The given matrix will become singular when x = 4. For a matrix to be singular, its determinant must be equal to 0. Computing the determinant of the given 3x3 matrix yields a quadratic equation in x with a solution of x = 4.

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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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

46. ENGINEERING MATHEMATICS
Page 38 TARGATE EDUCATION GATE-(EE/EC)
(C) No maxima, but a minima at x = 3
(D) A maxima at x = 1, but no minima
AA [GATE-ME-2005-IITB]
29. The function f(x) = 3 2
2 3 36 2
x x x
   has
its maxima at
(A) x = - 2 only
(B) x = 0 only
(C) x = 3 only
(D) both x = - 2 and x = 3
A [GATE-EC-2007-IITK]
30. Consider the function f(x) = 2
2.
x x
  the
maximum value of f(x) in the closed interval
[-4, 4] is
(A) 18 (B) 10
(C) – 2.25 (D) indeterminate
C [GATE-IN-2008-IISc]
31. Consider the function
2
6 9.
y x x
   The
maximum value of y obtained when x varies
over the internal 2 to 5 is
(A) 1 (B) 3
(C) 4 (D) 9
A [GATE-EC-2008-IISc]
32. For real values of x, the minimum value of
the function ( ) exp( ) exp( )
f x x x
   is :
(A) 2
(B) 1
(C) 0.5
(D) 0
A [GATE-EC-2010-IITG]
33. If 1/
y x
e x
 then y has a
(A) Maximum at x = e
(B) Minimum at x = e
(C) Maximum at x = 1
e
(D) Minimum at x = 1
e
C [GATE-IN-2007-IITK]
34. For real x, the maximum value of
sin
cos
x
x
e
e
is
(A) 1 (B) e
(C) 2
e (D) 
B [GATE-EE-2007-IITK]
35. Consider the function  
2
2
( ) 4
f x x
 
where x is a real number. Then the function
has
(A) Only one minimum
(B) Only two minima
(C) Three minima
(D) Three maxima
D [GATE-EC-2007-IITK]
36. Which one of the following functions is
strictly bounded ?
(A)
1
x
(B) x
e
(C) 2
x (D)
2
x
e
A [GATE-EC-2006-IITKGP]
37. As x increased from  to  , the function
( )
1
x
x
e
f x
e


(A) Monotonically increases
(B) Monotonically decreases
(C) Increases to a maximum value and then
decreases
(D) Decreases to a minimum value and then
increases.
A49 [GATE-EC-2014-IITKGP]
38. The maximum value of the determinant
among all 2 2
 real symmetric matrices with
trace 14 is________.
39. The plot of a function ( )
f x is shown in the
following figure. A possible expression for
the function ( )
f x is :
(A)  
exp | |
x (B)
1
exp
x

 
 
 
(C) exp(x) (D)
1
exp
x
 
 
 
D
40. Let 2
( ) 4 3.
f x x x
   Consider the following
statements :
1. ( )
f x is increasing in (2, )

2. ( )
f x is decreasing in ( , 2)
 

47. TOPIC 2.2 – MAXIMA AND MINIMA
www.targate.org Page 39
3. ( )
f x is has a stationary point at x = 2
Which of these statements are correct?
(A) 1 and 2 (B) 1 and 3
(C)2 and 3 (D) 1, 2 and 3
D
41. The function ( )
f x = 3 2
9 9 3
kx x x
   is
increasing in each interval, then (saddle point
should not be included)
(A) 3
k (B) 3
k
(C) 3
k (D) 3
k
C
42. What is the minimum value of the function
0
( ) ( 1)
x
f x t dt
 
 where x > 0 ?
(A) 1 (B) 1/2
(C) –1/2 (D) 5/2
B
43. The maximum value of
2
2
1
, 0
x
x
x
 

 
 
is
equal to
(A) e (B) 1/e
e
(C) e e (D) 1/ e
D[GATE-CE-2004-IITD]
44. The function
3 2
( ) 2 3 36 10
f x x x x
   
has a maximum at x =
(A) 3 (B) 2
(C) – 3 (D) - 2
D
45. The minimum value of
3 2
( ) 2 3 36 10
f x x x x
    is :
(A) 0 (B) 13

(C) 17
 (D) 71

B
46. A maximum value of
ln
( )
x
f x
x
 is :
(A) e (B) 1
e
(C) 1
e  (D) 1
e 
C
47. The function
2
( )
f x x
 has minimum at x =
(A) e (B) 1
e
(C) 0 (D) e + 1
D
48. The minimum value of ( ) ln
f x x x
 is :
(A) e (B) 1
e
(C) e
 (D) 1
e

B
49. The maximum value of ( ) x
f x xe
 is :
(A) e (B) 1
e
(C) 1 (D) e

D
50. Match the following list :
ListI
Function
List-II
Maximum Value of
( )
f x at x =
(A)
log
( )
x
f x
x
 (i)
1
e
(B)
1
( ) x
f x
x

(ii) 1

(C)
1
( ) x
f x x

(iii) e
(D)
1
( )
f x x
x
 
(iv) e

(A) (A)-(iv) , (B)-(iii), (C)-(ii), (D)-(i)
(B) (A)-(iv), (B)-(i), (C)-(iii), (D)-(ii)
(C) (A)-(iv), (B)-(i), (C)-(ii), (D)-(iii)
(D) (A)-(iii), (B)-(i), (C)-(iv), (D)-(ii)
B
51. The minimum distance from the point (4, 2)
to the parabola 2
8 ,
y x
 is
(A) 2 (B) 2 2
(C) 2 (D) 3 2
C
52. The shortest distance of the point (0, c),
where 0 5,
c
  from the parabola
2
y x
 is :
(A) 4 1
c  (B)
4 1
2
c 
(C)
4 1
2
c 
(D) None of these
C [GATE-EE-2011-IITM]
53. The function f(x) = 2
2 3
x x
  has

48. ENGINEERING MATHEMATICS
Page 40 TARGATE EDUCATION GATE-(EE/EC)
(A) A maxima at x = 1
and a minima at
x = 5
(B) A maxima at x = 1 and
a minima at
x = - 5
(C) Only a maximum at x = 1
(D) Only a minima at x = 0
A [GATE-EC-2014-IITKGP]
54. For 0 ,
t
   the maximum value of the
function f (t) = e-t
– 2e-2t
occurs at
(A) T = loge 4 (B) T = loge 2
(C) T = 0 (D) T = log e 8
C [GATE-EC-2014-IITKGP]
55. For a right angled triangle if the sum of the
lengths of the hypotenuse and a side is kept
constant in order to have maximum area of
the triangle, the angle between the
hypotenuse and the side is
(A) 120
(B) 360
(C) 600
(D) 450
A1 [GATE-EC-2015-IITK]
56. The maximum area (in square units) of a
rectangle whose vertices lie on the ellipse x2
+ 4y2
= 1 is ______.
D [GATE-EE-2010-IITG]
57. At t = 0, the function f(t) =
sin t
t
has
(A) A minimum (B) A discontinuity
(C) A point of inflection (D) A Maximum
AB [GATE-EC-2015-IITK]
58. Which one of the following graphs describes
the function
x 2
f(x) e (x x 1)

   ?
(A)
(B)
(C)
(D)
A9 [GATE-EE-2015-IITK]
59. If the sum of the diagonal
elements of a 2 2
 diagonal
matrix is –6, then the
maximum possible value of
determinant of the matrix is ______.
AA [GATE-ME-2015-IITK]
60. At x= 0, the function f(x) = |x| has
(A) a minimum
(B) a maximum
(C) a point of inflexion
(D) neither a maximum nor minimum
AB
61. The interval of increment of the function
( ) tan(2 / 7)
x
f x x e
    is :
(A) (0, )
 (B) ( ,0)

(C) (1, )
 (D) ( ,1)

AC
62. The function ( ) x
f x x
 decreases on the
interval
(A) (0, )
e (B) (0,1)
(C) (0,1/ )
e (D) none of these
AB
63. The function 2
( ) 2log( 2) 4 1
f x x x x
    
increases on the interval
(A) (1,2) (B) (2,3)
(C) (1,3) (D) (2,4)
AA
64. If the function 2
( ) 2 5
f x x kx
   is [1, 2],
then k lies in the interval
(A) ( ,4)
 (B) (4, )

(C) ( ,8)
 (D) (8, )


49. TOPIC 2.2 – MAXIMA AND MINIMA
www.targate.org Page 41
AA
65. The function 2
( ) x
f x x e
 is monotonic
decreasing when
(A) [0,2]
x R
 
(B) 0 2
x
 
(C) 2 x
  
(D) 0
x 
AA
66. The function ( ) cos 2
f x x x
   is
monotonic decreasing when
(A) 1/ 2
  (B) 1/ 2
 
(C) 2
  (D) 2
 
AB
67. Function 3
( ) 27 5
f x x x
   is
monotonically increasing (excluding
stationary point) when
(A) 3
x   (B) | | 3
x 
(C) 3
x   (D) | | 2
x 
AD
68. Function 3 2
( ) 2 9 12 29
f x x x x
    is
monotonically decreasing when
(A) 2
x  (B) 2
x 
(C) 3
x  (D) 1 2
x
 
AD
69. Function ( ) | | | 1|
f x x x
   is monotonically
increasing when
(A) 0
x  (B) 1
x 
(C) 1
x  (D) 0 1
x
 
AB
70. In the interval (1, 2), function
( ) 2| 1| 3| 2|
f x x x
    is
(A) increasing
(B) decreasing
(C) constant
(D) none of these
AC
71. If the function ( ) cos | | 2
f x x ax b
  
increases along the entire number scale, then
(A) a b
 (B)
1
2
a b

(C)
1
2
a   (D)
3
2
a  
AA
72. The function ( )
1 | |
x
f x
x


is :
(A) strictly increasing
(B) strictly decreasing
(C) neither increasing nor decreasing
(D) none of these
AD
73. Function ( ) x
f x a
 is increasing on R, if
(A) 0
a  (B) 0
a 
(C) 0 1
a
  (D) 1
a 
AB
74. Function ( ) loga
f x x
 is increasing on R, if
(A) 0 1
a
  (B) 1
a 
(C) 1
a  (D) 0
a 
AD
75. If the function 2
( ) 5
f x x kx
   is
increasing on [2, 4] then
(A) (2, )
k   (B) ( ,2)
k  
(C) (4, )
k   (D) ( ,4)
k  
AA
76. The function ( ) / 2 sin
f x x x
   defined on
[ / 3, / 3]
  is
(A) increasing (B) decreasing
(C) constant (D) none of these
A0.25 [GATE-CS-2014-IITKGP]
77. Let S be a sample space and two mutually
exclusive events A and B be such that
A B S
  . If P(.) denotes the probability of
the event, the maximum value of P(A)P(B) is
..................
AA [GATE-CE-2005-IITB]
78. Consider the circle | 5 5 | 2
z i
   in the
complex number plane (x, y) with z = x + iy.
The minimum distance from the origin to the
circle is :
(A) 5 2 2
 (B) 54
(C) 34 (D) 5 2
AC
79. A rectangular sheet of metal of length 6
metres and width 2 metres is given. Four
equal squares are removed from the corners.
The sides of this sheet are now turned up to
form an open rectangular box. Find
approximately, the height of the box (in
metre), such that the volume of the box is
maximum.

50. ENGINEERING MATHEMATICS
Page 42 TARGATE EDUCATION GATE-(EE/EC)
(A) 2.2 (B) 1.9
(C) 0.45 (D) 3.1
A32 [GATE-MN-2017-IITR]
80. A rectangle has two of its corners on the x
axis and the other two on the parabola
2
12
y x
  . The largest area of the rectangle
is __ .
A1 [GATE-MT-2017-IITR]
81. The function 3
( ) 3
f x x x
  has a minimum
at x = ___________.
A11.5 to 12.5 [GATE-EE-2018-IITG]
82. Let 3 2
( ) 3 7 5 6
f x x x x
    . The maximum
value of ( )
f x over the interval [0, 2] is
_______ (up to 1 decimal place).
AD [GATE-CE-2018-IITG]
83. At the point x = 0, the function 3
( )
f x x
 has
(A) local maximum
(B) local minimum
(C) both local maximum and minimum
(D) neither local maximum nor local
minimum
AB [GATE-CY/CH-2018-IITG]
84. For 0 2
x
   , sin x and cos x are both
decreasing functions in the interval ______.
(A) 0,
2

 
 
 
(B) ,
2

 

 
 
(C)
3
,
2

 

 
 
(D)
3
,2
2

 

 
 
AB T2.2 [GATE-CE-2019-IITM]
85. Which one of the following is NOT a correct
statement ?
(A) The function ,( 0)
x
x x  , has the global
maxima at x e

(B) The function ,( 0)
x
x x  , has the global
minima at x e

(C) The function 3
x , has neither global
minima nor global maxima
(D) The function | |
x has the global
minima at 0
x 
A450 T2.2.1 [GATE-AR-2019-IITM]
86. In a site map, a rectangular residential plot
measures 150mm×40mm , and the width of
the front road in the map measures 16 mm.
Actual width of the road is 4 m. If the
permissible F.A.R. is 1.2, the maximum built-
up area for the residential building will be
______m2
.
AA T2.2.1 [GATE-AE-2019-IITM]
87. The maximum value of the function
( ) x
f x xe
 (where x is real) is
(A) 1/ e (B) 2
2 / e
(C)
1/2
( ) / 2
e
(D) 
AC T2.2.1 [GATE-EY-2019-IITM]
88. Which of the following is correct about first
and second derivates at points P, Q and R for
( ) sin( )
f x x
 shown below?
(A)
2
2
0; 0; 0
P Q R
df df d f
dx dx dx
  
