
 
i , the inverse of matrix P is
(A)
4 3
1
4 3
24

 
 
 
 
i i
i i
(B)
4 3
1
4 3
25

 
 
 
 
i i
i i
(C)
4 3
1
4 3
24
 
 
 

 
i i
i i
(D) None
AB [GATE-EE-2005-IITB]
78. If
1 0 1
R 2 1 1
2 3 2

 
 
 
 
 
 
, then the top row of
1
R
is :
(A)  
5 6 4 (B)  
5 3 1

(C)  
2 0 1
 (D)  
2 1 0

AA [GATE-EE-1998-IITD]
79. If
5 0 2
A 0 3 0
2 0 1
 
 
 
 
 
then
1
A
(A)
1 0 2
1
0 0
3
2 0 5

 
 
 
 
 

 
(B)
5 0 2
1
0 0
3
2 0 1
 
 
 
 
 
 
(C)
1 1
0
5 2
1
0 0
3
1
0 1
2
 
 
 
 
 
 
 
 
 
(D)
1 1
0
5 2
1
0 0
3
1
0 1
2
 
 
 
 
 
 
 

 
 
AA [GATE-CE-1997-IITM]
80. If A and B are two matrices and if AB exist
then BA exists
(A) Only if A has many rows as B has
columns
(B) Only if both A and B are square
matrices
(C) Only if A and B are skew matrices
(D) Only if A and B are symmetric
AA [GATE-PI-2008-IISc]
81. Inverse of
0 1 0
1 0 0
0 0 1
 
 
 
 
 
is :
(A)
0 1 0
1 0 0
0 0 1
 
 
 
 
 
(B)
0 1 0
1 0 0
0 0 1

 
 

 
 

 
(C)
0 1 0
0 0 1
0 0 1
 
 
 
 

 
(D)
0 1 0
0 0 1
1 0 0

 
 

 
 

 
AA [GATE-CE-1997-IITM]
82. Inverse of matrix
0 1 0
0 0 1
1 0 0
 
 
 
 
 
is:
(A)
0 0 1
1 0 0
0 1 0
 
 
 
 
 
(B)
1 0 0
0 0 1
0 1 0
 
 
 
 
 
(C)
1 0 0
0 1 0
0 0 1
 
 
 
 
 
(D)
0 0 1
0 1 0
1 0 0
 
 
 
 
 
AA [GATE-PI-1994-IITKGP]
83. The matrix
1 4
1 5

 
 

 
is an inverse of the
matrix
5 4
1 1

 
 

 
(A) True (B) False
AB [GATE-CS-2004-IITD]
84. Let A, B, C, D be n n
 matrices, each with
non-zero determinant, If ABCD = 1, then
1
B
is
(A) 1 1 1
D C A
  
(B) CDA
(C) ADC
(D) Does not necessarily exist
[GATE-CS-1994-IITKGP]
85. The inverse of matrix
1 0 1
1 1 1
0 1 0
 
 

 
 
 
is :

18. ENGINEERING MATHEMATICS
Page 10 TARGATE EDUCATION GATE-(EE/EC)
1
1 1 1
1
A 0 0 2
2
1 1 1


 
 
  
 

 
ANS :
C [GATE-ME-2006-IITKGP]
86. Multiplication of matrices E and F is G.
Matrices E and G are E =
cos sin 0
sin cos 0
0 0 1
θ θ
θ θ

 
 
 
 
 
and G =
1 0 0
0 1 0
0 0 1
 
 
 
 
 
.
What is the matrix F?
(A)
cos sin 0
sin cos 0
0 0 1
θ θ
θ θ

 
 
 
 
 
(B)
cos cos 0
cos sin 0
0 0 1
θ θ
θ θ
 
 

 
 
 
(C)
cos sin 0
sin cos 0
0 0 1
θ θ
θ θ
 
 

 
 
 
(D)
sin cos 0
cos sin 0
0 0 1
θ θ
θ θ

 
 
 
 
 
AC [GATE-PE-2018-IITG]
87. The inverse of the matrix
1 3
1 2
 
 
 
is,
(A)
2 3
1 1
 
 
 
(B)
2 1
3 1

 
 

 
(C)
2 3
1 1

 
 

 
(D)
2 3
1 1

 
 

 
AD T1.3 [GATE-CS-2019-IITM]
88. Let X be a square matrix. Consider the
following two statemtns on X.
I. X is invertible.
II. Determinant of X is non-zero.
Which one of the following is TRUE?
(A) I implies II; II does not imply I.
(B) II implies I; I does not imply II.
(C) I does not imply II; II does not imply I.
(D) I and II are equivalent statements.
AC T1.3 [GATE-CE-2019-IITM]
89. The inverse of the matrix
2 3 4
4 3 1
1 2 4
 
 
 
 
 
is
(A)
10 4 9
15 4 14
5 1 6
 
 
 

 
 
 
 
(B)
10 4 9
15 4 14
5 1 6

 
 
 
 
 

 
(C)
4 9
2
5 5
4 14
3
5 5
1 6
1
5 5
 

 
 
 
 
 
 
 

 
 
(D)
4 9
2
5 5
4 14
3
5 5
1 6
1
5 5
 
 
 
 
 

 
 
 
 
 
 
AB T1.3 [GATE-PI-2019-IITM]
90. For any real, square and non-singular matrix
B, the
1
det 
B is
(A) Zero (B)
1
(det )
B
(C) (det )
 B (D) det B
A6 T1.3 [GATE-TF-2019-IITM]
91. The value of k for which the matrix
2
3 1
k
 
 
 
does not have an inverse is ______.
AC [GATE-EC-2016-IISc]
92. Let M4
= I, (where I denotes the identity
matrix) and M ≠ I, M2
≠ I and M3
≠ I. Then,
for any natural number k, M−1
equals :
(A) M4k + 1
(B) M4k + 2
(C) M4k +3
(D) M4k
**********
Eigen Values & Vectors
0.99to1.01 [GATE-EC-2014-IITKGP]
93. A real (4x4) matrix A satisfies the equation
A2
= I, where I is the (4x4) identity matrix
the positive eigen value of A is ------.
AA [GATE-ME-2016-IISc]
94. The condition for which the eigenvalues of
the matrix
2 1
1
A
k
 
  
 
are positive, is
(A) 1/ 2
k  (B) 2
k  
(C) 0
k  (D) 1/ 2
k  

19. TOPIC 1 - LINEAR ALGEBRA
www.targate.org Page 11
A2 [GATE-ME-2016-IISc]
95. The number of linearly independent
eigenvectors of matrix
2 1 0
0 2 0
0 0 3
A
 
 
  
 
 
is
_________.
A6 [GATE-CS-2014-IITKGP]
96. The product of non-zero eigen values of the
matrix
0 0 0
1 1
0 0
1 1 1
0 0
1 1 1
0 0
1 1 1
0 0 0
1 1
 
 
 
 
 
 
 
 
is ___________
AC [GATE-PE-2016-IISc]
97. Consider the matrix,
5 3
M
3 5
 
  
 
. The
normalized eigen-vector corresponding to
the smallest eigen-value of the matrix M is
(A)
3
2
1
2
 
 
 
 
 
 
(B)
3
2
1
2
 
 
 
 

 
 
(C)
1
2
1
2
 
 
 

 
 
 
(D)
1
2
1
2
 
 
 
 
 
 
A15.0 [GATE-CS-2016-IISc]
98. Two eigenvalues of a 3 3
 real matrix P are
 
2 1
  and 3. The determinant of P is
___________ .
A0.164-0.126 [GATE-CS-2016-IISc]
99. Suppose that the eigenvalues of matrix A are
1, 2, 4. The determinant of  
1 T
A
is__________
A0.99-1.01 [GATE-MT-2016-IISc]
100. For the transformtation shown below, if one
of the eigenvalues is 6, the other eigenvalue
of the matrix is _______
5 2
2 2
X x
Y y

     

     

     
AA [GATE-PI-2016-IISc]
101. The eigenvalues of the matrix
0 1
1 0
 
 

 
are
(A) i and i
 (B) 1 and -1
(C) 0 and 1 (D) 0 and -1
AA [GATE-TF-2016-IISc]
102. The eigen values and eigne vectors of
3 4
4 3
 
 

 
are
(A) 5
 and
1
2
 
 
 
,
1
2
 
 
 
respectively
(B) 3
 and
1
2
 
 
 
,
2
1
 
 
 
respectively
(C) 4
 and
1
2
 
 
 
,
2
1
 
 
 
respectively
(D) 5
 and
1
1
 
 
 
,
2
1
 
 
 
respectively
[GATE-CE-1998-IITD]
103. Obtain the eigen values and eigen vectors of
8 4
A
2 2

 
  
 
.
1
2
Solution :
1
for 4,X K
1
2
for 6,X K
1
 
   
 
 
   
 
C [GATE-IN-2009-IITR]
104. The eigen values of a 2 2
 matrix X are -2
and -3. The eigen values of matrix
1
( ) ( 5 )
X I X I

  are
(A) – 3, - 4 (B) -1, -2
(C) -1, -3 (D) -2, -4
A3.0 [GATE-BT-2016-IISc]
105. The positive Eigen value of the following
matrix is ______________.
2 1
5 2
 
 

 
A0.95-1.05 [GATE-EC-2016-IISc]
106. The value of for which the matrix
3 2 4
9 7 13
6 4 9
A
x
 
 
  
 
   
 
has zero as an eigenvalue is ______
AD T1.2 [GATE-ME-2019-IITM]
107. In matrix equation    
[ ]
A X R
 ,
 
4 8 4 2
[ ] 8 16 4 , 1
4 4 15 4
A X
   
 
 
    
 
 
 

   
and
 
32
16
64
R
 
 
  
 
 

20. ENGINEERING MATHEMATICS
Page 12 TARGATE EDUCATION GATE-(EE/EC)
One of the eigenvalues of matrix [ ]
A is
(A) 4 (B) 8
(C) 15 (D) 16
A2.9-3.1 [GATE-EC-2016-IISc]
108. The matrix
0 3 7
2 5 1 3
0 0 2 4
0 0 0
a
A
b
 
 
 

 
 
 
has det(A) =
100 and trace(A) = 14.
The value of |a − b| is ________.
AD [GATE-EC-2016-IISc]
109. Consider a 2 2
 square matrix
x
A

 
  
 
 
where x is unknown. If the eigenvalues of
the matrix A are (σ + jω) and (σ − jω) , then
x is equal to
(A) j
  (B) j
 
(C)   (D)  
A–6 [GATE-IN-2016-IISc]
110. Consider the matrix
2 1 1
2 3 4
1 1 2
A
 
 
  