(B)
2 2
2 2
0; 0; 0
R
P Q
d f d f df
dx dx dx
  
(C)
2 2 2
2 2 2
0; 0; 0
P Q R
d f d f d f
dx dx dx
  
(D)
2 2 2
2 2 2
0; 0; 0
P Q R
d f d f d f
dx dx dx
  
AC T2.2.2 [GATE-TF-2019-IITM]
89. For x in [0, ]
 , the maximum value of
(sin cos )
x x
 is
(A)
1
2
(B) 1
(C) 2 (D) 2
******
Double Variable
B [GATE-PI-2007-IITK]
90. For the function f(x, y) =
2 2
x y
 defined on
R2
, the point (0, 0) is :

51. TOPIC 2.2 – MAXIMA AND MINIMA
www.targate.org Page 43
(A) A local minimum
(B) Neither a local minimum
(nor) a local maximum.
(C) A local maximum
(D) Both a local minimum and a local
maximum
B [GATE-ME-2002-IISc]
91. The function f(x, y) = 2 3
2 2
x xy y
  has
(A) Only one stationary
point at (0, 0)
(B) Two stationary points
at (0, 0) and
1 1
,
6 3
 

 
 
(C) Two stationary points at (0, 0) and
(1, -1)
(D) No stationary point.
A [GATE-CE-2010-IITG]
92. Given a function
2 2
( , ) 4 6 8 4 8
f x y x y x y
    
The optimal value of ( , )
f x y
(A) Is a minimum equal to 10/3
(B) Is a maximum equal to 10/3
(C) Is a minimum equal to 8/3
(D) Is a maximum equal to 8/3
B [GATE-EC-1998-IITD]
93. The continuous function f(x, y) is said to
have saddle point at (a, b) if
(A) ( , ) ( , ) 0
x y
f a b f a b
  2
0
xy xx yy
f f f
 
(B) ( , ) 0, ( , ) 0,
x y
f a b f a b
  2
0
xy xx yy
f f f
 
(C) ( , ) 0, ( , ) 0,
x y
f a b f a b
  2
0
xy xx yy
f f f
 
(D) ( , ) 0, ( , ) 0,
x y
f a b f a b
  2
0
xy xx yy
f f f
 
AA [GATE-EC-1993-IITB
94. The function   2
f x,y x y 3xy 2y x
    has
(A) No local extremum
(B) One local maximum but no local
minimum
(C) One local minimum but no local
maximum
(D) One local minimum and one local
maximum
C
95. Stationary point is a point where, function
f(x,y) have,
(A) 0
f
x
 

(B) 0
f
y
 

(C) 0 & 0
f f
x y
 
 
 
(D) 0 & 0
f f
x y
 
 
 
AB
96. For function f(x,y) to have minimum value at
(a,b) value,
(A) 2
0
rt s
  and 0
r 
(B) 2
0
rt s
  and 0
r 
(C) 2
0
rt s
  and 0
r 
(D) 2
0
rt s
  and 0
r 
AA
97. For function f(x,y) to have maximum value at
(a,b),
(A) 2
0
rt s
  and 0
r 
(B) 2
0
rt s
  and 0
r 
(C) 2
0
rt s
  and 0
r 
(D) 2
0
rt s
  and 0
r 
AB
98. For function f(x,y) to have no extremum
value at (a,b),
(A) 2
0
rt s
 
(B) 2
0
rt s
 
(C) 2
0
rt s
 
(D) 2
0
rt s
 
AB
99. Find the minimum value of
2 2
( , ) 6 12
f x y x y x
   
(A) – 3 (B) 3
(C) – 9 (D) 9
AC
100. Find the maximum or minimum value
of
2 2 3
( , ) 4 3
f x y y xy x x
   
(A) minimum at (0, 0)
(B) maximum at (0, 0)
(C) minimum at (2/3, –4/3)
(D) maximum at (2/3, –4/3)
AA
101. Find the minimum value of
 
3 1 1
xy a
x y
 
(A) 2
3a (B) 2
a
(C) a (D) 1

52. ENGINEERING MATHEMATICS
Page 44 TARGATE EDUCATION GATE-(EE/EC)
AB
102. Divide 120 into three parts so that the sum of
their products taken two at a time is
maximum. If x, y, z are two parts, find value
of x, y and z
(A) 40, 40, 40
x y z
  
(B) 38, 50, 32
x y z
  
(C) 50, 40, 30
x y z
  
(D) 80, 30, 50
x y z
  
-------0000-------

53. TOPIC 2.2 – MAXIMA AND MINIMA
www.targate.org Page 45
Answer :
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
C D C D B A D C B #
11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
* D C 1.0 5.0 3.0 B B –13 0.0
21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
A D * C C B B A A A
31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
C A A C B D A 49 C D
41. 42. 43. 44. 45. 46. 47. 48. 49. 50.
D C B D D B C D B D
51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
B C C A C 1 D B 9 A
61. 62. 63. 64. 65. 66. 67. 68. 69. 70.
B C B A A A B D D B
71. 72. 73. 74. 75. 76. 77. 78. 79. 80.
C A D B D A * A C 32
81. 82. 83. 84. 85. 86. 87. 88. 89. 90.
1 * D B B 450 A C C B
91. 92. 93. 94. 95. 96. 97. 98. 99. 100.
B A B A C B A B B C
101. 102.
A B
11. –5.1- –4.9
23. –0.1 to 0.1
77. 0.25
82. 11.5 to 12.5

54. Page 46 TARGATE EDUCATION GATE-(EE/EC)
2.3
Limits
Limit, Continuity, Diff. Checkup
AD [GATE-ME-2014-IITKGP]
1. If a function is continuous at a point,
(A) The limit of the function
may not exist at the point
(B) The function must be
derivable at the point
(C) The limit of the function at the point
tends to infinity
(D) The limit must exist at the point and the
value of limit should be same as the value
of the function at that point
A [GATE-EC-1996-IISc]
2. If a function is continuous at a point its first
derivative
(A) May or may not exist
(B) Exists always
(C) Will not exist
(D) Has a unique value
AB [GATE-EC-2016-IISc]
3. Given the following statements about a
function : R R
f  , select the right option:
P : If f(x) is continuous at 0
x x
 , then it is
also differentiable at 0
x x
 .
Q : If f(x) is continuous at 0
x x
 , then it may
not be differentiable at 0
x x
 .
R : If f(x) is differentiable at 0
x x
 , then it is
also continuous at 0
x x
 .
(A) P is true, Q is false, R is false
(B) P is false, Q is true, R is true
(C) P is false, Q is true, R is false
(D) P is true, Q is false, R is true
AD [GATE-EE-2018-IITG]
4. Let f be a real-valued function of a real
variable defined as 2
( )
f x x
 for 0
x  , and
2
( )
f x x
  for 0
x  . Which one of the
following statements is true?
(A) ( )
f x is discontinuous at 0
x  .
(B) ( )
f x is continuous but not differentiable
at 0
x  .
(C) ( )
f x is differentiable but its first
derivative is not continuous at 0
x  .
(D) ( )
f x is differentiable but its first
derivative is not differentiable at 0
x  .
B [GATE-ME-1995-IITK]
5. The function f(x) = | 1|
x  on
the interval [ 2,0]
 is _________
(A) Continuous and
differentiable
(B) Continuous on the interval but not
differentiable at all points
(C) Neither continuous nor differentiable
(D) Differentiable but not continuous
C [GATE-CE-2011-IITM]
6. What should be the value of λ such that the
function defined below is continuous at ?
2
π
x=
cos
2
( ) 2
1
2
λ x π
if x
π
x
f x
π
if x





 




(A) 0 (B) 2π
(C) 1 (D)
2
π
C[GATE-ME-2010-IITG]
7. The function | 2 3 |
y x
 
(A) is continuous  x R
 and differential
x R
 
(B) is continuous x R
  and differential
x R
  except at x =
3
2
(C) is continuous x R
  and differential
x R
  except at x =
2
3

55. TOPIC 2.3 – LIMITS
www.targate.org Page 47
(D) is continuous x R
  and except at x =
3 and differential x R
 
D[GATE-ME-2002-IISc]
8. Which of the following functions is not
differentiable in the domain [-1, 1]?
(A) f(x) = x2
(B) f(x) = x – 1
(C) f(x) = 2
(D) f(x) = maximum (x, – x)
B
9. Consider f(x) =
2
0, 0
, 0







x
x
x
x
(A) f(x) is disconitinious everywhere
(B) f(x) is conitinious everywhere
(C) f’(x) = exist in (-1,1)
(D) f’(x) = exist in (-2,2)
AD
10. Consider f(x) =
, 3
4, 3
3 5, 3
x x
x
x x

 




  

f(x) is continues at x=3,then  will be :
(A) 4 (B) 3
(C) 2 (D) 1
AC [GATE-EC-2012-IITD]
11. Consider the function  
f x x
 in the
interval 1 x 1
   . At the point x = 0, f(x) is
(A) Continuous and differentiable
(B) Non-continuous and differentiable
(C) Continuous and non-differentiable
(D) Neither continuous nor differentiable
AA [GATE-CS-2013-IITB]
12. Which of the following function is continuous
at x= 3?
(A)  
2 if x 3
f x x 1 if x 3
x 3
if x 3
3

 

  

 
 

(B)  
4 if x 3
f x
8 x if x 3


 
 

(C)  
x 3 if x 3
f x
8 x if x 3
 

 
 

(D)   3
1
f x if x 3
x 27
 

AB [GATE-CS-1998-IITD]
13. Consider the function y x
 in the interval
 
1,1
 . In this interval, the function is:
(A) Continuous and differentiable
(B) Continuous but not differentiable
(C) Differentiable but not continuous
(D) Neither continuous nor differentiable
AD [GATE-PI-2016-IISc]
14. At x=0, the function
 
2
sin ,
x
f x
L

  
, 0
x L
   
(A) continuous and differentiable
(B) not continuous and not differentiable
(C) not continuous but differentiable
(D) continuous but not differentiable
AB [GATE-EE-2017-IITR]
15. A function f(x) is defined as
 
x
2
e x 1
f x ,
Inx ax bx, x 1
 
 
  

where x R

which of the following statements is TRUE?
(A) f(x) is NOT differentiable at x =1 for any
values of a and b
(B) f(x) is differentiable at x = 1 for the
unique values of a and b
(C) f(x) is differentiable at x = 1 for all the
values of a and b such that a + b = e
(D) f(x) is differentiable at x = 1 for all values
of a and b.
AB [GATE-MT-2017-IITR]
16. If | |
( ) x
f x e
 then a x = 0, the function f(x)
is :
(A) continuous and differentiable.
(B) continuous but not differentiable.
(C) neither continuous nor differentiable.
(D) not continuous but differentialble.
AD [GATE-TF-2018-IITG]
17. If
4 , 2
( )
4, 2
x x
f x
kx x
 

 
 

is
a continuous function for all

56. ENGINEERING MATHEMATICS
Page 48 TARGATE EDUCATION GATE-(EE/EC)
real values of x, then f(8)
is equal to _______________.
AD [GATE-PI-2018-IITG]
18. A real-valued function y of real variable x is
such that 5| |
y x
 . At x = 0, the function is
(A) discontinuous but differentiable
(B) both continuous and differentiable
(C) discontinuous and not differentiable
(D) continuous but not differentiable
AC [GATE-ME-2016-IISc]
19. The values of x for which the function
 
2
2
3 4
3 4
x x
f x
x x
 

 
is NOT continuous are
(A) 4 and -1 (B) 4 and 1
(C) -4 and 1 (D) -4 and -1
AB [GATE-PE-2016-IISc]
20. The function  
1
1 | |
f x
x


is :
(A) Continuous and differentiable
(B) continuous but not differentiable
(C) not continuous but differentiable
(D) not continuous and not differentiable
D [GATE-CS-1996-IISc]
21. The formula used to compute an
approximation for the second derivative of a
function f at a point 0
x is :
(A) 0 0
( ) ( )
2
f x h f x h
  
(B) 0 0
( ) ( )
2
f x h f x h
h
  
(C) 0 0 0
2
( ) 2 ( ) ( )
f x h f x f x h
h
   
(D) 0 0 0
2
( ) 2 ( ) ( )
f x h f x f x h
h
   
C [GATE-IN-2007-IITK]
22. Consider the function
3
( ) | |,
f x x
 where x is
real. Then the function f(x) at x = 0 is
(A) Continuous but not differentiable
(B) Once differentiable but not twice.
(C) Twice differentiable but not thrice.
(D) Thrice differentiable
B
23. If f(x) =
1
x
x 
then f’(0)
(A) 0 (B) 1
(C) 2 (D) 3
C [GATE-PI-2010-IITG]
24. If (x) = sin| |
x then the value of
df
dx
at
4
π
x


is :
(A) 0 (B)
1
2
(C)
1
2
 (D) 1
B [GATE-CE-2018-IITG]
25. Which of the following function(s) is an
accurate description of the graph for the
range(s) indicated?
(i) 2 4for 3 1
y x x
     
(ii) | 1| for 1 2
y x x
    
(iii) | | 1 1 2
y x for x
    
(iv) 1 for 2 3
y x
  
(A) (i), (ii) and (iii) only.
(B) (i), (ii) and (iv) only.
(C) (i) and (iv) only.
(D) (ii) and (iv) only.
******
Limits
Single Variable
AC [GATE-MN-2017-IITR]
26. Which one of the following plots represents
the relationship xy = c, which c is a positive
constant

57. TOPIC 2.3 – LIMITS
www.targate.org Page 49
(A) I (B) II
(C) III (D) IV
AC [GATE-ME-2016-IISc]
27.
 
3
0
log 1 4
1
e
x
x
x
Lt
e



is equal to
(A) 0 (B)
1
12
(C)
4
3
(D) 1
AC [GATE-ME-2016-IISc]
28. 2
lim 1
x
x x x

   is :
(A) 0
(B) 
(C) 1/2
(D) -
A1.0 [GATE-PE-2016-IISc]
29. The value of
0
1
lim
sin
x
x
e
x

 

 
 
is equal
to_________.
A1.0 [GATE-CS-2016-IISc]
30.
 