 
  
 
whose eigenvalues are 1,−1 and 3. Then
Trace of (A3
− 3A2
) is _______.
AD [GATE-CE-2016-IISc]
111. If the entries in each column of a square
matrix add up to 1, then an eigen value of
is :
(A) 4 (B) 3
(C) 2 (D) 1
A3.0 [GATE-EE-2016-IISc]
112. Consider a 3 × 3 matrix with every element
being equal to 1. Its only non-zero
eigenvalue is ____.
AA [GATE-EE-2016-IISc]
113. Let the eigenvalues of a 2 x 2 matrix A be 1,
–2 with eigenvectors x1 and x2 respectively.
Then the eigenvalues and eigenvectors of the
matrix 2
3 4
A A I
  would, respectively, be
(A) 1 2
2,14; ,
x x
(B) 2 1 2
2,14; ,
x x x

(C) 1 2
2,0; ,
x x
(D) 1 2 1 2
2,0; ,
x x x x
 
AC [GATE-AG-2016-IISc]
114. Eigen values of the matrix
5 3
1 4
 
 
 
are
(A) -6.3 and -2.7 (B) -2.3 and -6.7
(C) 6.3 and 2.7 (D) 2.3 and 6.7
AA,D [GATE-EE-2016-IISc]
115. A 3 × 3 matrix P is such that, P3
= P. Then
the eigenvalues of P are
(A) 1, 1, −1
(B) 1, 0.5 + j0.866, 0.5 − j0.866
(C) 1, −0.5 + j0.866, − 0.5 − j0.866
(D) 0, 1, −1
C [GATE-IN-2014-IITKGP]
116. For the matrix A satisfying the equation
given below, the eigen values are
1 2 3 1 2 3
[ ] 7 8 9 4 5 6
4 5 6 7 8 9
A
   
   

   
   
   
(A) (1 , )
j j
 (B) (1, 1, 0)
(C) (1,1,−1) (D) (1,0,0)
A [GATE-ME-2007-IITK]
117. If a square matrix A is real and symmetric
then the Eigen values
(A) Are always real
(B) Are always real and positive
(C) Are always real and non-negative
(D) Occur in complex conjugate pairs
C [GATE-EC-2010-IITG]
118. The Eigen values of a skew-symmetric
matrix are
(A) Always zero
(B) Always pure imaginary
(C) Either zero (or) pure imaginary
(D) Always real
A [GATE-IN-2001-IITK]
119. The necessary condition to diagonalizable a
matrix is that
(A) Its all Eigen values should be distinct
(B) Its Eigen values should be independent
(C) Its Eigen values should be real
(D) The matrix is non-singular
B [GATE-PI-2007-IITK]
120. If A is square symmetric real valued matrix
of dimension 2n, then the eigen values of A
are
(A) 2n distinct real values
(B) 2n real values not necessarily distinct
(C) n distinct pairs of complex conjugate
numbers
(D) n pairs of complex conjugate numbers,
not necessarily distinct

21. TOPIC 1 - LINEAR ALGEBRA
www.targate.org Page 13
C [GATE-CE-2004-IITD]
121. The eigen values of the matrix
4 2
2 1

 
 

 
are
(A) 1, 4 (B) – 1, 2
(C) 0, 5 (D) None
B [GATE-CS-2005-IITB]
122. What are the Eigen values of the following 2
x 2 matrix?
2 1
4 5

 
 

 
(A) – 1, 1 (B) 1, 6
(C) 2, 5 (D) 4, -1
C [GATE-EE-2009-IITR]
123. The trace and determinant of a 2x2 matrix
are shown to be -2 and -35 respectively. Its
eigen values are
(A) -30, -5 (B) -37, -1
(C) -7, 5 (D) 17.5, -2
A [GATE-CE-2002-IISc]
124. Eigen values of the following matrix are
1 4
4 1

 
 

 
(A) 3, -5 (B) -3, 5
(C) -3, -5 (D) 3, 5
C [GATE-EC-2008-IISc]
125. All the four entries of 2 x 2 matrix
P = 11 12
21 22
p p
p p
 
 
 
are non-zero and one of the
Eigen values is zero. Which of the following
statement is true ?
(A) 11 22 12 21 1
P P P P
 
(B) 11 22 12 21 1
P P P P
 
(C) 11 22 21 12 0
P P P P
 
(D) 11 22 12 21 0
P P P P
 
B [GATE-CE-2008-IISc]
126. The eigen values of the matrix
[P] =
4 5
2 5
 
 

 
are
(A) – 7 and 8 (B) – 6 and 5
(C) 3 and 4 (D) 1 and2
A [GATE-ME-2006-IITKGP]
127. Eigen values of a matrix S =
3 2
2 3
 
 
 
are 5
and 1. What are the Eigen values of the
matrix S2
= SS?
(A) 1 and 25 (B) 6, 4
(C) 5, 1 (D) 2, 10
A [GATE-EC-2013-IITB]
128. The minimum eigenvalue of the following
matrix is
3 5 2
5 12 7
2 7 5
 
 
 
 
 
(A) 0 (B) 1
(C) 2 (D) 3
B [GATE-CE-2007-IITK]
129. The minimum and maximum Eigen values
of Matrix
1 1 3
1 5 1
3 1 1
 
 
 
 
 
are –2 and 6
respectively. What is the other Eigen value?
(A) 5 (B) 3
(C) 1 (D) -1
A [GATE-EE-1998-IITD]
130. A =
2 0 0 1
0 1 0 0
0 0 3 0
1 0 0 4

 
 
 
 
 

 
the sum of the Eigen
Values of the matrix A is :
(A) 10 (B) – 10
(C) 24 (D) 22
C [GATE-PI-2005-IITB]
131. The Eigen values of the matrix M given
below are 15, 3 and 0. M =
8 6 2
6 7 4
2 4 3

 
 
 
 
 

 
, the
value of the determinant of a matrix is
(A) 20 (B) 10
(C) 0 (D) – 10
C [GATE-ME-2008-IISc]
132. The matrix
1 2 4
3 0 6
1 1 p
 
 
 
 
 
has one eigen value to
3. The sum of the other two eigen values is
(A) p (B) p – 1
(C) p – 2 (D) p – 3
A [GATE-IN-2010-IITG]
133. A real nxn matrix A = ij
a
 
  is defined as
follows
,
0,
ij
a i i j
otherwise
  






The sum of all n eigen values of A is :

22. ENGINEERING MATHEMATICS
Page 14 TARGATE EDUCATION GATE-(EE/EC)
(A)
( 1)
2
n n 
(B)
( 1)
2
n n 
(C)
( 1)(2 1)
2
n n n
 
(D) 2
n
A17 [GATE-EC-2015-IITK]
134. The value of p such that the vector
1
2
3
 
 
 
 
 
is an
eigenvector of the matrix
4 1 2
P 2 1
14 4 10
 
 
 
 

 
is _.
AB [GATE-EE-2015-IITK]
135. The maximum value of ‘a’ such that the
matrix
3 0 2
1 1 0
0 2
 
 
 

 
 

 
a
has three linearly
independent real eigenvectors is
(A)
2
3 3
(B)
1
3 3
(C)
1 2 3
3 3

(D)
1 3
3 3

A2 [GATE-ME-2015-IITK]
136. The lowest eigen value of the 2 2
 matrix
4 2
1 3
 
 
 
is ______
AD [GATE-CH-2012-IITD]
137. Consider the following  
2 2
 matrix
4 0
0 4
 
 
 
Which one of the following vectors is NOT a
valid eigen vectors of the above matrix?
(A)
1
0
 
 
 
(B)
2
1

 
 
 
(C)
4
3
 
 

 
(D)
0
0
 
 
 
AD [GATE-EC-2009-IITR]
138. The eigen values of the following matrix are:
1 3 5
3 1 6
0 0 3

 
 
 
 
 
 
(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
AA [GATE-EC-2006-IITKGP]
139. The eigen values and the corresponding
eigen vectors of a 2 2
 matrix are given by
Eigen value Eigen vector
1 8
  1
1
v
1
 
  
 
2 4
  2
1
v
1
 
  

 
The matrix is :
(A)
6 2
2 6
 
 
 
(B)
4 6
6 4
 
 
 
(C)
2 4
4 2
 
 
 
(D)
4 8
8 4
 
 
 
AC [GATE-EC-2006-IITKGP]
140. For the matrix
4 2
2 4
 
 
 
the eigen value
corresponding to the eigen vector
101
101
 
 
 
is:
(A) 2 (B) 4
(C) 6 (D) 8
AC [GATE-EC-2005-IITB]
141. Given matrix
4 2
4 3

 
 
 
the eigen vector is :
(A)
3
2
 
 
 
(B)
4
3
 
 
 
(C)
2
1
 
 

 
(D)
1
2

 
 
 
AB [GATE-EC-2000-IITKGP]
142. The eigen value of the matrix
2 1 0 0
0 3 0 0
0 0 2 0
0 0 1 4

 
 
 
 

 

 
are
(A) 2, -2, 1, -1 (B) 2, 3, -2, 4
(C) 2, 3, 1, 4 (D) None of these
AD [GATE-EC-1998-IITD]
143. The eigen value of the matrix
0 1
A
1 0
 
  
 
are
(A) 1, 1 (B) -1, -1
(C) j, -j (D) 1, -1
A1/3 [GATE-EE-2014-IITKGP]
144. A system matrix is given as follows
0 1 1
A 6 11 6
6 11 5

 
 
  
 
 
 
 
The absolute value of the ratio of the
maximum eigen value to the minimum eigen
value is _______.

23. TOPIC 1 - LINEAR ALGEBRA
www.targate.org Page 15
AA [GATE-EE-2014-IITKGP]
145. Which one of the following statements is
true for all real symmetric matrices?
(A) All the eigen values are real.
(B) All the eigen values are positive.
(C) All the eigen values are distinct.
(D) Sum of all the eigen values is zero.
AC [GATE-BT-2014-IITKGP]
146. The eigen values of
1 4
A
2 3

 
  

 
are:
(A) 2 i
 (B) -1 , -2
(C) 1 2i
  (D) Non- existent
AD [GATE-EE-2013-IITB]
147. A matrix has eigen values -1 and -2. The
corresponding eigen vectors are
1
1
 
 

 
and
1
2
 
 

 
respectively. The matrix is :
(A)
1 1
1 2
 
 
 
 
(B)
1 2
2 4
 
 
 
 
(C)
1 0
0 2

 
 

 
(D)
0 1
2 3
 
 
 
 
AB [GATE-EE-2008-IISc]
148. Let P be a 2 2
 real orthogonal matrix and
x

is a real vector  
T
1 2
x x with length
 
1
2 2 2
1 2
x x x
 

. Then which one of the
following statements is correct?
(A) Px x

 
where at least one vector
satisfies Px x

 
(B) Px x

 
for all vectors x

(C) Px x

 
where at least one vector
satisfies Px x

 
(D) No relationship can be established
between x

and Px

AA [GATE-EE-2007-IITK]
149. The linear operation L(x) is defined by the
cross product L(x)= b x,
 where b =  
T
010
and  
T
1 2 3
x x x are three dimensional
vectors. The 3 3
 matrix M of this operation
satisfies
 
1
2
3
x
L x M x
x
 
 
  
 
 
Then the eigen values of M are
(A) 0, +1, -1 (B) 1, -1, 1
(C) i, -i, 1 (D) i, -i, 0
AD [GATE-EE-2002-IISc]
150. The eigen values of the system represented
by
0 1 0 0
0 0 1 0
X
0 0 0 1
0 0 0 1
 
 
 

 
 
 
are
(A) 0, 0, 0, 0 (B) 1, 1, 1, 1
(C) 0, 0, 0, 1 (D) 1, 0, 0, 0
AC [GATE-EE-1998-IITD]
151. The vector
1
2
1
 
 
 
 

 
is an eigen vector of
2 2 3
A 2 1 6
1 2 0
 
 
 
 
 
 
 
 
one of the eigen values
of A is
(A) 1 (B) 2
(C) 5 (D) -1
A(-3,-2,-1) [GATE-EE-1995-IITK]
152. Given the matrix
0 1 0
A 0 0 1
6 11 6
 
 
  
 
 
 
. Its
eigen values are ___________.
AA [GATE-EE-1994-IITKGP]
153. The eigen values of the matrix
a 1
a 1
 
 
 
are
(A) (a+1), 0 (B) a, 0
(C) (a-1), 0 (D) 0, 0
AA [GATE-ME-2014-IITKGP]
154. One of the eigen vector of the matrix
5 2
9 6

 
 

 
is :
(A)
1
1

 
 
 
(B)
2
9

 
 
 
(C)
2
1
 
 

 
(D)
1
1
 
 
 

24. ENGINEERING MATHEMATICS
Page 16 TARGATE EDUCATION GATE-(EE/EC)
AD [GATE-ME-2014-IITKGP]
155. Consider a 3 3
 real symmetric S such that
two of its eigen values are a 0,b 0
  with
respective eigen vectors
1
2
3
x
x
x
 
 
 
 
 
,
1
2
3
y
y
y
 
 
 
 
 
. If
a b
 then 1 1 2 2 3 3
x y x y x y
  equals
(A) a (B) b
(C) ab (D) 0
AC [GATE-ME-2013-IITB]
156. The eigen values 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
AB [GATE-ME-2012-IITD]
157. For the matrix
5 3
A
1 3
 
  
 
, ONE of the
normalized eigen vectors is given as
(A)
1
2
3
2
 
 
 
 
 
 
(B)
1
2
1
2
 
 
 
 

 
 
(C)
3
10
1
10
 
 
 
 

 
 
(D)
1
5
2
5
 
 
 
 
 
 
AC [GATE-ME-2011-IITM]
158. Eigen values of real symmetric are always
(A) Positive (B) Negative
(C) Real (D) Complex
AA [GATE-ME-2010-IITG]
159. One of the eigen vectors of the matrix
2 2
A
1 3
 
  
 
is :
(A)
2
1
 
 

 
(B)
2
1
 
 
 
(C)
4
1
 
 
 
(D)
1
1
 
 

 
AB [GATE-ME-2008-IISc]
160. The eigen vector of the matrix
1 2
0 2
 
 
 
are
written in the form
1
a
 
 
 
and
1
b
 
 
 
. What is
a+b?
(A) 0 (B)
1
2
(C) 1 (D) 2
AB [GATE-ME-2004-IITD]
161. The sum of the eigen values of the given
matrix is :
1 1 3
1 5 1
3 1 1
 
 
 
 
 
(A) 5 (B) 7
(C) 9 (D) 18
AC [GATE-ME-2003-IITM]
162. For matrix
4 1
1 4
 
 
 
the eigen values are
(A) 3 and -3 (B) -3 and -5
(C) 3 and 5 (D) 5 and 0
AC [GATE-ME-1996-IISc]
163. The eigen values of
1 1 1
1 1 1
1 1 1
 
 
 
 
 
are:
(A) 0, 0, 0 (B) 0, 0, 1
(C) 0, 0, 3 (D) 1, 1, 1
AA [GATE-CE-2014-IITKGP]
164. The sum of eigen value of the matrix, [M] is
where  
215 650 795
M 655 150 835
485 355 550
 
 
  
 
 
(A) 915 (B) 1355
(C) 1640 (D) 2180
AA [GATE-CE-2014-IITKGP]
165. Which one of the following statements is
TRUE about every n n
 matrix with only
real Eigen values?
(A) If the trace of the matrix is positive and
the determinant of the matrix is
negative, at least one of its eigen values
is negative.
(B) If the trace of the matrix is positive, all
its eigen values is positive.
(C) If the determinant of the matrix is
positive, all its eigen values is positive.
(D) If the product of the trace and
determinant of the matrix is positive, all
its eigen values are positive.
AB [GATE-CE-2012-IITD]
166. The eigen value of the matrix
9 5
5 8
 
 
 
are:
(A) -2.42 and 6.86 (B) 3.48 and 13.53
(C) 4.70 and 6.86 (D) 6.86 and 9.50

25. TOPIC 1 - LINEAR ALGEBRA
www.targate.org Page 17
AB [GATE-CE-2007-IITK]
167. The minimum and maximum eigen values of
the matrix
1 1 3
1 5 1
3 1 1
 
 
 
 
 
are -2 and -6 and
respectively. What is the other eigen value?
(A) 5 (B) 3
(C) 1 (D) -1
AB [GATE-CE-2006-IITKGP]
168. For given matrix
2 2 3
A 2 1 6
1 2 0

 
 
  
 
 
 
, one of
the eigen values is 3. The other two eigen
values are
(A) 2, -5 (B) 3, -5
(C) 2, 5 (D) 3, 5
AD [GATE-CE-2001-IITK]
169. The eigen values of the matrix
5 3
2 9
 
 
 
are:
(A) (5.13, 9.42) (B) (3.85,2.93)
(C) (9.00, 5.00) (D) (10.16, 3.84)
AD [GATE-IN-2013-IITB]
170. One pairs of eigen vectors corresponding to
the two eigen values of the matrix
0 1
1 0

 
 
 
is :
(A)
1
,
j
 
 

 
j
1
 
 

 
(B)
0
1
 
 
 
1
0

 
 
 
(C)
1
,
j
 
 
 
0
1
 
 
 
(D)
1
,
j
 
 
 
j
1
 
 
 
AB [GATE-IN-2011-IITM]
171. Given that
2 2 3
A 2 1 6
1 2 0
 
 
 
 
 
 
 
 
has eigen
values -3, -3, 5. An eigen vector
corresponding to the eigen values 5 is
 
T
1 2 1
 . One of the eigen vectors of the
matrix
3
M is :
(A)  
T
1 8 1
 (B)  
T
1 2 1

(C)
T
3
1 2 1
 

 
(D)  
T
1 1 1

AA [GATE-IN-2010-IITG]
172. A real matrix n n
 matrix ij
A a
 
   is
defined as follows :
ij
a i
 ; if i j

=0; otherwise
The summation of all eigen values of A is :
(A)
 
n 1
n
2

(B)
 
n 1
n
2

(C)
  
n 1 2n 1
n
6
 
(D)
2
n
AA [GATE-PI-1994-IITKGP]
173. For the following matrix
1 1
2 3

 
 
 
the
number of positive roots is/are
(A) One (B) Two
(C) Four (D) can’t be found
AB [GATE-PI-2011-IITM]
174. The eigen values of the following matrix are
10 4
18 12

 
 

 
(A) 4, 9 (B) 6, -8
(D) 4, 8 (D) -6, 8
AOrthogonal [GATE-CS-2014-IITKGP]
175. The value of the dot product of the eigen
vectors corresponding to any pair of
different eigen values of a 4 4
 symmetric
definite positive matrix is___________.
AD [GATE-CS-2012-IITD]
176. Let A be the 2 2
 matrix with elements
11 12 21
a a a 1
    and 22
a 1
  . Then the
eigen value of the matrix
19
A are
(A) 1024 and -1024
(B) 1024 2 and 1024 2

(C) 4 2 and 4 2

(D) 512 2 and 512 2

AA [GATE-CS-2011-IITM]
177. Consider the matrix given below:
1 2 3
0 4 7
0 0 3
 
 
 
 
 
Which one of the following options provides
the CORRECT values of the eigen values of
the matrix?
(A) 1, 4, 3 (B) 3, 7, 3
(C) 7, 3, 2 (D) 1, 2, 3
AD [GATE-CS-2001-IITK]
178. Consider the following matrix
2 3
A
x y
 
  
 
.
If the eigen values of A are 4 and 8, then
(A) x = 4, y = 10
(B) x =5, y = 8

26. ENGINEERING MATHEMATICS
Page 18 TARGATE EDUCATION GATE-(EE/EC)
(C) x = -3, y = 9
(D) x = -4, y = 10
AA [GATE-CS-2008-IISc]
179. How many of the following matrices have an
eigen value 1?
1 1 0 1 1 1
, ,
0 0 0 0 1 1

     
     
     
and
1 0
1 1

 
 

 
(A) One (B) Two
(C) Three (D) Four
AC [GATE-CS-2007-IITK]
180. Let A be a 4 4
 matrix with eigen values -
5, -2, 1, 4. Which of the following is an
eigen value of
A I
I A
 
 
 
where I is the 4 4

identity matrix?
(A) -5 (B) -7
(C) 2 (D) 1
AA [GATE-CS-2003-IITM]
181. Obtain the eigen values of the matrix
1 2 34 49
0 2 43 94
A
0 0 2 104
0 0 0 1
 
 
 

 

 

 
(A) 1, 2, -2, -1 (B) -1, -2, -1, -2
(C) 1,2, 2, 1 (D) None
AA [GATE-CS-2013-IITB]
182. Let A be the matrix
3 1
1 2
 
 
 
. What is the
maximum value of T
X AX where the
maximum is taken over all x that are unit
eigen vectors of A?
(A) 5 (B)
5 5
2

(C) 3 (D)
5 5
2

AA [GATE-CS-2006-IITKGP]
183. What are the eigen values of the matrix P
given below ?
a 1 0
1 a 1
0 1 a
 
 
 
 
 
(A) a, a 2
 , a 2
 (B) a, a, a
(C) 0, a, 2a (D) -a, 2a, 2a
AC [GATE-BT-2013-IITB]
184. One of the eigen values of
10 4
P
18 12