4
sin 4
lim
4
x
x
x




__________.
A25.0 [GATE-PI-2016-IISc]
31.
2
5
0
1
lim
x
x
e
x

 

 
 
is equal to __________.
A0.5 [GATE-IN-2016-IISc]
32.  
2 2
lim 1
n
n n n

   is ___________ .
B [GATE-ME-1995-IITK]
33.
0
1
lim sin ______
x
x x


(A)  (B) 0
(C) 1 (D) Does not exist
AC T2.D [GATE-CE-2001-IITK]
34. Limit of the following series as x approaches
2

is
3 5 7
( )
3! 5! 7!
x x x
f x x
        
(A)
2
3
π
(B)
2
π
(C)
3
π
(D) 1
A [GATE-ME-2003-IITM]
35.
2
0
sin
lim ____
x
x
x


(A) 0 (B) 
(C)  (D) – 1
B [GATE-ME-2007-IITK]
36.
2
3
0
1
2
lim
x
x
x
e x
x

 
  
 
  
(A) 0 (B)
1
6
(C)
1
3
(D) 1
A [GATE-CS-2008-IISc]
37.
sin
lim ______
cos
x
x x
x x




(A) 1 (B) – 1
(C)  (D) 
C [GATE-IN-2005-IITB]
38.
0
sin
lim
x
x
x

is :
(A) Indeterminate (B) 0
(C) 1 (D) 
B [GATE-ME-2008-IISc]
39. The value of
1/3
8
2
lim
8
x
x
x



is
(A)
1
16
(B)
1
12
(C)
1
8
(D)
1
4
D [GAT -ME-2011-IITM]
40. What is
0
sin
lim
θ
θ
θ

equal to?
(A) θ (B) sinθ
(C) 0 (D) 1

58. ENGINEERING MATHEMATICS
Page 50 TARGATE EDUCATION GATE-(EE/EC)
A [GATE-CE-1997-IITM]
41.
0
sin
lim ,
θ
mθ
θ

where m is an integer, is one of
the following:
(A) m (B) m 
(C) mθ (D) 1
A [GATE-EC-2007-IITK]
42.
0
sin( / 2)
lim
θ
θ
θ

(A) 0.5
(B) 1
(C) 2
(D) Not defined
C [GAT -ME-2011-IITM]
43. What is
cos
Lim


 equal to?
(A) θ (B) sinθ
(C) 0 (D) 1
B [GATE--2004-IITD]
44. The value of the function.
3 2
3 2
0
( ) lim
2 7
x
x x
f x
x x




is _____
(A) 0 (B)
1
7

(C)
1
7
(D) 
C [GATE-PI-2008-IISc]
45. The value of the expression
0
sin( )
lim x
x
x
e x

 
 
 
is
(A) 0 (B)
1
2
(C) 1 (D)
1
1 e

C [GATE-IN-1999-IITB]
46.
5
0
1 1
lim _____
10 1
j x
jx
x
e
e






(A) 0
(B) 1.1
(C) 0.5
(D) 1
D [GATE-CE-1999-IITB]
47. Limit of the function,
2
lim
n
n
n n


is _____
(A) 1
2
(B) 0
(C) 
(D) 1
B [GATE-CE-2002-IISc]
48. Limit of the following sequence as n is
___________ 1/n
x n

(A) 0 (B) 1
(C)  (D) - 
C [GATE-PI-2007-IITK]
49. What is the value of
/4
cos sin
lim
/ 4
x π
x x
x π



(A) 2
(B) 0
(C) 2

(D) Limit does not exist
A [GATE-CE-2000-IITKGP]
50. Value of the function  
lim
x a
x a
x a


 is
________
(A) 1 (B) 0
(C)  (D) a
A
51. The
0
2
sin
3
lim
x
x
x

 
 
  is :
(A)
2
3
(B) 1
(C)
1
4
(D)
1
2
B
52. The value of
1/3
8
2
lim
( 8)
x
x
x



is :
(A)
1
16
(B)
1
12
(C)
1
8
(D)
1
4
B
53.
2
2 2
sin 1
lim
cos
x
x
x x



is :
(A)  (B) 0
(C) 1 (D) None of these

59. TOPIC 2.3 – LIMITS
www.targate.org Page 51
B
54.
2
1/
2
2
0
1 5
lim
1 3
x
x
x
x

 

 

 
is :
(A) 1/2
e
(B)
2
e
(C)
2
e
(D) e
B [GATE-CS-2010-IITG]
55. What is the value of
2
1
lim 1
n
n n
 
 

 
 
?
(A) 0 (B) 2
e
(C) 1/2
e
(D) 1
AC [GATE-ME-2015-IITK]
56. The value of
 
2
x 0 4
1 cos x
lim
2x


is
(A) 0 (B)
1
2
(C)
1
4
(D) undefined
A– 0.333 [GATE-ME-2015-IITK]
57. The value of
x 0
sin x
lim
2sin x x cos x


 
 

 
is ________
AD [GATE-IN-2005-IITB]
58. Given a real-valued continuous function f(t)
defined over [0, 1],  
t
0
t 0
1
lim f x dx
t
  :
(A) 
(B) 0
(C) f(1)
(D) f(0)
AD [GATE-IN-2001-IITK]
59.
x
4
sin 2 x
4
lim
x
4



 

 
 


equals
(A) 0 (B)
1
2
(C) 1 (D) 2
AB [GATE-PI-2012-IITD]
60. 2
x 0
1 cos x
lim
x


 
 
 
is
(A)
1
4
(B)
1
2
(C) 1 (D) 2
AA [GATE-CS-1995-IITK]
61.
 
3
2
2
x
x cos x
lim
x sin x
 


=___________.
(A) 
(B) 0
(C) 2
(D) Does not exist
C [GATE-EC-2014-IITKGP]
62. The value of
1
lim 1
x
x x

 

 
 
is
(A) Ln 2 (B) 1.0
(C) e (D) ∞
AB [GATE-ME-2014-IITKGP]
63.
 
2x
x 0
e 1
lim
sin 4x

 

 
 
 
is equal to
(A) 0 (B) 0.5
(C) 1 (D) 2
AA [GATE-ME-2014-IITKGP]
64.
x 0
x sin x
lim
1 cosx



is
(A) 0 (B) 1
(C) 3 (D) Not defined
AB [GATE-ME-2012-IITD]
65. 2
x 0
1 cos x
lim
x


 
 
 
is
(A)
1
4
(B)
1
2
(C) 1 (D) 2
AC [GATE-ME-2000-IITKGP]
66.
2
x 1
x 1
lim
x 1

 

 

 
is :
(A)  (B) 0
(C) 2 (D) 1
AA [GATE-ME-1995-IITK]
67.
 
3
2
2
x
x cos x
lim
x sin x
 


equal
(A)  (B) 0
(C) 2 (D) Does not exist

60. ENGINEERING MATHEMATICS
Page 52 TARGATE EDUCATION GATE-(EE/EC)
AD [GATE-ME-1994-IITKGP]
68. The value of
x
sin x
lim
x

 
 
 
(A)  (B) 2
(C) 1 (D) 0
A1 [GATE-ME-1993-IITB
69.
   
 
x
x 0
x e 1 2 cosx 1
lim
x 1 cosx

  

is_______.
AC [GATE-CE-2014-IITKGP]
70.
2
x 0
x sinx
lim
x

 

 
 
equal to
(A)  (B) 0
(C) 1 (D) 
AA [GATE-CE-2014-IITKGP]
71. The expression
a
a 0
x 1
lim
a


is equal to
(A) log x (B) 0
(C) xlogx (D) 
AC [GATE-CS-2017-IITR]
72. The value of
7 5
3 2
x 1
x 2x 1
lim
x 3x 2

 
 
(A) is 0 (B) is -1
(C) is 1 (D) doesnot exist
A1 [GATE-BT-2017-IITR]
73.
 
x 0
sin x
lim
x

is :
AD [GATE-ME-2017-IITR]
74. The value of
3
0
sin( )
lim
x
x x
x


is
(A) 0 (B) 3
(C) 1 (D) –1
AA [GATE-MN-2017-IITR]
75. The value of
2 1
2 2 1
2
2 1
lim
2 3 2
x
x
x
x x
x x



 
 
 
 
 
(A)
1
2
(B)
3
2
(C) 1 (D) 0
A31.5-32.5 [GATE-PE-2017-IITR]
76. The value of
 
4
0
2 16
lim
x
x
x

 
is ______.
AB [GATE-CH-2018-IITG]
77. The figure which represents
sin x
y
x
 for
0
x  (x in radians) is
(A)
(B)
(C)
(D)
A16 T2.3.2 [GATE-BT-2019-IITM]
78. The solution of
2
8
64
lim
8
x
x
x

 

 

 
is _______.
A0.16 to 0.17 T2.3.2 [GATE-AE-2019-IITM]
79. The value of the following limit is _____
(round off to 2 decimal places).
3
0
sin
lim

  

AC T2.3.2 [GATE-CS-2019-IITM]
80. Compute
4
2
3
81
lim
2 5 3
x
x
x x


 
(A) 1
(B) 53/12

61. TOPIC 2.3 – LIMITS
www.targate.org Page 53
(C) 108/7
(D) Limit does not exist
A2 T2.3.2 [GATE-ST-2019-IITM]
81. Let :
f  be defined by
2
( ) (3 4)cos
f x x x
  .
Then 2
0
( ) ( ) 8
lim
h
f h f h
h

  
is equal to …
AA T2.3.2 [GATE-CE-2019-IITM]
82. Which one of the following is correct ?
(A)
0
sin 4
lim 2
sin 2
x
x
x

 

 
 
and
0
tan
lim 1
x
x
x

 

 
 
(B)
0
sin 4
lim 1
sin 2
x
x
x

 

 
 
and
0
tan
lim 1
x
x
x

 

 
 
(C)
0
sin 4
lim
sin 2
x
x
x

 
 
 
 
and
0
tan
lim 1
x
x
x

 

 
 
(D)
0
sin 4
lim 2
sin 2
x
x
x

 

 
 
and
0
tan
lim
x
x
x

 
 
 
 
AD T2.3.2 [GATE-CE-2019-IITM]
83. The following inequality is true for all x close
to 0.
2
sin
2 2
3 1 cos
x x x
x
  

What is the value of
0
sin
lim
1 cos
x
x x
x
 
?
(A) 0 (B) 1/2
(C) 1 (D) 2
AA T2.3.2 [GATE-CH-2019-IITM]
84. The value of the expression
2
tan
lim
x
x
x


is
(A)  (B) 0
(C) 1 (D) 1

AC T2.3.2 [GATE-TF-2019-IITM]
85. The value of 2
0
1
lim
x
x
e x
x

 
is
(A)
1
2
 (B) 0
(C)
1
2
(D) 1
A0.49 to 0.51 T2.3.2 [GATE-PE-2019-IITM]
86. The value of 2
0
( 1)sin
lim
2
x
x x
x x



is _______
(round off to 2 decimal places).
Double Variable
AA [GATE-PI-2015-IITK]
87. The value of
2
( x, y) (0,0)
x xy
lim
x y



is
(A) 0 (B)
1
2
(C) 1 (D) 
AD [GATE-CE-2016-IISc]
88. What is the value of 2 2
0
0
lim
x
y
xy
x y



?
(A) 1
(B) -1
(C) 0
(D) Limit does not exist
AC [GATE-MA-2017-IITR]
89. Let 2
f :R R
 be defined by
 
2
2 2
y
sin x y , x 0
f x,y x
0, x 0
  
 
  
   
 

Then, at (0,0),
(A) f is continuous and the directional
derivative of f does NOT exist in some
direction
(B) f is NOT continuous and the directional
derivative of f exist in all directions
(C) f is NOT differentiable and the directional
derivative of f exist in all direction
(D) f is differentiable
AC
90.
( , ) (1,1)
4 2
lim _____
x y
x y
x y




.
(A) 1 (B) 2
(C) 3 (D) Does not exist
AB
91. 2 2
( , ) (4,2)
lim
x y
xy
x y



____.
(A) 1/5 (B) 2/5
(C) 3/5 (D) Does not exist
AD
92. 2 2
( , ) (0,0)
lim ln( )
x y
x y

  _____.
(A) 3.5 (B) 4.5
(C) 5.5 (D) Does not exist
AC

62. ENGINEERING MATHEMATICS
Page 54 TARGATE EDUCATION GATE-(EE/EC)
93.
2
( , ) (2,4)
2
lim
2
x y
y xy
y x




_____.
(A) 2 (B) 3
(C) 4 (D) Does not exist
-------0000-------

63. TOPIC 2.3 – LIMITS
www.targate.org Page 55
Answer :
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
D A B D B C C D B D
11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
C A B D B B D D C B
21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
D C B C B C C C 1.0 1.0
31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
25.0 0.5 B C A B A C B D
41. 42. 43. 44. 45. 46. 47. 48. 49. 50.
A A C B C C D B C A
51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
A B B B B C * D D B
61. 62. 63. 64. 65. 66. 67. 68. 69. 70.
A C B A B C A D 1 C
71. 72. 73. 74. 75. 76. 77. 78. 79. 80.
A C 1 D A * B 16 * C
81. 82. 83. 84. 85. 86. 87. 88. 89. 90.
2 A D A C * A D C C
91. 92. 93.
B D C
57. –0.333
76. 31.5-32.5
79. 0.16 to 0.17
86. 0.49 to 0.51

64. Page 56 TARGATE EDUCATION GATE-(EE/EC)
2.4
Integral & Differential Calculas
Single Integration
B [GATE-EC-2007-IITK]
1. The following plot shows a function y which
varies linearly with x. The value of the
integral
2
1
I ydx
  is :
(A) 1.0 (B) 2.5
(C) 4.0 (D) 5.0
AC [GATE-EC-2016-IISc]
2. Consider the plot of f(x) versus x as shown
below.
Suppose
5
( ) ( )
x
F x f y dy