 
  

 
is
(A) 2 (B) 4
(C) 6 (D) 8
AC [GATE-EC-2017-IITR]
185. Consider the 5 5
 matrix
1 2 3 4 5
5 1 2 3 4
A 4 5 1 2 3
3 4 5 1 2
2 3 4 5 1
 
 
 
 

 
 
 
 
It is given that A has only one real eigen
value. Then the real eigenvalue of A is :
(A) – 2.5 (B) 0
(C) 15 (D) 25
AC [GATE-EE-2017-IITR]
186. The matrix
3 1
0
2 2
A 0 1 0
1 3
0
2 2
 
 
 
 
 
 
 
 
has three
distinct eigenvalues and one of its
eigenvectors is
1
0
1
 
 
 
 
 
. Which one of the
following can be another eigenvector of A?
(A)
0
0
1
 
 
 
 

 
(B)
1
0
0

 
 
 
 
 
(C)
1
0
1
 
 
 
 

 
(D)
1
1
1
 
 

 
 
 
AA [GATE-EE-2017-IITR]
187. The eigenvalues of the matrix given below
are
0 1 0
0 0 1
0 3 4
 
 
 
 
 
 
(A) (0, –1, –3) (B) (0, –2, –3)
(C) (0, 2, 3) (D) (0, 1, 3)
AC [GATE-AG-2017-IITR]
188. Characteristic equation of the matrix
2 2
2 1
 
 
 
 
with Eigen value  is :

27. TOPIC 1 - LINEAR ALGEBRA
www.targate.org Page 19
(A) 2
3 4 0
     (B) 2
3 2 0
    
(C) 2
3 0
    (D) 2
3 0
   
AA [GATE-CE-2017-IITR]
189. Consider the matrix
5 1
4 1

 
 
 
. Which one of
the following statements is TRUE for the
eigenvalues and eigenvectors of this matrix?
(A) Eigenvalue 3 has a multiplicity of 2 and
only one independent eigenvector exists
(B) Eigenvalue 3 has a multiplicity of 2 and
two independent eigenvectors exist
(C) Eigenvalue 3 has a multiplicity of 2 and
no independent eigenvector exists
(D) Eigenvalues are 3 and -3 and two
independent eigenvectors exist.
AA [GATE-CE-2017-IITR]
190. Consider the following simultaneous
equations (with 1 2
c and c being constants):
1 2 1
3x 2x c
 
1 2 2
4x x c
 
The characteristic equation for these
simultaneous equations is
(A) 2
4 5 0
    
(B) 2
4 5 0
    
(C) 2
4 5 0
    
(D) 2
4 5 0
    
A5 [GATE-CS-2017-IITR]
191. If the characteristic polynomial of a
3 3
 matrix M over (the set of real
numbers) is 3 2
4 a 30,a
       , and
one eigenvalue of M is 2, then the largest
among the absolute values of the
eigenvalues of M is ________.
AD [GATE-GG-2017-IITR]
192. Which one of the following sets of
vectors 
1 2 3
v ,v ,v is linearly dependent?
(A) 1 2 3
(0, 1,3), (2,0,1),
v v v
   
( 2, 1,3)
 
(B) 1 2 3
(2, 2,0), (0,1, 1),
v v v
    
(0,4,2)
(C) 1 2 3
(2,6,2), (2,0, 2),
v v v
    (0,4,2)
(D) 1 2 3
(1,4,7), (2,5,8),
v v v
   (3,6,9)
AC [GATE-IN-2017-IITR]
193. The eigen values of the matrix
1 1 5
A 0 5 6
0 6 5

 
 
  
 

 
are
(A) -1, 5 , 6 (B) 1, 5 j6
 
(C) 1, 5 j6
 (D) 1, 5, 5
AB [GATE-ME-2017-IITR]
194. The product of eigenvalues of the matrix P is
2 0 1
4 3 3
0 2 1
P
 
 
 
 
 

 
(A) –6 (B) 2
(C) 6 (D) –2
A5 [GATE-ME-2017-IITR]
195. The determinant of a 2 2
 matrix is 50. If
one eigenvalue of the matrix is 10, the other
eigenvalue is ________.
A0 [GATE-ME-2017-IITR]
196. Consider the matrix
50 70
70 80
A
 
  
 
whose
eigenvectors corresponding to eigenvalues
1
 and 2
 are 1
1
70
50
x
 
  
 
 
and
2
2
80
70
x
 
 
  
 
, respectively. The value of
1 2
T
x x is _________ .
A17 [GATE-TF-2018-IITG]
197. If
3 1
1 3
A
 
  
 
, then the sum of all
eigenvalues of the matrix 2 1
4
M A A
  is
equal to ________.
AA [GATE-PH-2018-IITG]
198. The eigenvalues of a Hermitian matrix are
all
(A) real
(B) imaginary
(C) of modulus one
(D) real and positive
A24.5 to 25.5 [GATE-PI-2018-IITG]
199. The diagonal elements of a 3-by-3 matrix are
–10, 5 and 0, respectively. If two of its
eigenvalues are –15 each, the third
eigenvalue is ______.
AA [GATE-IN-2018-IITG]
200. Let N be a 3 by 3 matrix with real number
entries. The matrix N is such that 2
0
N  .
The eigen values of N are
(A) 0, 0, 0 (B) 0,0,1
(C) 0,1,1 (D) 1,1,1

28. ENGINEERING MATHEMATICS
Page 20 TARGATE EDUCATION GATE-(EE/EC)
AD [GATE-CE-2018-IITG]
201. The matrix
2 4
4 2

 
 

 
has
(A) real eigenvalues and eigenvectors
(B) real eigenvalues but complex
eigenvectors
(C) complex eigenvalues but real
eigenvectors
(D) complex eigenvalues and eigenvectors
AC [GATE-EC-2018-IITG]
202. Let M be a real 4 4
 matrix. Consider the
following statements:
S1: M has 4 linearly independent
eigenvectors.
S2: M has 4 distinct eigenvalues.
S3: M is non-singular (invertible).
Which one among the following is TRUE?
(A) S1 implies S2 (B) S1 implies S3
(C) S2 implies S1 (D) S3 implies S2
AD [GATE-CS-2018-IITG]
203. Consider a matrix P whose only
eigenvectors are the multiples of
1
4
 
 
 
.
Consider the following statements.
(I) P does not have an inverse
(II) P has a repeated eigenvalue
(III) P cannot be diagonalized
Which one of the following options is
correct?
(A) Only I and III are necessarily true
(B) Only II is necessarily true
(C) Only I and II are necessarily true
(D) Only II and III are necessarily true
A3 [GATE-CS-2018-IITG]
204. Consider a matrix T
A uv
 where
1
2
u
 
  
 
,
1
1
v
 
  
 
. Note that T
v denotes the transpose
of v. The largest eigenvalue of A is ____.
AA T1.4 [GATE-MT-2019-IITM]
205. One of the eigenvalues for the following
matrix is _______.
2
8
a
a
 
 
 
(A) 4
a  (B) 4
a
 
(C) 4 (D) 4

A2 T1.4 [GATE-AE-2019-IITM]
206. One of the eigenvalues of the following
matrix is 1.
2
1 3
x
 
 

 
The other eigenvalue is _____.
A12 T1.4 [GATE-CS-2019-IITM]
207. Consider the following matrix :
1 2 4 8
1 3 9 27
1 4 16 64
1 5 25 125
R
 
 
 

 
 
 
The absolute value of the product of Eigen
values of R is ______.
AD T1.4 [GATE-CE-2019-IITM]
208. Euclidean norm (length) of the vector
[4 2 6]T
  is :
(A) 12 (B) 24
(C) 48 (D) 56
AB T1.4 [GATE-XE-2019-IITM]
209. If
3 2 4
2 0 2
4 2 3
Q
 
 
  
 
 
and 1 2 3
( )
P v v v
 is
the matrix 1 2
,
v v and 3
v are linearly
independent eigenvectors of the matrix Q,
then the sum of the absolute values of all the
elements of the matrix
1
P QP

is
(A) 6 (B) 10
(C) 14 (D) 22
AB T1.4 [GATE-ME-2019-IITM]
210. Consider the matrix
1 1 0
0 1 1
0 0 1
P
 
 
  
 
 
The number of distinct eigenvalues of P is
(A) 0 (B) 1
(C) 2 (D) 3
AA T1.4 [GATE-TF-2019-IITM]
211. The eigenvalues of the matrix
3 0 0
0 2 3
0 1 2
 
 

 
 

 
are

29. TOPIC 1 - LINEAR ALGEBRA
www.targate.org Page 21
(A) –1,1,3 (B) –3,2,–2
(C) 3,2,–1 (D) 3,2,1
AA T1.4 [GATE-PE-2019-IITM]
212. Let 1
 and 2
 be the two eigenvalues of the
matrix
0 1
1 1
A

 
  
 
. Then, 1 2
   and
1 2
  , are respectively
(A) 1 and 1 (B) 1 and –1
(C) -1 and 1 (D) –1 and –1
A10 T1.4 [GATE-IN-2019-IITM]
213. A 33 matrix has eigen values 1, 2 and 5.
The determinant of the matrix is _______ .
AD T1.4 [GATE-EE-2019-IITM]
214. M is a 2  2 matrix with eigenvalues 4 and
9. The eigenvalues of 2
M are
(A) 4 and 9 (B) 2 and 3
(C) −2 and −3 (D) 16 and 81
AC T1.4 [GATE-EE-2019-IITM]
215. Consider a 2  2 matrix 1 2
[ ]
M v v
 ,
where, 1
v and 2
v are the column vectors.
Suppose 1 1
2
T
T
u
M
u
  
  
 
, where 1
T
u and 2
T
u are
the row vectors, Consider the following
statements:
Statement 1: 1 1 1
T
u v  and 2 2 1
T
u v 
Statement 2: 1 2 0
T
u v  and 2 1 0
T
u v 
Which of the following options is correct?
(A) Statement 2 is true and statement 1 is
false
(B) Both the statements are false
(C) Statement 1 is true and statement 2 is
false
(D) Both the statements are true
A3 T1.4 [GATE-EC-2019-IITM]
216. The number of distinct eigenvalues of the
matrix
2 2 3 3
0 1 1 1
0 0 3 3
0 0 0 2
A
 
 
 

 
 
 
is equal to ___.
**********
Rank
C [GATE-EE-2014-IITKGP]
217. Two matrices A and B are given below:
p q
A
r s
 
  
 
;
2 2
2 2
p q pr qs
B
pr qs r s
 
 
 
 
 