  . Which one of the
following is a graph of ( )
F x ?
(A)
(B)
(C)
(D)
AB [GATE-ME-2014-IITKGP]
3. The value of the integral
   
   
2
2
2
0
x 1 sin x 1
dx
x 1 cos x 1
 
  
 is :
(A) 3
(B) 0
(C) -1
(D) – 2
AB [GATE-CE-2011-IITM]
4. What is the value of definite integral
a
0
x
dx
x a x
 
 ?
(A) 0 (B)
a
2
(C) a (D) 2a
D [GATE-CS-2011-IITM]
5. Given 1,
i   what will be the evaluation of
the definite integral 2
0
cos sin
cos sin
π
x i x
dx
x i x


 ?
(A) 0 (B) 2
(C) – i (D) i
A[GATE-ME-2005-IITB]
6. 6 7
(sin sin )
a
a
x x dx


 is equal to( a)
(A) 6
0
2 sin
a
xdx
 (B) 7
0
2 sin
a
xdx

(C) 6 7
0
2 (sin sin )
a
x x dx

 (D) None
D
7. Value of
/2
0
(sin cos )log(sin cos )
π
x x x x dx
 
 :
(A)  (B) 1
(C) / 2
 (D) 0

65. TOPIC 2.4 – INTEGRAL & DIFFERENTIAL CALCULAS
www.targate.org Page 57
B
8. The value of the integral | |
a
a
x dx

 is equal
to
(A) a (B) 2
a
(C) 0 (D) 2a
A0.5 T2.5.3 [GATE-AE-2019-IITM]
9. A function ( )
f x is defined by
1
( ) ( | |)
2
f x x x
  . The value of
1
1
( )
f x dx


is (round off to 1 decimal place.
C
10. After evaluating
/2
0
log(tan ) ,
π
x dx
 the value
of given integral will be :
(A) / 4
π (B) / 2
π
(C) 0 (D) 1
B
11. What is the value of
2
4
0
( ) (sin )
x x dx



(A) 1
 (B) 0
(C) 1 (D) π
D
12. The value of integral
/2
0
log tan
π
xdx
 is :
(A) 2 log 2
π
 (B) log 2
π

(C) 1 (D) 0
C [GATE-CE-2002-IISc]
13. The value of the following definite integral
is :
2
2
sin 2
1 cos
π
π
x
x


 dx
(A) 2ln 2
 (B) 2
(C) 0 (D) 2
(ln 2)
A
14. The value of 2 2
:
a
a
x a x dx



(A) 0 (B) 1
(C) 1 (D) 1

AC [GATE-ME-2013-IITB]
15. The value of the definite integral
 
e
1
xIn x dx
 is
(A)
3
4 2
e
9 9
 (B)
3
2 4
e
9 9

(C)
3
2 4
e
9 9
 (D)
3
4 2
e
9 9

AD [GATE-ME-2011-IITM]
16. If f(x) is an even function and ‘a’ is a
positive real number, then  
a
a
f x dx

 equals
(A) 0 (B) a
(C) 2a (D)  
a
0
2 f x dx

AD [GATE-MN-2016-IISc]
17. 4
X C
 is the general integral of
(A)
3
3 x dx
 (B) 3
1
4
x dx

(C)
3
x dx
 (D)
3
4 x dx

A1.0 [GATE-PE-2016-IISc]
18. The value of the definite integral  
1
In
e
x dx
 is
equal to _________.
A0.99-1.01 [GATE-MT-2016-IISc]
19. The value of the integral
/2
0
sin
x xdx


 _________
A3.13-3.15 [GATE-AE-2016-IISc]
20. The value of definite integral
 
0
x sin x dx

 is________.
A2.090 to 2.104 T2.5.1 [GATE-ME-2019-IITM]
21. The value of the following definite integral is
___ (round off to three decimal places)
 
1
ln
e
x x dx

A1.65-1.75 [GATE-BT-2016-IISc]
22. The value of the integral
0.9
0
(1 )(2 )
dx
x x
 
 is
______ .
A2.58 [GATE-AG-2016-IISc]
23. The value of the integral,
2
4
2
2
1
1
x
I dx
x



 is
___________
AA [GATE-ME-2006-IITKGP]
24. Assuming i 1
  and t is a real number,
it
3
0
e dt

 is :

66. ENGINEERING MATHEMATICS
Page 58 TARGATE EDUCATION GATE-(EE/EC)
(A)
3 1
i
2 2
 (B)
3 1
i
2 2

(C)
1 3
i
2 2
 
  
 
 
(D)
1 3
i 1
2 2
 
 
 
 
 
A( 2
2 cos
dy
x x
dx
 ) [GATE-ME-1995-IITK]
25. Given
2
x
1
y cos tdt
  then
dy
dx
is
_______________.
AA [GATE-CE-2001-IITK]
26. Value of the integral 2
4
0
I cos xdx

  is :
(A)
1
8 4

 (B)
1
8 4


(C)
1
8 4

  (D)
1
8 4

 
A4 [GATE-CS-2012-IITD]
27. If
2
0
x sin x dx k

 
 , then the value of k is
equal to ______________.
AD [GATE-CH-2013-IITB]
28. Evaluate x
dx
e 1

 (Note: C is a constant of
integration)
(A)
x
x
e
C
e 1


(B)
 
x
x
In e 1
C
e


(C)
x
x
e
In C
e 1
 

 

 
(D)  
x
In 1 e C

 
AC [GATE-CH-2010-IITG]
29. For a function g(x), if g(0) = 0,  
g ' 0 2
 then
 
g x
0
x 0
2t
lim dt
x
  is equal to
(A)  (B) 2
(C) 0 (D) 
C
30. The value of  
2
2
| | | 1 |
x x dx

 
 is :
(A) 7 (B) 5
(C) 9 (D) 10
B
31. The value of integral
100
0
| sin |
π
x dx
 is :
(A) 100 (B) 200
(C) 100 (D) 200
AA
32.
/2
2
0
cos
π
xdx
 is equal to
(A)
4
π
(B)
2
π
(C) π (D) 2π
A
33. If
2
0
( ) ,
x
x tdt
   then
d
dx

is :
(A) 2
2 x (B) x
(C) 0 (D) 1
B[GATE-EE-2010-IITG]
34. The value of the quantity P, where P =
1
0
,
x
xe dx
 is equal to
(A) 0 (B) 1
(C) e (D) 1/e
D[GATE-CS-2009-IITR]
35.
/4
0
(1 tan ) / (1 tan )
π
x x dx
 
 evaluates to
(A) 0 (B) 1
(C) ln 2 (D)
1
ln 2
2
C[GATE-EC-2006-IITKGP]
36. The integral 3
0
sin
π
θdθ
 is given by
(A) 1/2 (B) 2/3
(C) 4/3 (D) 8/2
C
37. The integral
3/2
/2
3/2 3/2
0
sin
sin cos
π x
dx
x x

 is
equal to
(A) 0 (B) 1
(C) / 4
π (D) / 2
π
A
38. The value of integral
1
0
2 sin
2 4
πt π
dt
 

 
 

is :
(A) 0 (B) 1
(C) – 1 (D) 2
A
39. The value of integral
2 2
4
dx
x x

 is given
by :
(A)
2
4
4
x
C
x

  (B)
2
4
4
x
C
x



67. TOPIC 2.4 – INTEGRAL & DIFFERENTIAL CALCULAS
www.targate.org Page 59
(C)
2
4
4
x
C
x

 (D)
4
4
x
C
x


40. The value of integral 6
sec xdx
 is given
by :
(A) 5 3
1 2
tan tan tan
5 3
x x x C
  
(B) 5 3 2
1 2
tan tan tan
5 3
x x x C
  
(C) 4 2
1 2
tan tan tan
5 3
x x x C
  
(D) 5 3
1 2
tan tan tan
5 3
x x x C
  
D [GATE-EE-2007-IITK]
41. The integral
2
0
1
sin( )cos
2
π
t τ τdτ
π

 equals
(A) sin cos
t t (B) 0
(C) (1/ 2)cost (D) (1/ 2)sin t
A [GATE-PI-2008-IISc]
42. The value of the integral
/2
/2
( cos )
π
π
x x dx

 is :
(A) 0 (B) 2
π
(C) π (D) 2
π
C [GATE-ME-1994-IITKGP]
43. The integration of log xdx
 has the value
(A) ( log 1)
x x  (B) log x x

(C) (log 1)
x x  (D) None
AD [GATE-AG-2017-IITR]
44. 2 2
( )
I a x dx
 
 is :
(A) 2 2 1
0.5 sin
x
a x
a

 
 
   
 
 
 
(B) 2 2 1
0.5 sin
x
x a x
a

 
 
   
 
 
 
(C) 2 2 2 1
0.5 sin
x
a x a
a

 
 
   
 
 
 
(D) 2 2 2 1
0.5 sin
x
x a x a
a

 
 
   
 
 
 
AB [GATE-CE-2017-IITR]
45. Let x be a continuous variable defined over
the interval  
,
  and  
x
x e
f x e

 
 . The
integral    
g x f x dx
  is equal to
(A)
x
e
e

(B)
x
e
e


(C)
x
e
e
(D) x
e
AB [GATE-CE-2018-IITG]
46. The value of the integral 2
0
cos
x x dx

 is
(A) 2
/ 8
 (B) 2
/ 4

(C) 2
/ 2
 (D) 2

A 0.27 to 0.30 [GATE-CS-2018-IITG]
47. The value of
/4
2
0
cos( )
x x dx

 correct to three
decimal places (assuming that 3.14
  ) is
____.
AC [GATE-EC-2017-IITR]
48. The values of the integrals
1 1
3
0 0 ( )
x y
dy dx
x y
 

 

 
  and
1 1
3
0 0 ( )
x y
dx dy
x y
 

 

 
 
are
(A) same and equal to 0.5
(B) same to equal to – 0.5
(C) 0.5 and – 0.5 respectively
(D) – 0.5 and 0.5 respectively
Simple Improper Integration
C
49. The value of integral
1
2/3
1
:
dx
x


(A) 6 (B) 6

(C) Does not exist (D) None of above
C [GATE-EE-2005-IITB]
50. If 3
1
,
S x dx


  then S has the value
(A)
1
3

(B)
1
4
(C)
1
2
(D) 1
B [GATE-EC/IN-2005-IITB]
51. The value of the integral
1
2
1
1
dx
x

 is :
(A) 2
(B) does not exists
(C) –2
(D) 

68. ENGINEERING MATHEMATICS
Page 60 TARGATE EDUCATION GATE-(EE/EC)
B [GATE-EE-2010-IITG]
52. Consider the following integral
4
1
x dx



(A) Diverges (B) converges to 1/3
(C) Converges to 1/4 (D) converges to 0
A2.0 [GATE-EC-2016-IISc]
53. The integral
1
0 (1 )
dx
x


is equal to ________.
AC [GATE-CH-2011-IITM]
54. The value of the improper integral 2
dx
1 x

 
 ,
is :
(A) 2
  (B) 0
(C)  (D) 2
B
55.
1
1
| |
x
dx
x

 is equal to
(A) 2 (B) 0
(C) 1 (D)
1
2
D[GATE-ME-2010-IITG]
56. The value of the integral 2
1
dx
x

 
 is :
(A) π
 (B) π / 2

(C) / 2
π (D) π
D [GATE-ME-2008-IISc]
57. Which of the following integrals is
unbounded ?
(A)
/4
0
tan
π
xdx
 (B) 2
0
1
1
dx
x



(C)
0
x
xe dx


 (D)
1
0
1
1
dx
x


AA T2.5.1 [GATE-AG-2019-IITM]
58. 2 2
0 ( 1)
dx
I
x



 has the value
(A) 0.785 (B) 0.915
(C) 1.000 (D) 1.245
AA T2.5.1 [GATE-PH-2019-IITM]
59. The value of the integral 2 2
cos( )
kx
dx
x a



 ,
where 0
k  and 0
a  , is
(A) ka
e
a


(B)
2 ka
e
a


(C)
2
ka
e
a


(D)
3
2
ka
e
a


AA [GATE-CE-2017-IITR]
60. Consider the following definite integral:
 
2
1
1
2
0
sin x
I dx
1 x




The value of integral is
(A)
3
24

(B)
3
12

(C)
3
48

(D)
3
64

B
61. The value of 2
0 (1 )(1 )
xdx
x x

 

(A) / 2
 (B) / 4

(C) 0 (D) π
A
62. The value of 2 2
0
4
4
adx
x a


 will be :
(A) π (B) / 2

(C) / 2
 (D) π

C
63. The value of integral
2
1
1
0 2
1
x
e
xdx
x


 is :
(A)
1
2
e
(B)
1
2
e
(C) 1
e (D) 1
e
Laplace form of Integration
AB [GATE-CE-2016-IISc]
64. The value of 2
0 0
1 sin
1
x
dx dx
x x
 


  is :
(A)
2

(B) 
(C)
3
2

(D) 1
AA T7 [GATE-CE-2019-IITM]
65. The Laplace transform of sinh (at) is
(A) 2 2
a
s a

(B) 2 2
a
s a

(C) 2 2
s
s a

(D) 2 2
s
s a


69. TOPIC 2.4 – INTEGRAL & DIFFERENTIAL CALCULAS
www.targate.org Page 61
AC T7 [GATE-PE-2019-IITM]
66. The Laplace transform of the function
( ) t
f t e
 is given by
(A) 2
1
( 1)
s 
(B)
1
1
s 
(C)
1
1
s 
(D) 2
1
( 1)
s 
AC T7 [GATE-IN-2019-IITM]
67. The output y(t) of a system is related to its
input ( )
x t as
0
( ) ( 2)
t
y t x d
   
 ,
where, x(t) = 0 and y(t) = 0 for 0
t  . The
transfer function of the system is :
(A)
1
s
(B)
2
(1 )
s
e
s