If the rank of matrix A is N, then the rank of
matrix B is :
(A) N /2 (B) N-1
(C) N (D) 2 N
A [GATE-PI-1994-IITKGP]
218. If for a matrix, rank equals both the number
of rows and number of columns, then the
matrix is called
(A) Non-singular (B) singular
(C) Transpose (D) Minor
A [GATE-EE-2007-IITK]
219. 1 2 3
, , ,........ m
q q q q are n-dimensional vectors
with m < n. This set of vectors is linearly
dependent. Q is the matrix with
1 2 3
, , ,....... m
q q q q as the columns. The rank of
Q is
(A) Less than m (B) m
(C) Between m and n (D) n
A [GATE-EC-1994-IITKGP]
220. The rank of (m x n) matrix (m < n) cannot be
more than
(A) m (B) n
(C) mn (D) None
B [GATE-CE-2000-IITKGP]
221. Consider the following two statements.
(I) The maximum number of linearly
independent column vectors of a matrix
A is called the rank of A.
(II) If A is n n
 square matrix then it will
be non-singular if rank of A = n
(A) Both the statements are false
(B) Both the statements are true
(C) (I) is true but (II) is false
(D) (I) is false but (II) is true
AB [GATE-EE-2016-IISc]
222. Let A be a 4 × 3 real matrix with rank 2.
Which one of the following statement is
TRUE?
(A) Rank of
T
A A is less than 2.
(B) Rank of
T
A A is equal to 2.
(C) Rank of
T
A A is greater than 2.
(D) Rank of
T
A A can be any number
between 1 and 3.
C [GATE-CS-2002-IISc]
223. The rank of the matrix
1 1
0 0
 
 
 
is
(A) 4 (B) 2
(C) 1 (D) 0

30. ENGINEERING MATHEMATICS
Page 22 TARGATE EDUCATION GATE-(EE/EC)
C [GATE-CS-1994-IITKGP]
224. The rank of matrix
0 0 3
9 3 5
3 1 1

 
 
 
 
 
is :
(A) 0 (B) 1
(C) 2 (D) 3
A [GATE-EE-1995-IITK]
225. The rank of the following (n+1) x (n+1)
matrix, where ‘a’ is a real number is :
2
2
2
1 . . .
1 . . .
.
.
1 . . .
n
n
n
a a a
a a a
a a a
 
 
 
 
 
 
 
 
(A) 1
(B) 2
(C) n
(D) depends on the value of a
AC [GATE-IN-2015-IITK]
226. Let A be an n n
 matrix with rank r (0 < r <
n). Then AX = 0 has p independent
solutions, where p is
(A) r (B) n
(C) n – r (D) n + r
AA [GATE-EE-2008-IISc]
227. If the rank of a  
5 6
 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) T
QQ will be invertible.
(D) T
Q Q will be invertible
AB [GATE-EE-2007-IITK]
228.  
T
1 2 n
X x ,x ........x
 is an n-tuple non-zero
vector. The n n
 matrix T
V X.X
 .
(A) Has rank zero (B) Has rank 1
(C) Is orthogonal (D) Has rank n
AC [GATE-EE-1994-IITKGP]
229. A 5 7
 matrix has all its entries equal to -1.
The rank of the matrix is
(A) 7 (B) 5
(C) 1 (D) 0
AB [GATE-ME-1994-IITKGP]
230. Rank of the matrix
0 2 2
7 4 8
7 0 4
 
 
 
 
 
 
is 3.
(A) True (B) False
A2 [GATE-CE-2014-IITKGP]
231. The rank of the matrix
6 0 4 4
8 18
2 14
0 10
14 14
 
 

 
 


 
is__________.
AB [GATE-IN-2013-IITB]
232. The dimension of the null space of the
matrix
0 1 1
1 1 0
1 0 1
 
 

 
 
 
 
is
(A) 0 (B) 1
(C) 2 (D) 3
AD [GATE-IN-2009-IITR]
233. 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
range space P is:
(A) 0 (B) 1
(C) 2 (D) 3
AB [GATE-IN-2007-IITK]
234. Let ij
A a
 
   , 1 i, j n
  , with n 3
 and
ij
a i.j
 . Then the rank of A is
(A) 0 (B) 1
(C) n-1 (D) n
AC [GATE-IN-2000-IITKGP]
235. The rank of matrix
1 2 3
A 3 4 5
4 6 8
 
 
  
 
 
is
(A) 0 (B) 1
(C) 2 (D) 3
AC [GATE-CS-1994-IITKGP]
236. The rank of matrix
0 0 3
9 3 5
3 1 1

 
 
 
 
 
(A) 0 (B) 1
(C) 2 (D) 3
AC [GATE-BT-2012-IITD]
237. What is the rank of the following matrix?

31. TOPIC 1 - LINEAR ALGEBRA
www.targate.org Page 23
5 3 1
6 2 4
14 10 0

 
 

 
 
 
(A) 0 (B) 1
(C) 2 (D) 3
AC [GATE-EC-2017-IITR]
238. The rank of the matrix
5 10 10
M 1 0 2
3 6 6
 
 
  
 
 
is
(A) 0 (B) 1
(C) 2 (D) 3
A4 [GATE-EC-2017-IITR]
239. The rank of the matrix
1 1 0 0 0
0 0 1 1 0
0 1 1 0 0
1 0 0 0 1
0 0 0 1 1

 
 

 
 

 

 
 

 
is ______ .
A2 [GATE-CS-2017-IITR]
240. Let
1 1 1
P 2 3 4
3 2 3

 
 
 
 
 

 
and
1 2 1
Q 6 12 6
5 10 5
  
 
 
  
 
 
be two matrices.
Then the rank of P+Q is __________.
A1 [GATE-IN-2017-IITR]
241. If v is a non-zero vector of dimension 3 1,

then the matrix A = vvT
has rank = _______.
AA [GATE-MN-2017-IITR]
242. If the rank of the following matrix is less
than 3, the values of x are
1
1
1
x x
A x x
x x
 
 
  
 
 
(A) 1, –1/2 (B) 1, 1/2
(C) 2, –1/4 (D) 2, –3/4
AB [GATE-GG-2018-IITG]
243. The maximum number of linearly
independent rows of an m n
 matrix G
where m > n is
(A) m. (B) n.
(C) m – n. (D) 0.
AB [GATE-ME-2018-IITG]
244. The rank of the matrix
4 1 1
1 1 1
7 3 1
 
 
 
  
 
 

 
is
(A) 1 (B) 2
(C) 3 (D) 4
AB [GATE-CE-2018-IITG]
245. The rank of the following matrix is
1 1 0 2
2 0 2 2
4 1 3 1

 
 
 
 
 
(A) 1 (B) 2
(C) 3 (D) 4
A–AC [GATE-AG-2018-IITG]
246. Rank of a matrix
5 3 3 1
3 2 2 1
2 1 2 8
 
 
 
 
 
 

 
is
(A) 1 (B) 2
(C) 3 (D) 4
A3 T1.5 [GATE-EE-2019-IITM]
247. The rank of the matrix,
0 1 1
1 0 1
1 1 0
 
 
  
 
 
M , is
**********
Homogenous & Linear Eqn.
B [GATE-EE-2014-IITKGP]
248. Given a system of equations:
1
2
2 2
5 3
x y z b
x y z b
  
  
Which of the following is true regarding its
solutions?
(A) The system has a uniqne solution for
any given b1 and b2
(B) The system will have infinitely many
solutions for any given b1 and b2
(C) Whether or not a solution exists
depends on the given b1 and b2
(D) The system would have no solution for
any values of b1 and b2
D [GATE-EE-2013-IITB]
249. The equation 1
2
2 2 0
1 1 0
x
x
  
   

 
   

   
 
has
(A) No solution
(B) Only one solution 1
2
0
0
x
x
   

   
 
 

32. ENGINEERING MATHEMATICS
Page 24 TARGATE EDUCATION GATE-(EE/EC)
(C) Non-zero unique solution
(D) Multiple solutions
AC[GATE-ME-2011-IITM]
250. Consider the following system of equations
1 2 3 2 3
2 0, 0
x x x x x
     and 1 2 0
x x
  .
This system has
(A) A unique solution
(B) No solution
(C) Infinite number of solution
(D) Five solutions
B [GATE-CS-1996-IISc]
251. Let AX = B be a system of linear equations
where A is an m n matrix B is an 1
m
column matrix which of the following is
false?
(A) The system has a solution, if
( ) ( / )
ρ A ρ A B

(B) If m = n and B is a non – zero vector
then the system has a unique solution
(C) If m < n and B is a zero vector then the
system has infinitely many solutions.
(D) The system will have a trivial solution
when m = n , B is the zero vector and
rank of A is n.
B [GATE-EE-1998-IITD]
252. A set of linear equations is represented by
the matrix equations Ax = b. The necessary
condition for the existence of a solution for
this system is :
(A) must be invertible
(B) b must be linearly dependent on the
columns of A
(C) b must be linearly independent on the
columns of A
(D) None
B [GATE-IN-2007-IITK]
253. Let A be an n x n real matrix such that A2
= I
and Y be an n-dimensional vector. Then the
linear system of equations Ax = Y has
(A) No solution
(B) unique solution
(C) More than one but infinitely many
dependent solutions.
(D) Infinitely many dependent solutions
B [GATE-ME-2005-IITB]
254. A is a 3 4
 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
B [GATE-EC-2014-IITKGP]
255. Thesystem of linear equation
2 1 3 5
3 0 1 4
1 2 5 14
a
b
c
    
    
 
    
    
    
has
(A) A unique solution
(B) Infinitely many solutions
(C) No solution
(D) Exactly two solutions
D [GATE-IN-2006-IITKGP]
256. A system of linear simultaneous equations is
given as AX = b
Where A =
1 0 1 0
0 1 0 1
1 1 0 0
0 0 0 1
 
 
 
 
 
 
& b =
0
0
0
1
 
 
 
 
 
 
Then the rank of matrix A is
(A) 1 (B) 2
(C) 3 (D) 4
B
257. A system of linear simultaneous equations is
given as Ax b

Where A =
1 0 1 0
0 1 0 1
1 1 0 0
0 0 0 1
 
 
 
 
 
 
& b =
0
0
0
1
 
 
 
 
 
 
Which of the following statement is true?
(A) x is a null vector
(B) x is unique
(C) x does not exist
(D) x has infinitely many values
AA [GATE-EC-1994-IITKGP]
258. Solve the following system
1 2 3 3
x x x
  
1 2 0
x x
 
1 2 3 1
x x x
  
(A) Unique solution
(B) No solution
(C) Infinite number of solutions
(D) Only one solution
C [GATE-ME-1996-IISc]
259. In the Gauss – elimination for a solving
system of linear algebraic equations,
triangularization leads to

33. TOPIC 1 - LINEAR ALGEBRA
www.targate.org Page 25
(A) diagonal matrix
(B) lower triangular matrix
(C) upper triangular matrix
(D) singular matrix
AD [GATE-ME-2016-IISc]
260. The solution to the system of equations
2 5 2
4 3 30
x
y
     