(C)
2s
e
s

(D) 2
1 s
e
s


A9 T7 [GATE-IN-2019-IITM]
68. The output of a continuous-time system y(t) is
related to its input x(t) as
1
( ) ( ) ( 1)
2
y t x t x t
   . If the Fourier
transforms of x(t) and y(t) are ( )
X  and
( )
Y  respectively and
2
| (0)| 4
X  , the value
of
2
| (0)|
Y is _____.
AD [GATE-EE-2016-IISc]
69. he value of the integral sin 2
2
t
dt
t



 
 

 
 is
equal to
(A) 0 (B) 0.5
(C) 1 (D) 2
A3 [GATE-EC-2015-IITK]
70. The value of the integral
sin(4 t)
12cos(2 t) dt
4 t





 is _____.
AB
71. valuate
0
sin t
t


(A) π (B)
2
π
(C)
4
π
(D)
3
π
B
72. The integral
0
sin
t
e t
dt
t


 is given by :
(A)
2
π
(B)
4
π
(C)
6
π
(D)
8
π
Beta and Gama Integration
AC [GATE-EC-2010-IITG]
73. The integral
2
x
2
1
e dx
2




 is equal to
(A)
1
2
(B)
1
2
(C) 1 (D) 
AB [GATE-IT-2006-IITKGP]
74. The following definite integral evaluates to
2
x
0 20
e dx
 
 
 
 


(A)
1
2
(B) 5
(C) 10
(D) 
A [GATE-EC-2005-IITB]
75. The value of the integral
2
0
1
exp
8
2
x
I dx
π
  
 
 
 
 is :
(A) 1 (B) π
(C) 2 (D) 2π
C
76. The integral 3
0
sin
t
te tdt


 is given by :
(A)
1
50
(B)
2
50
(C)
3
50
(D)
4
50
AC [GATE-CH-2012-IITD]
77. If a is a constant, then the value of the
integral 2 ax
0
a xe dx


 ,
(A)
1
a
(B) a
(C) 1 (D) 0
0.6 [GATE-EE-1994-IITKGP]
78. The value of
3
1/ 2
0
.
y
e y dy


 is ________
A0.43-0.45 [GATE-PH-2017-IITR]
79. The integral
2
2
0
x
x e dx


 is equal to ______ (up
to two decimal places).

70. ENGINEERING MATHEMATICS
Page 62 TARGATE EDUCATION GATE-(EE/EC)
AB [GATE-CE-2013-IITB]
80. The value of 4 3
6
0
cos 3 sin 6 d

  

(A) 0 (B)
1
15
(C) 1 (D)
8
3
**********
Area & Volume Calculation
Area Calculation
A [GATE-EC-2008-IISc]
81. The value of the integral of the function g(x,
y) =
3 4
4 10
x y
 along the straight line
segment from the point (0, 0) to the point (1,
2) in the x-y plane is
(A) 33 (B) 35
(C) 40 (D) 56
AB [GATE-CE-2016-IISc]
82. The area of the region bounded by the
parabola
2
1
y x
  and the straight line
+ y = 3 is :
(A)
59
6
(B)
9
2
(C)
10
3
(D)
7
6
B [GATE-ME/PI-2004-IITD]
83. The area enclosed between the parabola y =
x2
ad the straight line y = x is _____
(A) 1/8 (B) 1/6
(C) 1/3 (D) 1/2
A [GATE-ME-2009-IITR]
84. The area enclosed between the curves
2
4
y x
 and 2
4
x y
 is
(A)
16
3
(B) 8
(C)
32
3
(D) 16
B [GATE-ME-1995-IITK]
85. The area bounded by the parabola 2
2 y x

and the lines 4
x y
  is equal to _________
(A) 6 (B) 18
(C)  (D) None
B [GATE-CE-1997-IITM]
86. Area bounded by the curve y = x2
and the
lines x = 4 and y = 0 is given by
(A) 64 (B)
64
3
(C)
128
3
(D)
128
4
AA [GATE-PI-2012-IITD]
87. The area enclosed between the straight line y
= x and the parabola 2
y x
 in the x – y plane
is :
(A)
1
6
(B)
1
4
(C)
1
3
(D)
1
2
AC [GATE-ME-2017-IITR]
88. A parametric curve defined by
cos , sin
2 2
u u
x y
 
   
 
   
   
in the range
0 1
u
  is rotated about the X-axis by 360
degrees. Area of the surface generated is
(A)
2

(B) 
(C) 2 (D) 4
89. Area of the ellipse
2 2
2 2
1
x y
a b
  , is ….
(A) ab
 (B) / ab

(C) 2 2
/ a b
 (D) none of these
AC
90. The surface area of the sphere
2 2 2
2 4 8 2 0
x y z x y z
       is …..
(A) 72 (B) 82
(C) 92 (D) 29
AA
91. Area between the parabolas 2
4
y x
 and
2
4
x y
 is …….
(A) 16/3 (B)17/3
(C) 18/3 (D) None
Volume Calculation
AB T2.5.3 [GATE-ME-2019-IITM]
92. A parabola
2
x y
 with 0 1
x
  is shown in
the figure. The volume of the solid of rotation
obtained by rotating the shaded area by 0
360
around the x-axis is

71. TOPIC 2.4 – INTEGRAL & DIFFERENTIAL CALCULAS
www.targate.org Page 63
(A)
4

(B)
2

C)  (D) 2
D [GATE-EE-2006-IITKGP]
93. The expression V =
2
2
1
H
o
h
πR dh
H
 

 
 
 for
the volume of a cone is equal to _______.
(A)
2
2
1
R
o
h
πR dr
H
 

 
 

(B)
2
2
1
R
o
h
πR dh
H
 

 
 

(C) 1
R
o
r
2πrH dh
R
 

 
 

(D)
2
0
1
R r
rH dr
R
 
 
 
 

D [GATE-ME-2010-IITG]
94. The parabolic arcy = , 1 2
x x
  is
revolved around the x-axis. The volume of
the solid of revolution is
(A)
4
π
(B)
2
π
(C)
3
4
π
(D)
3
2
π
AD [GATE-XE-2016-IISc]
95. The volume of the solid obtained by
revolving the curve
2
y x,0 x 1
   around y
–axis is :
(A)  (B) 2
(C)
2

(D)
8
5

AA [GATE-EE-1994-IITKGP]
96. The volume generated by revolving he area
bounded by the parabola 2
8
y x
 and the line
2
x about y-axis is
(A)
128
5
π
(B)
5
128π
(C)
127
5π
(D)
32
5

A10.0 [GATE-EC-2016-IISc]
97. A triangle in the xy-plane is bounded by the
straight lines 2x = 3y, y = 0 and x = 3. The
volume above the triangle and under the
plane x + y + z = 6 is __________.
862to866 [GATE-EC-2014-IITKGP]
98. The volume under the surface z(x,y) = x + y
and above the triangle in the x – y plane
defined by 0 y x
  and 
0 12
x
  is :
1.01 [GATE-EE-2017-IITR]
99. Let 2
R
I c xy dxdy
  , where R is the region
shown in the figure and 4
c 6 10
  . The
value of I equals ________.(Give the answer
up to two decimal places.)
A2 [GATE-MA-2017-IITR]
100. Let D be the region in
2
bounded by the
parabola 2
y 2x
 and the line y = x. Then
D
3xydxdy
 equals____________.
A0.70-0.85 [GATE-EC-2017-IITR]
101. A three dimensional region R of finite
volume is described by
2 2 3
x y z ;0 z 1,
   
Where x, y, z are real. The volume of R(up to
two decimal places) is _______
A20 [GATE-EC-2016-IISc]
102. The integral
1
( 10)
2 D
x y dxdy
 
  , where
D denotes the disc : 2 2
4
x y
  , evaluates to
_____ .
A6 T2.5.3 [GATE-MN-2019-IITM]
103. If area S, in the x-y plane, is bounded by a
triangle with vertices (0,0), (10,1) and (1,1),
the value of 2
S
xy y dxdy

 is _______.
AC T2.5.3 [GATE-CE-2019-IITM]
104. Consider the hemi-spherical tank of radius 13
m as shown in the figure (not drawn to scale).
What is the volume of water (in 3
m ) when
the depth of water at the centre of the tank is
6m ?

72. ENGINEERING MATHEMATICS
Page 64 TARGATE EDUCATION GATE-(EE/EC)
(A) 78 (B) 156
(C) 396 (D) 468
A [GATE-EE-2009-IITR]
105. If (x, y) is continuous function defined over
(x, y) [0,1] [0,1]
 Given two constraints,
2
x y
 and 2
,
y x
 the volume under f(x, y)
is
(A) 2
1
0
( , )
y x y
y x y
f x y dxdy
 
 
 
(B) 2 2
1 1
( , )
y x
y x x y
f x y dxdy
 
 
 
(C)
1 1
0 0
( , )
y x
y x
f x y dxdy
 
 
 
(D)
0 0
( , )
y x x y
x x
f x y dxdy
 
 
 
A [GATE-EE-2005-IITB]
106. Changing the order of integration in the
double integral I =
8 2
0 /4
( , )
x
f x y dy dx
  leads
to I = ( , ) .
s q
r p
f x y dy dx
  What is q?
(A) 4y (B) 16 y2
(C) x (D) 8
AC [GATE-IN-2015-IITK]
107. The double integral
a y
0 0
f (x,y)dxdy
  is
equivalent to
(A)
x y
0 0
f (x,y)dxdy
 
(B)
y
a
0 x
f(x, y) dxdy

(C)
a a
0 x
f (x,y)dxdy
 
(D)
a a
0 0
f (x,y)dxdy
 
A [GATE-ME-2004-IITD]
108. The volume of an object expressed in
spherical co-ordinates is given by
2 /3 1
2
0 0 0
sin
π π
V r drd dθ
  
  
The value of the integral
(A)
3
π
(B)
6
π
(C)
2
3
π
(D)
4
π
AD
109. Find the volume bounded by the xy-plane, the
paraboloid 2 2
2z x y
  and the cylinder
2 2
4
x y
  .
(A) 1 (B) 2
(C) 3 (D) 4
AB
110. Find the volume bounded by the cylinder
2 2
4
x y
  and the planes 4
y z
  and
0
z  .
(A) 6 (B) 16
(C) 26 (D) 36
AC
111. Calculate the volume of the solid bounded by
the planes 0, 0, 1
x y x y z
     and
0
z  .
(A)
1
2
(B)
1
4
(C)
1
6
(D)
1
8
AC
112. Find the volume cut from the sphere
2 2 2 2
x y z a
   by the cone 2 2 2
x y z
  .
(A) 3
(2 2) / 3
a
 
(B) 2
(2 2) / 2
a
 
(C) 3
(2 2) / 3
a
 
(D) 2
(2 2) / 3
a
 
Double and Triple Integration
AD T2.5.3 [GATE-PI-2019-IITM]
113. The solution of
1 1
a b
dxdy
x y
 is
(A) ln( )
ab (B) ln( / )
a b
(C) ln( ) ln( )
a b
 (D) ln( )ln( )
a b
A1.99 to 2.01 T2.5.3 [GATE-EC-2019-IITM]
114. The value of the integral
0
sin
y
x
dxdy
x
 
  , is
equal to _____.
D [GATE-EC-2000-IITKGP]
115.
/2 /2
0 0
sin( )
π π
x y dxdy

 
(A) 0 (B) π
(C)
2
π (D) 2

73. TOPIC 2.4 – INTEGRAL & DIFFERENTIAL CALCULAS
www.targate.org Page 65
A [GATE-CS-2008-IISc]
116. The value of
3
0 0
(6 )
x
x y dxdy
 
 is _____
(A) 13.5 (B) 27.0
(C) 40.5 (D) 54.0
D [GATE-IN-2007-IITK]
117. The value of
2 2
0 0
x y
e e dxdy
 
 
 is:
(A)
2
π
(B) π
(C) π (D)
4
π
AB [GATE-ME-2014-IITKGP]
118. The value of the integral
2 x
x y
0 0
e dydx

  is,
(A)  
1
e 1
2
 (B)  
2
2
1
e 1
2

(C)  
2
1
e e
2
 (D)
2
1 1
e
2 e
 

 
 
AD [GATE-ME-2000-IITKGP]
119.  
2 2
0 0
sin x y dxdy
 

 
(A)
2
2x (B) x
(C) 0 (D) 2
A64 [GATE-TF-2018-IITG]
120. The value of the integral
2
4 16
2 2
0 0
x
y x y dy dx


 
is __________.
A
121. By a change of variables x(u, v) = uv,
( , ) /
y u v v u
 in a double integral, the integral
( , )
f x y changes to  
, .
u
f uv
v
Then ( , )
u v
 is
_______ .
(A)
2v
u
(B) 2 u v
(C) 2
V (D) 1
122.
2
0 0
( )
x
x y dx dy

  = ….
(A)1 (B) 2
(C) 3 (D) 4
AC
123. 2 3
x y dxdy
 over the rectangle 0 1
x
  and
0 3
y
  is ……
(A) 12/3 (B) 20/8
(C) 27/4 (D) 10/5
AA
124.
1
2 2
0
( ) .............
x
x
x y dx dy
 
 
(A) 3/35 (B) 23/98
(C) 88/104 (D) none of these
AC
125. 2 2
( )
x y dx dy

 in the positive quadrant for
which 1
x y
  , is ….
(A) 9/5 (B) 8/4
(C) 1/6 (D) 2/4
AC
126. Evaluate
2 2 2
1 (1 ) (1 )
0 0 0
x x y
xyz dxdydz
  
   .
(A)
3
46
(B)
2
24
(C)
1
48
(D)
4
84
AA
127.
2 3 2
2
0 1 1
.....
xy zdzdydx 
  
(A) 26 (B) 42
(C) 84 (D) 16
AA
128. Evaluate
1
1 0
( )
z x z
x z
x y z dxdydz

 
 