   
 
 
     
is :
(A) 6, 2 (B) -6, 2
(C) -6, -2 (D) 6, -2
A14.9-15.1 [GATE-CH-2016-IISc]
261. A set of simultaneous linear algebraic
equations is represented in a matrix form as
shown below.
1
2
3
4
5
0 0 0 13 46
4
5 5 10 161
2 2
0 0 5 3 61
2
0 0 0 5 30
4
3 5 81
2 2 1
x
x
x
x
x
 
   
 
   
 
   
 
   

 
   
 
   
 
   
   
 
The value (rounded off to the nearest
integer) of 3
x is _________.
A1.00 [GATE-MN-2016-IISc]
262. The value of x in the simultaneous equations
is _______
3 2 3
x y z
  
2 3 3
x y z
   
2 4
x y z
  
AB [GATE-CE-2016-IISc]
263. Consider the following linear system
2 3
x y z a
  
2 3 3
x y z b
  
5 9 6
x y z c
  
This system is consistent if a, b and c satisfy
the equation
(A) 7a – b – c = 0
(B) 3a + b – c = 0
(C) 3a – b + c = 0
(D) 7a – b + c = 0
AA [GATE-PI-2016-IISc]
264. The number of solutions of the simultaneous
equations y = 3x + 3 and y = 3x+5 is
(A) zero (B) 1
(C) 2 (D) infinite
AB [GATE-AE-2016-IISc]
265. Consider the following system of linear
equations :
2x + y + z = 1
3x – 3y +3z = 6 x – 2y + 3z = 4
This system of linear equation has
(A) no solution (B) one solution
(C) two solutions (D) three solutions
A [GATE-CS-2004-IITD]
266. How many solutions does the following
system of linear equations have
5 1
x y
   
2
x y
 
3 3
x y
 
(A) Infinitely many
(B) Two distinct solutions
(C) Unique
(D) None
A2 [GATE-EC-2015-IITK]
267. Consider the system of linear equations :
x – 2y +3z = –1
x – 3y + 4z = 1 and
–2x +4y – 6z = k,
The value of k for which the system has
infinitely many solutions is _______.
AA [GATE-EE-2005-IITB]
268. In the matrix equation PX=Q, which of the
following is a 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 the matrix P
(B) Matrix Q must have only non-zero
elements
(C) Matrix P must be singular
(D) Matrix P must be square
AC [GATE-ME-2012-IITD]
269. x 2y z 4
   , 2x y 2z 5
   , x y z 1
  
The system of algebraic equations given
above has
(A) A unique solution 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 solutions
AB [GATE-ME-2008-IISc]
270. For what value of a, if any, will the
following system of equation in x,y and z
have solution?
2x + 3y = 4, x + y + z = 4, x + 2y – z = a

34. ENGINEERING MATHEMATICS
Page 26 TARGATE EDUCATION GATE-(EE/EC)
(A) Any real number
(B) 0
(C) 1
(D) There is no such value
AA [GATE-ME-2003-IITM]
271. Consider a system of simultaneous equations
1.5x + 0.5y + z = 2
4x + 2y + 3z = 9
7x + y + 5z = 10
(A) The solution is unique
(B) Infinitely many solutions exist
(C) The equations are inconsistent
(D) Finite many solution exist
[GATE-ME-1995-IITK]
272. Solve the system of equations: 2x + 3y + z =
9, 4x + y = 7, x – 3y – 7z = 6
Solution: A(X=1,Y=3,Z=-2)
AA [GATE-CE-2007-IITK]
273. For what values of  and  the following
simultaneous equation have an infinite
number of solutions?
x + y + z = 5, x + 3y + 3z = 9,
x + 2y +  z = 
(A) 2, 7 (B) 3, 8
(C) 8, 3 (D) 7, 2
AD [GATE-CE-2006-IITKGP]
274. Solution for the system defined by the set of
equation 4y + 3z = 8; 2x –z = 2 and 3x + 2y
= 5 is :
(A)
4
x 0,y 1,z
3
  
(B)
1
x 0,y ,z 2
2
  
(C)
1
x 1,y ,z 2
2
  
(D) Non- existent
AD [GATE-CE-2005-IITB]
275. Consider a non-homogeneous system of
linear equations represents 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.
AB [GATE-CE-2005-IITB]
276. Consider the following system of equations
in there real variable 1 2
x ,x and 3
x
1 2 3
2 3 1
x x x
  
1 2 3
3x 2x 5x 2
  
1 2 3
x 4x x 3
   
This system of equation has
(A) Has no solution
(B) A unique solution
(C) More than one but finite number of
solutions
(D) An infinite number of solutions
AB [GATE-IN-2005-IITB]
277. Let A be n n
 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
AC [GATE-PI-2010-IITG]
278. The value of q for which the following set of
linear algebra equations
2x + 3y = 0
6x + qy = 0
can have non-trivial solution is:
(A) 2 (B) 7
(C) 9 (D) 11
AB [GATE-PI-2009-IITR]
279. The value of 3
x obtained by solving the
following system of linear equations is
1 2 3
x 2x 2x 4
  
1 2 3
2x x x 2
   
1 2 3
x x x 2
   
(A) -12 (B) 2
(C) 0 (D) 12
A1 [GATE-CS-2014-IITKGP]
280. Consider the following system of equation
3x + 2y = 1
4x + 7z =1
x +y + z = 3
x – 2y +7z =0
The number of solutions for this system
is___________.
AD [GATE-CS-2008-IISc]
281. The following system of equations
1 2 3
x x 2x 1
  

35. TOPIC 1 - LINEAR ALGEBRA
www.targate.org Page 27
1 2 3
x 2x 3x 2
  
1 2 3
x 4x ax 4
  
has a unique solution. The only possible
value(s) for a is/are
(A) 0
(B) either 0 or 1
(C) one of 0, 1 and -1
(D) any real number other than 5
AB [GATE-CS-2003-IITM]
282. Consider the following system of linear
equation:
2 1 4 x
4 3 12 y 5
1 2 8 z 7
 
     
     
 
     
     

     
Notice that second and the third columns of
coefficient matrix are linearly dependent.
For how many values of  , does this system
of equations have many solutions?
(A) 0 (B) 1
(C) 2 (D) Infinitely many
AC [GATE-CS-2004-IITD]
283. What values of x, y and z satisfies the
following system of linear equations?
1 2 3 x 6
1 3 4 y 8
2 2 3 z 12
     
     

     
     
     
(A) x = 6, y = 3, z = 2
(B) x = 12, y =3, z = -4
(C) x= 6, x = 6, z = -4
(D) x = 12, y = -3, z = 0
AB [GATE-CH-2012-IITD]
284. Consider the following set of linear algebraic
equation
1 2 3
1 2
2 3
2 3 2
1
2 2 0
x x x
x x
x x
  
  
 
The system has
(A) A unique solution
(B) No solution
(C) An infinite number of solutions
(D) Only the trivial solution
AB [GATE-BT-2014-IITKGP]
285. The solution for the following set of
equations is :
5x 4y 10z 13
  
x 3y z 7
  
4x 2y 2 0
  
(A) x = 2, y = 1, z = 1
(B) x = 1, y = 2, z = 0
(C) x = 1, y = 0, z = 2
(D) x= 0, y = 1, z = 2
AD [GATE-BT-2014-IITKGP]
286. The solution to the following set of
equations is
2x 3y 4
 
4x 6y 0
 
(A) x = 0, y = 0 (B) x = 2, y = 0
(C) 4x = 6y (D) No solution
AB [GATE-BT-2013-IITB]
287. The solution of the following set of
equations is :
x 2y 3z 20
  
7x 3y z 13
  
x 6y 2z 0
  
(A) x = -2, y = 2, z = 8
(B) x = -2, y = -3, z = 8
(C) x = 2, y = 3, z = -8
(D) x = 8, y = 2, z = -3
AB [GATE-AE-2017-IITR]
288. Matrix 
2 0 2
A 3 2 7
3 1 5
 
 
  
 
 
and vector
 
4
b 4
5
 
 
  
 
 
are given. If vector {x} is the
solution to the system of equations
    
A x b
 , which of the following is true
for {x}:
(A) Solution does not exist
(B) Infinite solutions exist
(C) Unique solution exists
(D) Five possible solutions exist
AD [GATE-AE-2017-IITR]
289. Let matrix  
2 6
A
0 2

 
  
 
. Then for non-
trivial vector   1
2
x
x
x
 
  
 
, which of the
following is true for the value of
    
T
K x A x :

(A) K is always less than zero
(B) K is always greater than zero
(C) K is non-negative
(D) K can be anything

36. ENGINEERING MATHEMATICS
Page 28 TARGATE EDUCATION GATE-(EE/EC)
A4 [GATE-BT-2017-IITR]
290. The value of c for which the following
system is linear equations has an infinite
number of solutions is _________
1 2 x c
1 2 y 4
     

     
     
AC [GATE-IN-2018-IITG]
291. Consider the following system of linear
equations:
3 2 2
6 2
x ky
kx y
  
 
Here x and y are the unknowns and k is a real
constant. The value of k for which there are
infinite number of solutions is
(A) 3 (B) 1
(C) −3 (D) −6
A2 [GATE-EC-2018-IITG]
292. Consider matrix 2 2
2
k k
A
k k k
 
  

 
and
vector
1
2
x
x
x
 
  
 
. The number of distinct real
values of k for which the equation Ax = 0
has infinitely many solutions is _______.
A16 T1.6 [GATE-AE-2019-IITM]
293. The following system of equations
2 0,
2 0,
2 0
x y z
x y z
x y z
  
   
   
(A) has no solution
(B) has a unique solution.
(C) has three solutions.
(D) has an infinite number of solutions.
A6 T1.6 [GATE-XE-2019-IITM]
294. The value of  for which the system of
equations
3 3
2 0
2 7
x y z
x z
y z
  
 
   
has a solution is _____.
AC T1.6 [GATE-CH-2019-IITM]
295. A system of n homogenous linear equations
containing n unknowns will have non-trivial
solutions if and only if the determinant of
the coefficient matrix is
(A) 1 (B) –1
(C) 0 (D) 
AC T1.6 [GATE-ME-2019-IITM]
296. The set of equations
1
3 5
5 3 6
x y z
ax ay z
x y az
  
  
  
has infinite solutions, if a =
(A) – 3 (B) 3
(C) 4 (D) – 4
**********
Hamiltons
A0.9 to 1.1 [GATE-EE-2018- IITG]
297. Let
1 0 1
1 2 0
0 0 2
A

 
 
 
 