  
(A) 0 (B) 1
(C) 2 (D) 3
**********
Differential Calculus
D [GATE-CS-1995-IITK]
129. If at every point of a certain curve, the slope
of the tangent equals
2x
y

, the curve is
_________
(A) A straight line (B) A parabola
(C) A circle (D) An Ellipse
B [GATE-IN-2008-IISc]
130. Given y = 2
2 10
x x
  the value of
1
X
dy
dx 
is
equal to

74. ENGINEERING MATHEMATICS
Page 66 TARGATE EDUCATION GATE-(EE/EC)
(A) 0 (B) 4
(C) 12 (D) 13
A [GATE-PI-2009-IITR]
131. The total derivative of the function ‘xy’ is
(A) xdy ydx
 (B) xdx ydy

(C) dx dy
 (D) dx dy
A [GATE-ME-1998-IITD]
132. If
2
0
( )
x
x t dt
   then __________
d
dx


(A) 2
2x (B) x
(C) 0 (D) 1
D [GATE-ME-2008-IISc]
133. The length of the curvey y =
3/2
2
3
x between
x = 0 & x = 1 is
(A) 0.27 (B) 0.67
(C) 1 (D) 1.22
D [GATE-CE-2010-IITG]
134. A parabolic cable is held between two
supports at the same level. The horizontal
span between the supports is L.
The sag at the mid-span is h. The equation of
the parabola is y =
2
2
4 ,
x
h
L
where x is the
horizontal coordinate and y is the vertical
coordinate with
the origin at the centre of the cable. The
expression for the total length of the cable is
(A)
2 2
4
0
1 64
L h x
dx
L


(B)
2 2
/2
4
0
2 1 64
L h x
dx
L


(C)
2 2
/2
4
0
1 64
L h x
dx
L


(D)
2 2
/2
4
0
2 1 64
L h x
dx
L


AD [GATE-AG-2017-IITR]
135. Differentiation of
2
1 x
 gives
(A) 2
1
(1 )
x

(B)
2
1
1 x

(C)
2
1 x
x

(D)
2
1
x
x

AA [GATE-CE-2017-IITR]
136. The tangent to the curve represented by
y xInx
 is required to have 45 inclination
with the x-axis. The coordinates of the
tangent point would be
(A) (1,0) (B) (0,1)
(C) (1,1) (D)  
2, 2
AC [GATE-EY-2017-IITR]
137. Consider the function x
y e
 . The slope of
this function at 10
x  is :
(A) 0 (B) 10
(C) 10
e (D) 10
10e
A-2 [GATE-EY-2017-IITR]
138. The y intercept of the tangent of curve
3 2
1
y x x x
    at 1
x  is _____.
A0.40 to 0.45 [GATE-MN-2018-IITG]
139. Given 2
6
y x x
   , the value of (ln )
d
y
dx
at 2
x  is ___________.
AC T2.5.4 [GATE-EC-2019-IITM]
140. Consider a differentiable function f(x) on the
set of real numbers such that f(-1) = 0
and|f '(x)| 2
 . Given these conditions, which
one of the following inequalities is
necessarily true for all x [-2,2]
 ?
(A)
1
f(x) | x 1|
2
  (B)
1
f(x) | x |
2

(C) f(x) 2| x+1|
 (D) f(x) 2| x |

-----00000-----

75. TOPIC 2.4 – INTEGRAL & DIFFERENTIAL CALCULAS
www.targate.org Page 67
Answer :
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
B C B B D A D B 0.5 C
11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
B D C A C D D 1.0 * *
21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
* * * A * A 4 D C C
31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
B A A B D C C A A ?
41. 42. 43. 44. 45. 46. 47. 48. 49. 50.
D A C D B B * C C C
51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
B B 2.0 C B D D A A A
61. 62. 63. 64. 65. 66. 67. 68. 69. 70.
B A C B A C C 9 D 3
71. 72. 73. 74. 75. 76. 77. 78. 79. 80.
B B C B A C C 0.6 0.6 B
81. 82. 83. 84. 85. 86. 87. 88. 89. 90.
A B B A B B A C ? C
91. 92. 93. 94. 95. 96. 97. 98. 99. 100.
A B D D D A 10.0 * 1.01 2
101. 102. 103. 104. 105. 106. 107. 108. 109. 110.
* 20 6 C A A C A D B
111. 112. 113. 114. 115. 116. 117. 118. 119. 120.
C C D * D A D B D 64
121. 122. 123. 124. 125. 126. 127. 128. 129. 130.
A D C A C C A A D B
131. 132. 133. 134. 135. 136. 137. 138. 139. 140.
A A D D D A C -2 * C
19. 0.99-1.01
20. 3.13 to 3.15
21. 2.090 to 2.104
22. 1.65 to 1.75
23. 2.58
25.
2
2 cos
dy
x x
dx

47. 0.27 to 0.30
79. 0.43 to 0.45
98. 862-866
101. 0.70-0.85
114. 1.99 to 2.01
138. -2
139. 0.40-0.45

76. Page 68 TARGATE EDUCATION GATE-(EE/EC)
2.5
Series
Taylor Series Expansion
AA T2.6 [GATE-CE-2019-IITM]
1. For a small value of h, the Taylor series
expansion for  
f x h
 is
(A)
2 3
( ) '( ) "( ) '"( ) ...
2! 3!
h h
f x hf x f x f x
    
(B)
2 3
( ) '( ) "( ) '"( ) ...
2! 3!
h h
f x hf x f x f x
    
(C)
2 3
( ) '( ) "( ) '"( ) ...
2! 3
h h
f x hf x f x f x
    
(D)
2 3
( ) '( ) "( ) '"( ) ...
2 3
h h
f x hf x f x f x
    
D [GATE-EE-1998-IITD]
2. A discontinuous real function can be
expressed as
(A) Taylor’s series and
Fourier’s series
(B) Taylor’s series and
not by Fourier’s series
(C) Neither Taylor’s series nor Fourier’s
series
(D) Not by Taylor’s series, but by Fourier’s
series
A4.94 to 4.96 T2.6 [GATE-MN-2019-IITM]
3. For a function ( )
f x , (1) 5
f  and
'(1) 5
f   . Ignoring all higher order terms in
Taylor series, the value of the function at x =
1.01 (rounded off to two decimal places) is
_________.
A [GATE-EC-2008-IISc]
4. Which of the following functions would have
only odd powers of x in its Taylor series
expansion about the point 0?
x 
(A)
3
sin( )
x (B)
2
sin( )
x
(C)
3
cos( )
x (D)
2
cos( )
x
AB [GATE-CE-2012-IITD]
5. The infinite series
2 3 4
x x x
1 x ........
2! 3! 4!
     corresponds to
(A) sec x (B) x
e
(C) cos x (D) 2
1 sin x

AB [GATE-CE-1997-IITM]
6. For real values of x, cos(x) can be written in
one of the forms of a convergent series given
below :
(A)  
2 3
x x x
cos x 1 ........
1! 2! 3!
    
(B)  
2 4 6
x x x
cos x 1 ........
2! 4! 5!
    
(C)  
3 5 7
x x x
cos x x ........
3! 5! 7!
    
(D)  
2 2 3
x x x
cos x x ........
1! 2! 3!
    
AB [GATE-EC-2005-IITB]
7. In the Taylor series expansion of x
e sin x

about the point x, the coefficient of
 
2
x   is:
(A) e
(B) 0.5e
(C) e 1

 (D) e 1


AA [GATE-EC-2007-IITK]
8. For the function x
e
, the linear
approximation around x = 2 is :
(A)   2
3 x e

(B) 1 – x
(C)   2
3 2 2 1 2 x e
 
  
 
(D) 2
e
A [GATE-EC-2014-IITKGP]
9. The Taylor series expansion of 3 sin x + 2 cos
x is
(A)
3
2
2 3 ........
2
x
x x
   
(B)
3
2
2 3 ........
2
x
x x
   

77. TOPIC 2.5 – SERIES
www.targate.org Page 69
(C)
3
2
2 3 ........
2
x
x x
   
(D)
3
2
2 3 ........
2
x
x x
   
AB [GATE-ME-2010-IITG]
10. The infinite series
 
3 5 7
x x x
f x x .......
3! 5! 7!
     converges to
(A)  
cos x (B) sin(x)
(C) sinh(x) (D) x
e
D [GATE-CE-1998-IITD]
11. The Taylor’s series expansion of sinx is :
(A)
2 4
1
2! 4!
x x
  (B)
2 4
1
4! 4!
x x
 
(C)
2 5
3! 5!
x x
x   (D)
3 5
3! 5!
x x
x  
B [GATE-ME-2011-IITM]
12. A series expansion for the function sinθ is
______
(A)
2 4
1 ........
2! 4!
θ θ
  
(B)
3 6
........
3! 5!
θ θ
θ   
(C)
2 3
1 ........
2! 3!
θ θ
θ
   
(D)
3 5
.....
3! 5!
 
   
B
13. tan
4
x

 

 
 
when expanded in Taylor’s
series, gives
(A) 2 3
4
1 ....
3
x x x
   
(B) 2 3
8
1 2 2 ...
3
x x x
   
(C)
2 4
1 ...
2! 4!
x x
  
(D) None of these
D [GATE-CE-2000-IITKGP]
14. The Taylor expansion of sinx about / 6
x π

is given by
(A)
2 3
1 3 3
....
2 2 6 12 6
π π
x x
   
    
   
   
(B)
3 5 7
...
3! 5! 7!
x x x
x    
(C)
3 5
1 1
.......
6 3! 6 5! 6
π π π
x x x
     
     
     
     
(D)
2
1 3 1
...
2 2 6 4 6
π π
x x
   
    
   
   
D [GATE-EC-2009-IITR]
15. The Taylor series expansion of
sin x
x 

at
x π
 is given by
(A)
2
( )
1
3!
x π

    
(B)
2
( )
1
3!
x π

     
(C)
2
( )
1
3!
x π

    
(D)
2
( )
1
3!
x π

     
C [GATE-EC-2008-IISc]
16. In the Taylor series expansion of ex
about x =
2, the coefficient of (x – 2)4
is :
(A)
1
4!
(B)
4
2
4!
(C)
2
4!
e
(D)
4
4!
e
B [GATE-EE-1995-IITK]
17. The third term in the taylor’s series expansion
of x
e about ‘a’ would be _______
(A) ( )
a
e x a

(B) 2
( )
2
a
e
x a

(C)
2
a
e
(D) 3
( )
6
a
e
x a

C [GATE-EC-2007-IITK]
18. For | | 1,
x  coth( )
x can be approximated as
(A) x
(B) 2
x
(C)
1
x
(D) 2
1
x

78. ENGINEERING MATHEMATICS
Page 70 TARGATE EDUCATION GATE-(EE/EC)
AC [GATE-EC-2017-IITR]
19. Let  
2
x x
f x e 
 for real x. From among the
following, choose the Taylor series
approximation of f(x) around x = 0, which
includes all powers of x less than or equal to
3,
(A) 2 3
1 x x x
  
(B) 2 3
3
1 x x x
2
  
(C) 2 3
3 7
1 x x x
2 6
  
(D) 2 3
1 x 3x 7x
  
AB [GATE-PE-2018-IITG]
20. The Taylor series expansion of the function,
1
( )
1
f x
x



around 0
x  (up to 4th
order
term) is:
(A) 2 3 4
1 x x x x
   
(B) 2 3 4
1 x x x x
    
(C) 2 3 4
1 x x x x
    
(D) 2 3 4
1 2 3 4
x x x x
    
A–0.01 – 0.01 [GATE-EC-2018-IITG]
21. Taylor series expansion of
2
2
0
( )
t
x
f x e dt
 
 
 
 
 
around x = 0 has the form
2
0 1 2
( ) ...
f x a a x a x
   
The coefficient 2
a (correct to two decimal
places) is equal to _______.
**********
Convergence Test
AA [GATE-CE-1999-IITB]
22. The infinite series
 
 
2
n 1
n!
2n !