 

 
and
3 2
4 5
B A A A I
    , where I is the 3 3

identity matrix. The determinant of B is
_____ (up to 1 decimal place).
Statement for Linked Answer Questions for next
two problems
Cayley-Hamilton Theorem states that a square
matrix satisfies its own characteristic equation.
Consider a matrix
A =
3 2
1 0

 
 

 
AA [GATE-EE-2007-IITK]
298. A satisfies the relation
(A) -1
A 3I 2A 0
  
(B) 2
A 2A 2I 0
  
(C) (A I)(A 2I)
 
(D) exp (A) = 0
AA [GATE-EE-2007-IITK]
299. 9
A equals
(A) 511 A + 510 I (B) 309 A + 104 I
(C) 154 A + 155 I (D) exp (9A)
AB [GATE-EC-2012-IITD]
300. Given that
5 3
A
2 0
 
 
  
 
and
1 0
I
0 1
 
  
 
,
the value of
3
A is:
(A) 15A + 12I (B) 19A + 30I
(C) 17A + 15I (D) 17A + 21I
AD [GATE-EE-2008-IISc]
301. The characteristic equation of a
 
3 3
 matrix P is defined as
  3 2
I P 2 1 0
          

37. TOPIC 1 - LINEAR ALGEBRA
www.targate.org Page 29
If I denotes identity matrix, then the inverse
of matrix P will be :
(A)  
2
P P 2I
  (B)  
2
P P I
 
(C)  
2
P P I
   (D)  
2
P P 2I
  
**********
Geometrical Transformation
AC [GATE-PI-2015-IITK]
302. Match the linear transformation matrices
listed in the first column to their
interpretations in the second column.
P.
1 0
0 0
 
 
 
1. Stretch in the y-axis
Q.
0 0
0 1
 
 
 
2. Uniform stretch in x and
y-axis
R.
1 0
0 3
 
 
 
3. Projection in x-axis
S.
4 0
0 4
 
 
 
4. Projection in y-axis
(A) P-1,Q-2, R-3, S-4
(B) P-2,Q-3, R-4, S-1
(C) P-3,Q-4, R-1, S-2
(D) P-4,Q-1, R-2, S-3
AD [GATE-IN-2009-IITR]
303. The matrix
0 0 1
P 1 0 0
0 1 0
 
 
  
 
 
rotates a vector
about the axis
1
1
1
 
 
 
 
 
by an angle of
(A) 30 (B) 60
(C) 90 (D) 120
AC T1 [GATE-ME-2019-IITM]
304. The transformation matrix for mirroring a
point in x – y plane about the line y x
 is
given by
(A)
1 0
0 1
 
 

 
(B)
1 0
0 1

 
 
 
(C)
0 1
1 0
 
 
 
(D)
0 1
1 0

 
 

 
AB T1 [GATE-PH-2019-IITM]
305. During a rotation, vectors along the axis of
rotation remain unchanged. For the rotation
matrix
0 1 0
0 0 1
1 0 0
 
 

 
 

 
, the unit vector along
the axis of rotation is :
(A)  
1 ˆ
ˆ ˆ
2 2
3
i j k
  (B)  
1 ˆ
ˆ ˆ
3
i j k
 
(C)  
1 ˆ
ˆ ˆ
3
i j k
  (D)  
1 ˆ
ˆ ˆ
2 2
3
i j k
 
AD [GATE-IN-2017-IITR]
306. The figure shows a shape ABC and its
mirror image 1 1 1
A B C across the horizontal
axis (X-axis). The coordinate transformation
matrix that maps ABC to 1 1 1
A B C is :
(A)
0 1
1 0
 
 

 
(B)
0 1
1 0
 
 
 
(C)
1 0
0 1

 
 
 
(D)
1 0
0 1
 
 

 
------0000-------

38. ENGINEERING MATHEMATICS
Page 30 TARGATE EDUCATION GATE-(EE/EC)
Answer :
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
B D B C A D D A B B
11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
A C A C A B D C D *
21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
–6 D C * * * D B B A
31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
B B C A C A A 1 B B
41. 42. 43. 44. 45. 46. 47. 48. 49. 50.
88 23 A A D 16 A B C A
51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
0 D A C # A D A D C
61. 62. 63. 64. 65. 66. 67. 68. 69. 70.
C D 5.5 C 2 C A A A B
71. 72. 73. 74. 75. 76. 77. 78. 79. 80.
A D * B A B A B A A
81. 82. 83. 84. 85. 86. 87. 88. 89. 90.
A A A B # C C D C B
91. 92. 93. 94. 95. 96. 97. 98. 99. 100.
6 C * A 2 6 C * * *
101. 102. 103. 104. 105. 106. 107. 108. 109. 110.
A A # C 3.0 * D * D –6
111. 112. 113. 114. 115. 116. 117. 118. 119. 120.
D 3.0 A C A,D C A C A B
121. 122. 123. 124. 125. 126. 127. 128. 129. 130.
C B C A C B A A B A
131. 132. 133. 134. 135. 136. 137. 138. 139. 140.
C C A 17 B 2 D D A C
141. 142. 143. 144. 145. 146. 147. 148. 149. 150.
C B D 1/3 A C D B A D
151. 152. 153. 154. 155. 156. 157. 158. 159. 160.
C * A A D C B C A B
161. 162. 163. 164. 165. 166. 167. 168. 169. 170.
B C C A A B B B D D
171. 172. 173. 174. 175. 176. 177. 178. 179. 180.
B A A B * D A D A C
181. 182. 183. 184. 185. 186. 187. 188. 189. 190.
A A A C C C A C A A
191. 192. 193. 194. 195. 196. 197. 198. 199. 200.
5 D C B 5 0 17 A * A
201. 202. 203. 204. 205. 206. 207. 208. 209. 210.
D C D 3 A 2 12 D B B
211. 212. 213. 214. 215. 216. 217. 218. 219. 220.
A A 10 D C 3 C A A A
221. 222. 223. 224. 225. 226. 227. 228. 229. 230.
B B C C A C A B C B
231. 232. 233. 234. 235. 236. 237. 238. 239. 240.
2 B D B C C C C 4 2
241. 242. 243. 244. 245. 246. 247. 248. 249. 250.
1 A B B B C 3 B D C
251. 252. 253. 254. 255. 256. 257. 258. 259. 260.
B B B B B D B A C D
261. 262. 263. 264. 265. 266. 267. 268. 269. 270.
* 1.00 B A B A 2 A C B
271. 272. 273. 274. 275. 276. 277. 278. 279. 280.
A # A D D B B C B 1
281. 282. 283. 284. 285. 286. 287. 288. 289. 290.
D B C B B D B B D 4
291. 292. 293. 294. 295. 296. 297. 298. 299. 300.
C 2 16 6 C C * A A B
301. 302. 303. 304. 305. 306.
D C D C B D
20. 2.8 to 3.0
24. 199 to 201
25. 10
26. 160
73. 0.25
93. 0.99 to 1.01
98. 15.0
99. 0.164 to 0.126
100. 0.99 to 1.01
106. 0.95 to 1.05
108. 2.9 to 3.1
152. (-3,-2,-1)
175. Orthogonal
199. 24.5 to 25.5
261. 14.9 to 15.1
297. 0.9 to 1.1

39. 2
Calculus

40. Page 32 TARGATE EDUCATION GATE-(EE/EC)
2.1
Mean Value Theorem
Rolle’s MVT
A
1. If the 3 2
( ) 11 6
f x ax bx x
    satisfies
the conditions of Rolle’s Theorem in [1, 3]
and
1
' 2 0
3
f
 
 
 
 
, then value of a and b
are respectively
(A) (1, 6)
 (B) 2,1

(C) 1,1/ 2
 (D) 1,6

C
2. Which of the following function satisfies the
conditions of Rolle’s theorem?
(A)
1 1 1
sin , x
x π π
  
(B)
tan
,0
x
x π
x
 
(C) ( 1),0 1
x x x
  
(D)
1
,0 1
x
x
x

 
D
3. The value of c in Rolle’s theorem, where
2 2
π π
c
   and ( ) cos
f x x
 is equal to
(A) / 4
π (B) / 3
π
(C) π (D) 0
A
4. Given that Rolle’s theorem holds for
3 2
( ) 6 5
f x x x kx
    on {1, 3} with
1
2 .
3
c  
The value of k is :
(A) 11 (B) 7
(C) 3 (D) – 3
C
5. Find C of the Rolle’s theorem for
( ) ( 1)( 2)
f x x x x
   in [1, 2]
(A) 1.5 (B)  
1 1/ 3

(C)  
1 1/ 3
 (D) 1.25
C
6. Find C of the Rolle’s theorem for
( ) sin
x
f x e x
 in [0, ]
π
(A) / 4
π (B) / 2
π
(C) 3 / 4
π (D) does not exist
A
7. Find C of Rolle’s theorem for
3 4
( ) ( 2) ( 3)
f x x x
   in [ 2,3]

(A) 1/ 7 (B) 2 / 7
(C) 1/ 2 (D) 3 / 2
B
8. Find C of Rolle’s theorem for
/2
( ) ( 3) x
f x x x e
  in [ 3,0].

(A) 1
 (B) 2

(C) 0.5
 (D) 0.5
C
9. Rolle’s theorem cannot be applied for the
function ( ) | 2 |
f x x
  in [-2, 0] because
(A) ( )
f x is not continuous in [ 2,0]

(B) ( )
f x is not differentiable in ( 2,0)

(C) ( 2) (0)
f f
 
(D) None of these
AB
10. Rolle’s Theorem holds for function
3 2
,
x bx cx
  1 2
x
  at the point 4/3
then value of b and c are respectively :
(A) 8, 5
 (B) 5,8

(C) 5, 8
 (D) 5, 8
 
B
11. Rolle’s theorem cannot be applied for the
function ( ) | |
f x x
 in [ 2,2]
 because

41. TOPIC 2.1 - MEAN VALUE THEOREM
www.targate.org Page 33
(A) ( )
f x is not continuous in [-2,2]
(B) ( )
f x is not differentiable in (-2, 2)
(C) ( 2) (2)
f f
 
(D) None of these
**********
Lagranges’s MVT
A2.6-2.7 [GATE-CH-2016-IISc]
12. The Lagrange mean-value
theorem is satisfied for
  3
5
f x x
  , in the interval
(1, 4) at a value (rounded off
to the second decimal place) of x equal
to________.
D[GATE-CE-2005-IITB]
13. A rail engine accelerates from its satisfactory
position for 8 seconds and travels a distance
of 280 m. According to the mean value
theorem, the speed motor at a certain time
during acceleration must read exactly
(A) 0 km/h
(B) 8 km/h
(C) 75 km/h
(D) 126 km/h
AB [GATE-EC-2015-IITK]
14. A function f(x) = 1 – x2
+ x3
is defined in the
closed interval [–1,1]. The value of x, in the
open interval (–1,1) for which the mean
value theorem is satisfied, is :
(A)
1
2

(B)
1
3

(C)
1
3
(D)
1
2
AB [GATE-EE-2010-IITG]
15. A function
2
y 5x 10x
  is defined over an
open interval x = (1, 2). At least at one point
in this interval,
dy
dx
is exactly.
(A) 20 (B) 25
(C) 30 (D) 35
C[GATE-ME-1994-IITKGP]
16. The value of  in the mean value theorem
of ( ) ( ) ( ) ( )
f b f a b a f
    for
(A) b a
 (B) b a

(C)
( )
2
b a

(D)
( )
2
b a

AA [GATE-ME-2018-IITG]
17. According to the Mean Value Theorem, for a
continuous function ( )
f x in the interval
[ , ]
a b , there exists a value  in this interval
such that ( )
b
a
f x dx 

(A) ( )( )
f b a
  (B) ( )( )
f b a
 
(C) ( )( )
f a b   (D) 0
A
18. If the function ( ) x
f x e
 is defined in [0,
1], then the value of c of the mean value
theorem is :
(A) log( 1)
e  (B) ( 1)
e 
(C) 0.5 (D) 0.5

A
19. Find C of Lagrange’s mean value theorem
for ( ) ( 1)( 2)( 3)
f x x x x
    in [1, 2]
(A) 2 1 3
 (B)  
2 1/ 3

(C)  
1 1/ 3
 (D)  
1 1/ 3

B
20. Find C of Lagrange’s mean value theorem
for ( ) log
f x x
 in [1, ]
e
(A) 2
e (B) 1
e 
(C) ( 1) / 2
e  (D) ( 1) / 2'
e 
A
21. Find C of Lagrange’s mean value for
2
( )
f x lx mx n
   in [ , ]
a b
(A) ( ) / 2
a b
 (B) ab
(C) 2 / ( )
ab a b
 (D) ( ) / 2
b a

A
22. Find C of Lagrange’s theorem mean value
theorem for 2
( ) 7 13 19
f x x x
   in
[ 11/ 7,13/ 7]

(A) 1/7 (B) 2/7
(C) 3/7 (D) 4/7
B
23. Find C of Lagrange’s mean value theorem
for ( ) x
f x e
 in [0, 1]
(A) 0.5
(B) log( 1)
e 

42. ENGINEERING MATHEMATICS
Page 34 TARGATE EDUCATION GATE-(EE/EC)
(C) log( 1)
e 
(D) log( 1) / ( 1)
e e
 
D
24. ( ) ( 2)( 2),1 4
f x x x x x
     will
satisfy mean value theorem at x =
(A) 1 (B) 2
(C) 13 (D) 7
A
25. For the curve 4 2
2 3,
y x x
   the tangent
at the point (1, 4) is parallel to the chord
joining the points (0, 3) and the point
(A) (2,31) (B) ( 2,31)

(C)
3
, 6
2
 
 
 
 
(D)
3 15
,
2 2
 
 
 
**********
Cauchy’s MVT
B
26. Find C of Cauchy’s mean value theorem for
( )
f x x
 and ( ) 1/
g x x
 in [ , ]
a b
(A) ( ) / 2
a b
 (B) ab
(C) 2 / ( )
ab a b
 (D) ( ) / 2
b a

C
27. Find C of Cauchy’s mean value theorem for
the function 1/x and 2
1/ x in [a, b]
(A) ( ) / 2
a b
 (B) ab
(C) 2 / ( )
ab a b
 (D) ( ) / 2
b a

B
28. Find C of Cauchy’s mean value theorem for
the functions sin x and cos x in
[ / 2,0]

(A) / 3
π
 (B) / 4
π

(C) / 6
π
 (D) / 8
π

B
29. Let ( )
f x and ( )
g x be differentiable
function for 0 1,
x
  such that
(0) 2,
f  (0) 0
g  (1) 6.
f  Let there
exist a real number c in (0, 1) such that
'( ) 2 '( ),
f c g c
 then (1)
g equals :
(A) 1 (B) 2
(C) – 2 (D) – 1
-------0000-------

43. TOPIC 2.1 - MEAN VALUE THEOREM
www.targate.org Page 35
Answer :
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
A C D A C C A B C B
11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
B * D B B C A A A B
21. 22. 23. 24. 25. 26. 27. 28. 29.
A A B D A B C D B
12. 2.6 to 2.7

44. Page 36 TARGATE EDUCATION GATE-(EE/EC)
2.2
Maxima and Minima
Single Variable
AC [GATE-EC-2012-IITD]
1. The maximum value
of   3 2
f x x 9x 24x 5
    in the interval [1,
6] is :
(A) 21 (B) 25
(C) 41 (D) 46
AD [GATE-ME-2005-IITB]
2. The right circular cone of largest volume that
can be enclosed by a sphere of 1m radius has
a height of _____ .
(A) 2 (B) 3
(C) 4/3 (D) 2/3
AC [GATE-EE-2011-IITM]
3. The function   2
f x 2x x 3
   has
(A) a maxima at x = 1 and minima at x = 5
(B) a maxima at x = 1 and minima at x = -5
(C) only a maxima at x = 1
(D) only a minimal at x = 1
AD [GATE-ME-2012-IITD]
4. At x = 0, the function   3
f x x 1
  has
(A) a maximum value
(B) a minimum value
(C) a singularity
(D) a point of inflection
AB [GATE-ME-2006-IITKGP]
5. Equation of line normal to function
   
2/3
f x x 8 1
   at P(0, 5) is:
(A) y = 3x – 5 (B) y = 3x + 5
(C) 3y = x + 15 (D) 3y = x – 15
AA [GATE-CE-2004-IITD]
6. The function   3 2
f x 2x 3x 36x 2
    has
its maxima at
(A) x = -2 only
(B) x = 0 only
(C) x = 3 only
(D) both x = -2 and x = 3
AD [GATE-CE-2002-IISc]
7. The following function has a local minima at
which the value of x
  2
f x x 5 x
 
(A)
2
5
 (B) 5
(C)
5
2
(D)
5
2

AC [GATE-CE-2004-IITD]
8. The maxima and minima of the function
3 2
f(x) 2x 15x 36x 10
    occur,
respectively at
(A) x = 3 and x = 2
(B) x = 1 and x = 3
(C) x = 2 and x = 3
(D) x = 3 and x = 4
AB [GATE-CS-2008-IISc]
9. A point on a curve is said to be an extremum
if it is a local minimum or a local maximum.
The number of distinct extrema for the curve
4 3 2
3x 16x 24x 37
   is:
(A) 0 (B) 1
(C) 2 (D) 3
[GATE-CS-1998-IITD]
10. Find the point of local maxima and minima if
any of the following function defined in
3 2
0 x 6, x 6x 9x 15
    
ANS: Maxima x= 1, Minima x = 3
A-5.1- -4.9 [GATE-ME-2016-IISc]
11. Consider the function   3 2
2 3
f x x x
  in the
domain [-1, 2]. The global minimum of f(x) is
_________.
AD [GATE-CE-2016-IISc]
12. The optimum value of the function
  2
4 2
f x x x
   is :
(A) 2(maximum)
(B) 2(minimum)

45. TOPIC 2.2 – MAXIMA AND MINIMA
www.targate.org Page 37
(C) -2(maximum)
(D) -2(minimum)
AC [GATE-PI-2016-IISc]
13. The range of values of k for which the
function
   
2 2 3 4
4 6 8
f x k x x x
   
has a local maxima at point x = 0 is :
(A) 2
k   or 2
k 
(B) 2
k   or 2
k 
(C) 2 2
k
  
(D) 2 2
k
  
A1.0 [GATE-AE-2016-IISc]
14. Let x be a positive real number. The function
  2
2
1
f x x
x
  has it minima at x = ____.
A5.0 [GATE-XE-2016-IISc]
15. Let   3 2
f x 2x 3x 69
   ,
5 x 5
   .
Find the point at which
f(x) has the maximum value at.
A3.0 [GATE-BT-2016-IISc]
16. Consider the equation
2
aS
V
S
b S
c

 
Given a = 4, b = 1 and c = 9, the positive
value of S at which V is maximum, will be
_______.
AB [GATE-ME-2007-IITK]
17. The minimum value of function
2
y x
 in the
interval [1, 5] is:
(A) 0 (B) 1
(C) 25 (D) undefined
AB [GATE-EC-2016-IISc]
18. As x varies from −1 to +3, which one of the
following describes the behaviour of the
function 3 2
( ) 3 1
f x x x
   ?
(A) f(x) increases
monotonically.
(B) f(x) increases, then decreases and
increases again.
(C) f(x) decreases, then increases and
decreases again.
(D) f(x) increases and then decreases.
A–13 [GATE-IN-2016-IISc]
19. Let :[ 1,1]
f   where f (x) = 2x3
− x4
−10.
The minimum value of f (x) is______.
A0.0 [GATE-EE-2016-IISc]
20. The maximum value attained by the function
f(x) = x(x − 1)(x − 2) in the interval [1, 2] is
____.
AA [GATE-AG-2016-IISc]
21. The function 2
( ) 6
f x x x
   is :
(A) minimum at x = ½
(B) maximum at x = ½
(C) minimum at x = – ½
(D) maximum at x = – ½
D[GATE-CS-2008-IISc]
22. A point on the curve is said to be an
extremum if it is a local minimum (or) a local
maximum. The number of distinct extreme
for the curve 4 3 2
3 6 24 37
x x x
   is
___________
(A) 0
(B) 1
(C) 2
(D) 3
-0.1to0.1 [GATE-EC-2014-IITKGP]
23. The maximum value of the function f (x) = ln
(1 + x) – x (where x > - 1) occurs at x ---------
---.
5.9to6.1 [GATE-EC-2014-IITKGP]
24. The maximum value of
  3 2
2 9 12 3
f x x x x
    in the interval
0 3
x
  is ----------.
C [GATE-EE-2014-IITKGP]
25. Minimum of the real valued function
2/3
( ) ( 1)
f x x
  occurs at x equal to
(A) ‒∞ (B) 0
(C) 1 (D) ∞
B [GATE-EE-2014-IITKGP]
26. The minimum value of the function
3 2
( ) 3 24 100
f x x x x
    in the interval [–
3, 3] is
(A) 20 (B) 28
(C) 16 (D) 32
B [GATE-EE-1994-IITKGP]
27. The function
2 250
y x
x
  at x = 5 attains
(A) Maximum (B) Minimum
(C) Neither (D) 1
A [GATE-ME-1995-IITK]
28. The function f(x) = 3 2
6 9 25
x x x
   has
(A) A maxima at x = 1 and minima at x = 3
(B) A maxima at x = 3 and a minima at x = 1