(A) Converges
(B) Diverges
(C) Is unstable
(D) Oscillates
AD T2.6 [GATE-XE-2019-IITM]
23. For the series 1
( 1)
,
2
n
n
n
x
x
n



    
 ,
which of the following statements is NOT
correct ?
(A) The series converges at x = -3
(B) The series converges at x = -1
(C) The series converges at x = 0
(D) The series converges at x = 1
D [GATE-EC-2014-IITKGP]
24. The series
0
1
( 1)( 2)...1
n
n n n


 

converges to
(A) 2 In 2 (B) 2
(C) 2 (D) e
B [GATE-CE-1998-IITD]
25. The infinite sires
1 1
1
2 3
       
(A) Converges
(B) Diverges
(C) Oscillates
(D) Unstable
B
26. The infinite sires
1 1 1 1
1
2 3 4 5
        
(A) Converges (B) Diverges
(C) Oscillates (D) Unstable
B [GATE-IN-2011-IITM]
27. The series 2
0
1
( 1)
4
α
m
m
m
x


 converges for
(A) 2 2
x
   (B) 1 3
x
  
(C) 3 1
x
   (D) 3
x
AC [GATE-AG-2018-IITG]
28. The type of the sequence
3
1
n
n
a
n
 
  

 
is
(A) oscillatory (B) bounded
(C) converging (D) diverging
AA
29. The Infinite Series
3
5
1 3
n
n
n

 

(A) Converges (B) Diverges
(C) Is Unstable (D) Oscillates
AA
30. For what value or p does the 3
1 2
p
n
n
n

 

converge ?
(A) 2
p  (B) 2
p 
(C) 1
p  (D) Always
diverges

79. TOPIC 2.5 – SERIES
www.targate.org Page 71
AA
31.
1
!( 1)!
(3 )!
n
n n
n




(A) Converges (B) Diverges
(C) Is Unstable (D) Oscillates
AB
32. Determine the range of x for convergence of
the series
3 3
4
0 1
n
n
n x
n

 

(A) | | 1
x  (B) | | 1
x 
(C) 2 3
x
  (D) 3
x 
AB
33. The Infinite Series
1
5
3
n
n
n n





(A) Converges (B) Diverges
(C) Is Unstable (D) Oscillates
**********
Miscellaneous
A25250.0 [GATE-MN-2018-IITG]
34. Sum of the series 5, 10, 15, ……….., 500 is
____________.
AB [GATE-CE-2018-IITG]
35. 2
times
...
n
a a a a a b
    


 and
2
times
...
m
b b b b ab
    


 , where a, b, n and m
are natural numbers. What is the value of
times times
... ...
n m
m m m m n n n n
  
       
  
  
  



 

 ?
(A) 2 2
2a b (B) 4 4
a b
(C) ( )
ab a b
 (D) 2 2
a b

AB T2.6 [GATE-BT-2019-IITM]
36. Which of the following are geometric series ?
P. 1, 6, 11, 16, 21, 26, ...
Q. 9, 6, 3, 0, -3, -6, ...
R. 1, 3, 9, 27, 81, ...
S. 4, -8, 16, -32, 64, ...
(A) P and Q only
(B) R and S only
(C) Q and S only
(D) P, Q and R only
AC [GATE-CE-2018-IITG]
37. Consider a sequence of numbers
1 2 3
, , ,..., n
a a a a where
1 1
2
n
a
n n
 

, for each
integer 0
n  . What is the sum of the first 50
terms ?
(A)
1 1
1
2 50
 
 
 
 
(B)
1 1
1
2 50
 
 
 
 
(C)
1 1 1
1
2 50 52
   
  
   
   
(D)
1 1
1
51 52
 
 
 
 
A0.32 to 0.32 [GATE-BT-2018-IITG]
38. If 2 3
1 ... 1.5
r r r
      , then,
2 3
1 2 3 4 ...
r r r
     = (up to two decimal
places) _____.
AD [GATE-CS-2018-IITG]
39. Which one of the following is a closed form
expression for the generating function of the
sequence { }
n
a , where 2 3
n
a n
  for all
0,1,2,...
n  ?
(A) 2
3
(1 )
x

(B) 2
3
(1 )
x
x

(C) 2
2
(1 )
x
x


(D) 2
3
(1 )
x
x


-----00000-----

80. ENGINEERING MATHEMATICS
Page 72 TARGATE EDUCATION GATE-(EE/EC)
Answer :
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
A D * A B B B A A B
11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
D B B D D C B C C B
21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
* A D D B B B C A A
31. 32. 33. 34. 35. 36. 37. 38. 39.
A B B * B B C * D
3. 4.94-4.96
21. -0.01
34. 25250.0
38. 0.32

81. www.targate.org Page 73
03
Differential Equations
Linearity/Order/Degree of DE
T1.1 AB [GATE-EC-2009-IITR]
1. The order of differential equation
3
2
4
2

 
  
 
 
x
d y dy
y e
dx
dx
is :
(A) 1 (B) 2
(C) 3 (D) 4
AB [GATE-EC-2005-IITB]
2. The following differential equation has
2
2
2
2
3 4 2
d y dy
y x
dx dx
 
   
 
 
(A) degree = 2, order = 1
(B) degree = 1, order = 2
(C) degree = 4, order = 3
(D) degree = 2, order = 3
AB [GATE-CE-2007-IITK]
3. The degree of differential equation
2
3
2
d x
2x 0
dt
  is:
(A) 0 (B) 1
(C) 2 (D) 3
A [GATE-CE-2010-IITG]
4. The order and degree of a differential
equation
3
3
2
3
4 0
d y dy
y
dx dx
 
  
 
 
are
respectively
(A) 3 and 2 (B) 2 and 3
(C) 3 and 3 (D) 3 and 1
B [GATE-ME-2007-IITK]
5. The differential equation
4 2
4 2
0
d y d y
P ky
dx dx
   is
(A) Linear of Fourth order
(B) Non – Linear of fourth order
(C) Non – Homogeneous
(D) Linear and Fourth degree
D [GATE-ME-1999-IITB]
6. The equation
2
2 8
2
( 4 ) 8
d y dy
x x y x
dx
dx
     is a
(A) partial differential equation
(B) non-linear differential equation
(C) non-homogeneous differential equation
(D) ordinary differential equation
AD [GATE-ME-2013-IITB]
7. The partial differential equation
2
2
u u u
u
dt dx dx
  
  is a
(A) Linear equation of order 2
(B) Non- linear equation of order 1
(C) Linear equation of order 1
(D) Non-linear equation of order 2
AB [GATE-ME-2010-IITG]
8. The Blasius equation,
3 2
3 2
d f f d f
0
d 2 d
 
 
, is a
(A) Second order nonlinear ordinary
differential equation
(B) Third order nonlinear ordinary
differential equation
(C) Third order linear ordinary linear
equation
(D) Mixed order nonlinear ordinary
differential equation
AC [GATE-ME-1995-IITK]
9. The differential equation
 
3 5 3
y" x sinx y' y cosx
   is :
(A) Homogeneous
(B) Non-linear
(C) Second order linear
(D) Non-homogeneous with constant
coeffiecients

82. ENGINEERING MATHEMATICS
Page 74 TARGATE EDUCATION GATE-(EE/EC)
AC [GATE-IN-2005-IITB]
10. The differential equation
3 2
2 2
2
2
dy d y
1 C
dx dx
   
 
 
   
 
 
   
 
is of
(A) 2nd
order and 3rd
degree
(B) 3rd
order and 2nd
degree
(C) 2nd
order and 2nd
degree
(D) 3rd
order and 3rd
degree
B [GATE-ME-1993-IITB
11. The differential
2
2
sin 0
d y dy
y
dx
dx
   is
(A) linear (B) non – linear
(C) homogeneous (D) of degree two
A [GATE-EC-1994-IITKGP]
12. Match each of the items A, B, C with an
appropriate item from 1, 2, 3, 4 and 5
(A)
2
1 2 3 4
2
d y dy
a a y a y a
dx
dx
  
(B)
3
1 2 3
3
d y
a a y a
dx
 
(C)
2
2
1 2 3
2
0
d y dy
a a x a x y
dx
dx
  
(1) Non – linear differential equation
(2) Linear differential equation with
constant coefficients
(3) Linear homogeneous differential
equation
(4) Non – linear homogeneous differential
equation
(5) Non – linear first order differential
equation
(A) A – 1, B – 2, C – 3
(B) A – 3, B – 4, C - 2
(C) A – 2, B – 5, C – 3
(D) A – 3, B – 1, C – 2
A1.0 [GATE-MN-2018-IITG]
13. The degree of the differential equation
2
3
2
2 0
d x
x
dt
  is ____________.
**********
First Order & Degree DE
Lebnitz Linear Form
AB [GATE-TF-2016-IISc]
14. The integrating factor of
 
2
2cos y 4x dx xsin ydy 0
   is :
(A) -x (B) x
(C) 2
x (D) - 2
x
AA [GATE-EE-2017-IITR]
15. Consider the differential
equation   
2 dy
t 81 5ty sin t
dt
   with
y(1) 2
  . There exists a unique solution for
this differential equation when t belongs to
the interval
(A) (-2, 2) (B) (-10, 10)
(C) (-10, 2) (D) (0, 10)
AD [GATE-EC-2012-IITD]
16. With initial condition x(1) = 0.5, the solution
of the differential equation,
dx
t x t
dt
  is
(A)
1
x t
2
  (B)
2 1
x t
2
 
(C)
2
t
x
2
 (C)
t
x
2

AD [GATE-CE-2014-IITKGP]
17. The integrating factor for the differential
equation 1
k t
2 1 0
dP
k P k L e
dt

  is
(A) 1
k t
e
(B) 2
k t
e
(C) 1
k t
e (D) 2
k t
e
AB [GATE-IN-2010-IITG]
18. Consider the differential equation
x
dy
y e
dx
  with y(0) = 1. The value of y(1)
is :
(A) 1
e e
 (B)  
1
1
e e
2


(C)  
1
1
e e
2

 (D)  
1
2 e e

A [GATE-EE-1994-IITKGP]
19. The solution of the differential equation
dy y
x
dx x
  with the condition that 1
y  at
x = 1 is :
(A)
2
2
3 3
 
x
y
x
(B)
1
2 2
x
y
x
 
(C)
2
3 3
x
y   (D)
2
2
3 3
x
y
x
 
B [GATE-ME-2006-IITKGP]
20. The solution of the differential equation
2
2 x
dy
xy e
dx

  with (0) 1
y  is :

83. TOPIC 3 – DIFFERENTIAL EQUATIONS
www.targate.org Page 75
(A)
2
(1 ) x
x e
 (B)
2
(1 ) x
x e

(C)
2
(1 ) x
x e
 (D)
2
(1 ) x
x e

D [GATE-ME-2005-IITB]
21. If 2 2 ln
2
dy x
x xy
dx x
 
 
 
 
and y(1) = 0 then
what is y(e)?
(A) e (B) 1
(C)
1
e
(D) 2
1
e
A [GATE-CE-2005-IITB]
22. Transformation to linear form by substituting
v = 1 n
y 
of the equation
( ) ( ) , 0
n
dy
P t y q t y n
dt
   will be
(A) (1 ) (1 )
dv
n pv n q
dt
   
(B) (1 ) (1 )
dv
n pv n q
dt
   
(C) (1 ) (1 )
dv
n pv n q
dt
   
(D) (1 ) (1 )
dv
n pv n q
dt
   
A0.51-0.53 [GATE-PH-2017-IITR]
23. Consider the differential equation
/ tan( ) cos( )
dy dx y x x
  . If (0) 0
y  ,
( / 3)
y  is _______. (up to two decimal
places).
A10.50 to 12.50 [GATE-PE-2018-IITG]
24. The variation of the amount of salt in a tank
with time is given by,
0.025 20
dx
x
dt
  ,
where, x is the amount of salt in kg and t is
the time in minutes. Given that there is no salt
in the tank initially, the time at which the
amount of salt increases to 200 kg is
__________ minutes. (rounded-off to two
decimal places)
AA [GATE-PE-2018-IITG]
25. Which one of the following is the integrating
factor (IF) for the differential equation,
2
(cos ) cos
dy
x y x
dx
  ?
(A) tan x
e (B) cos x
e
(C) tan x
e
(D) sin x
e
AC [GATE-ME-2018-IITG]
26. If y is the solution of the differential equation
3 3
0
dy
y x
dx
  , (0) 1
y  , the value of ( 1)
y 
is
(A) −2 (B) −1
(C) 0 (D) 1
AC [GATE-EC-2015-IITK]
27. Consider the different equation
dx
10 0.2x
dt
  with initial condition
x(0) 1
 . The response x(t) for t>0 is :
(A) 0.2t
2 e

(B) 0.2t
2 e

(C) 0.2 t
50 49e

(D) 0.2 t
50 49e

AD [GATE-EC-2014-IITKGP]
28. A system described by a linear, constant
coefficient, ordinary, first order differential
equation has exact solution given by y(t) for t
> 0, when the forcing function is x(t) and the
initial condition is y(0). If one wishes to
modify the system so that the solution
becomes -2y(t) for t > 0, we need to
(A) Change the initial condition to –y(0) and
the forcing function to x(t)
(B) Change the initial condition to 2y(0) and
the forcing function to –x(t)
(C) Change the initial condition to  
j 2 y 0
and the forcing function to  
j 2 x t
(D) Change the initial condition to -2y(0)
and the forcing function to -2x(t)
AD [GATE-ME-2009-IITR]
29. The solution of
4
dy
x y x
dx
  with the
condition  
6
y 1
5
 is :
(A)
4
x 1
y
5 x
  (B)
4
4x 4
y
5 5x
 
(C)
4
x
y 1
5
  (D)
5
x
y 1
5
 
AB T3.3 [GATE-BT-2019-IITM]
30. What is the solution of the differential
equation
dy x
dx y
 , with the initial condition, at
0
x  , 1
y  ?
(A)
2 2
1
x y
  (B)
2 2
1
y x
 
(C)
2 2
2 1
y x
  (D)
2 2
0
x y
 
AD T3.3 [GATE-CE-2019-IITM]
31. An ordinary differential equation is giaven
below.
( ln )
dy
x x y
dx
 

 
 

84. ENGINEERING MATHEMATICS
Page 76 TARGATE EDUCATION GATE-(EE/EC)
The solution for the above equation is
(Note : K denotes a constant in the options)
(A) ln
y Kx x
 (B)
x
y Kxe

(C)
x
y Kxe
 (D) ln
y K x

AD T3.3 [GATE-CH-2019-IITM]
32. The solution of the ordinary differential
equaqtion 3 1
dy
y
dx
  , subject to the initial
condition 1
y  at 0
x  , is
(A) /3
1
(1 2 )
3
x
e
 (B) /3
1
(5 2 )
3
x
e

(C) 3
1
(5 2 )
3
x
e
 (D) 3
1
(1 2 )
3
x
e

AB T3.3 [GATE-ME-2019-IITM]
33. The differential equation 4 5
dy
y
dx
  is valid
in the domain 0 1
x
  with (0) 2.25
y  . The
solution of the differential equation is
(A)
4
5
x
y e
  (B)
4
1.25
x
y e
 
(C)
4
5
x
y e
  (D)
4
1.25
x
y e
 
Variable Separable Form
T2.2 AD [GATE-PH-2016-IISc]
34. Consider the linear differential equation
dy
xy
dx
 . If y = 2 and x = 0, then the value of
y at x = 2 is given by
(A) 2
e
(B) 2
2e
(C) 2
e (D) 2 2
e
AC T3.2 [GATE-ME-2019-IITM]
35. For the equation 2
7 0
dy
x y
dx
  , if
(0) 3/ 7
y  , then the value of (1)
y is
(A) 7/3
7
3
e
(B) 3/7
7
3
e
(C) 7/3
3
7
e
(D) 3/7
3
7
e
A0.5 T3.2 [GATE-AE-2019-IITM]
36. The curve ( )
y f x
 is such that its slope is
equal to
2
y for all real x. If the curve passes
through (1, –1), the value of y at x = –2 is
____(round off to 1 decimal place).
AC [GATE-EC-2011-IITM]
37. The solution of differential equation
dy
ky,
dx
 y(0) = c is :
(A) ky
x ce
 (B) cy
x ke

(C) kx
y ce
 (D)
kx
y ce 

AB [GATE-EC-2008-IISc]
38. Which of the following is a solution to the
differential equation,
     
d
x t 3x t 0,x 0 2?
dt
  
(A)   t
x t 3e
 (B)   3t
x t 2e

(C)   2
3
x t t
2
 (D)   2
x t 3t

AA [GATE-EE-2005-IITB]
39. The solution of first order differential
equation x(t) = 3x(t),   0
x 0 x
 is:
(A)   3 t
0
x t x e 
 (B)   3
0
x t x e

(C)  
1
3
0
x t x e

 (D)   1
0
x t x e

AB [GATE-ME-2014-IITKGP]
40. The solution of the initial value problem
 
dy
2xy,y 0 2
dx
   is :
(A)
3
x
1 e
 (B)
2
x
2 e 
(C)
2
x
1 e
 (D)
2
x
2e
AD [GATE-CE-2011-IITM]
41. The solution of the ordinary differential
equation
dy
2y 0
dx
  for the boundary
condition y = 5 at x = 1 is :
(A)
2x
y e
 (B) 2 x
y 2e

(C) 2x
y 109.5e
 (D) 2x
y 36.95e

B [GATE-ME-1994-IITKGP]
42. For the differential equation 5 0
dy
y
dt
 
with (0) 1,
y  the general solution is :
(A) 5t
e (B)
5t
e
(C) 5
5 t
e
(D)
5 t
e 
AC [GATE-CE-2018-IITG]
43. The solution of the equation 0
dy
x y
dx
 
passing through the point (1,1) is
(A) x (B) 2
x
(C) 1
x
(D) 2
x
AA [GATE-EC-2018-IITG]
44. A curve passes through the point
( 1, 0)
x y
  and satisfies the differential

85. TOPIC 3 – DIFFERENTIAL EQUATIONS
www.targate.org Page 77
equation
2 2
2
dy x y y
dx y x

  . The equation that
describes the curve is
(A)
2
2
ln 1 1
y
x
x
 
  
 
 
(B)
2
2
1
ln 1 1
2
y
x
x
 
  
 
 
(C) ln 1 1
y
x
x
 
  
 
 
(D)
1
ln 1 1
2
y
x
x
 
  
 
 
C [GATE-ME-2007-IITK]
45. The solution of
2
dy
y
dx
 with initial value
y(0) = 1 is bounded in the internal is
(A) x
    (B) 1
x
  
(C) 1, 1
x x
  (D) 2 2
x
  
C [GATE-CE-1999-IITB]
46. If C is a constant, then the solution of
2
1
dy
y
dx
  is :
(A) sin( )
y x c
  (B) cos( )
y x c
 
(C) tan( )
y x c
  (D)
x
y e c
 
D [GATE-CE-2007-IITK]
47. The solution for the differential equation
2
dy
x y
dx
 with the condition that y = 1 at x =
0 is :
(A)
1
2x
y e
 (B)
3
ln( ) 4
3
x
y  
(C)
2
ln( )
2
x
y  (D)
3
3
x
y e

A [GATE-CE-2009-IITR]
48. Solution of the differential equation
3 2 0
dy
y x
dx
  represents a family of
(A) ellipses (B) circles
(C) parabolas (D) hyperbolas
AD [GATE-PI-2015-IITK]
49. The solution to 6yy'–25x=0 represents a
(A) family of circles
(B) family of ellipses
(C) family of parabolas
(D) family of hyperbolas
A [GATE-EC-2009-IITR]
50. Match each differential equation in Group I to
its family of solution curves from Group II.
Group I Group II
P: dy y
dx x

(1) Circles
Q: dy y
dx x


(2) Straight lines
R: dy x
dx y

(3) Hyperbolas
S: dy x
dx y


(A) P-2, Q-3, R-3, S-1
(B) P-1, Q-3, R-2, S-1
(C) P-2,Q-1,R-3, S-3
(D) P-3, Q-2, R-1, S-2
A [GATE-EE-2011-IITM]
51. With K as constant, the possible solution for
the first order differential equation 3x
dy
e
dx


is :
(A) 3
1
3
x
e K


 (B) 3
1
( 1)
3
x
e K
 
(C) 3
3 x
e K

  (D) kx
y Ce

AA [GATE-ME-2003-IITM]
52. The solution of the differential equation
2
dy
y 0
dx
  is :
(A)
1
y
x c


(B)
3
x
y c
3

 
(C) x
ce
(D) Unsolvable as equation is non-linear
AD [GATE-CE-2008-IISc]
53. Solution of dy x
dx y
  at x = 1 and y 3
 is :
(A) 2
x y 2
  (B) 2
x y 4
 
(C) 2 2
x y 2
   (D) 2 2
x y 4
 
AA [GATE-IN-2008-IISc]
54. Consider the differential equation
2
dy
1 y
dx
  . Which one of the following can
be a particular solution of this differential
equation?

86. ENGINEERING MATHEMATICS
Page 78 TARGATE EDUCATION GATE-(EE/EC)
(A)  
y tan x 3
 
(B) y tan x 3
 
(C)  
x tan y 3
 
(D) x tan y 3
 
D [GATE-ME-2011-IITM]
55. Consider the differential equation
2
(1 ) .
dy
y x
dx
  The general solution with
constant “C” is
(A)
2
tan
2
x
y C
 
 
 
 
(B) 2
tan
2
x
y C
 
 
 
 
(C) 2
tan
2
x
y C
 
 
 
 
(D)
2
tan
2
x
y C
 
 
 
 
AC [GATE-EC-2015-IITK]
56. The general solution of the differential
equation
dy 1 cos2y
dx 1 cos2x



is :
(A) tan y–cot x = c (c is a constant)
(B) tan x–cot y = c (c is a constant)
(C) tan y+cot x = c (c is a constant)
(D) tan x+cot y = c (c is a constant)
AC [GATE-ME-2015-IITK]
57. Consider the following differential equation
5
dy
y
dt
  ; initial condition: y = 2 at t = 0.
The value of y at t = 3 is
(A) –5e-10
(B) 2e-10
(C) 2e-15
(D) -15e2
A6 [GATE-TF-2018-IITG]
58. If ( )
y x is the solution of the differential
equation ' 8
yy x
 , (0) 2
y  , then the
absolute value of (2)
y is _________.
AD [GATE-MN-2018-IITG]
59. If c is a constant, the solution of the
differential equation 4 9 0
dy
y x
dx
  is
(A)
2 2
81 16
x y
c
  (B)
2 2
16 81
x y
c
 
(C)
2 2
9 4
x y
c
  (D)
2 2
4 9
x y
c
 
AD [GATE-CE-2017-IITR]
60. The solution of the equation
dQ
Q 1
dt
  with
Q = 0at t = 0 is
(A)   t
Q t e 1

  (B)   t
Q t 1 e
 
(C)   t
Q t 1 e
  (D)   t
Q t 1 e
 
Exact Differential Equation Form
T 2.3 AB [GATE-CE-1997-IITM]
61. For the differential equation
( , ) ( , ) 0
dy
f x y g x y
dx
  to be exact is
(A) f g
y x
 

 
(B) f g
x y
 

 
(C) f g
 (D)
2 2
2 2
f g
x y
 

 
C [GATE-ME-1994-IITKGP]
62. The necessary & sufficient for the differential
equation of the form M(x, y)dx + N(x, y) dy
= 0 to be exact is
(A) M = N (B)
M N
x y
 

 
(C)
M N
y x
 

 
(D)
2 2
2 2
M N
x y
 

 
AC [GATE-CE-1994-IITKGP]
63. The necessary and sufficient condition for the
differential equation of the form M(x, y)
 
dx N x, y dy 0
  to be exact is:
(A) Linear (B) Non-linear
(C) Homogeneous (D) of degree two
AA [GATE-AG-2016-IISc]
64. The general solution of the differential
equation 3 2 2 2
x y y
dy
e x e
dx
 
  is
(A) 2 3 3
1 1
( )
2 3
y x
C e e x
  
(B) 2 3 2
1
( )
3
y x
C e e x
  
(C) 2 3 2
1 1
( )
3 2
y x
C e e x
  
(D) 2 3 3
1
( )
3
y x
C e e x
  
MISCELLANEOUS
T2.4 AA [GATE-MA-2016-IISc]
65. Let y be the solution of
| |,
y y x x
   

87. TOPIC 3 – DIFFERENTIAL EQUATIONS
www.targate.org Page 79
 
1 0
y  
Then y(1) is equal to
(A) 2
2 2
e e
 (B) 2
2
2e
e

(C)
2
2
e
 (D) 2 2e

AA T3.4 [GATE-TF-2019-IITM]
66. One of the points which lies on the solution
curve of the following differential equation
2 2
2 ( ) 0
xydx x y dy
  
with the initial condition ( ) 1
y t  is
(A) (–1, 1) (B) (0, 0)
(C) (0, 1) (D) (2, 1)
AB T3.4 [GATE-IN-2019-IITM]
67. The curve y = f(x) is such that the tangent to
the curve at every point (x, y) has a Y-axis
intercept c, given by c = –y. then, f(x) is
proportional to
(A) x–1
(B) x2
(C) x3
(D) x4
AC T3.4 [GATE-EC-2019-IITM]
68. The families of curves represented by the
solution of the equation
n
dy x
dx y
 
  
 
For
1
n   and 1
n   , respectively, are
(A) Circles and Hyperbolas
(B) Parabolas and Circles
(C) Hyperbolas and Circles
(D) Hyperbolas and Parabolas
AD [GATE-EC-2017-IITR]
69. Which one of the following is the general
solution of the first order differential equation
 
2
dy
x y 1
dx
   ,
Where x, y are real?
(A) y = 1+ x +  
1
tan x c

 , where c is a
constant
(B)  
y 1 x tan x c ,
    where c is a
constant
(C)  
1
y 1 x tan x c ,

    where c is a
constant
(D)  
y 1 x tan x c ,
    where c is a
constant
AD [GATE-ME-2014-IITKGP]
70. The general solution of the differential
equation  
dy
cos x y ,
dx
  with c as a constant,
is :
(A)  
y sin x y x c
   
(B) x y
tan y c
2

 
 
 
 
(C) x y
cos x c
2

 
 
 
 
(D) x y
tan x c
2

 
 
 
 
AC T2.5.1 [GATE-MN-2019-IITM]
71. If ( )
f x is a polynomial function that passes
through origin, and ( ) '( )
g x f x
 , then
(A) '( ) ( )
g a f a

(B) '( ) '( )
g a f a

(C)
0
( ) ( )
a
g x dx f a


(D)
0
( ) ( )
a
f x dx g a


AA [GATE-ME-2014-IITKGP]
72. The matrix form of the linear system
dx
3x 5y
dt
  and
dy
4x 8y
dt
  is,
(A)
x 3 5 x
d
y 4 8 y
dt

     

     
     
(B)
x 3 8 x
d
y 4 5 y
dt
     

     

     
(C)
x 4 5 x
d
y 3 8 y
dt

     

     
     
(D)
x 4 8 x
d
y 3 5 y
dt
     

     

     
ATRUE [GATE-ME-1995-IITK]
73. A differential equation of the form
 
dx
f x, y
dy
 is homogeneous if the function
 
f x, y depends only the ratio
y
x
(or) x
y
(True/False)?
AA [GATE-BT-2017-IITR]
74. Growth of a microbe in a test tube is modeled
as dX X
rX 1
dt K
 
 
 
 
, where, X is the
biomass, r is the growth rate, and K is the
carrying capacity of the

88. ENGINEERING MATHEMATICS
Page 80 TARGATE EDUCATION GATE-(EE/EC)
environment  
r 0;K 0
  . If the value of
starting biomass is
K
,
100
which one of the
following graphs qualitatively represents the
growth dynamics:
(A)
(B)
(C)
(D)
A0 [GATE-MA-2017-IITR]
75. If ( )
x t and ( )
y t are the solutions of the
system
dx
y
dt
 and
dy
x
dt
  with the initial
conditions (0) 1
x  and (0) 1
y  , then
( / 2) ( / 2)
x y
   equals ________.
AC [GATE-CH-2018-IITG]
76. Consider the following two equations :
0
dx
x y
dt
  
0
dy
x y
dt
  
The above set of equations is represented by
(A)
2
2
0
d y dy
y
dt dt
  
(B)
2
2
0
d x dx
y
dt dt
  
(C)
2
2
0
d x dy
y
dt dt
  
(D)
2
2
0
d x dx
y
dt dt
  
AC [GATE-MA-2018-IITG]
77. The general solution of the differential
equation
2 2
'
xy y x y
   for 0
x 
is given by (with an arbitrary positive
constant k)
(A) 2 2 2
ky x x y
  
(B) 2 2 2
kx x x y
  
(C) 2 2 2
kx y x y
  
(D) 2 2 2
ky y x y
  
**********
Higher Order DE
T3.1 -1.05- -0.95 [GATE-ME-2016-IISc]
78. If  
y f x
 satisfies the boundary value
problem 9 0
y y
  ,  
0 0
y  ,
 
/ 2 2
y   , then  
/ 4
y  is ________.
AB T3.2 [GATE-AG-2019-IITM]
79. General solution to the differential equation
" 4 ' 5 0
y y y
   is
(A)
2
( cos sin )
x
e a x b x

(B)
2
( cos sin )
x
e a x b x


(C) ( cos2 sin2 )
x
e a x b x

(D) ( cos2 sin2 )
x
e a x b x


AB T3.2 [GATE-PH-2019-IITM]
80. For the differential equation
2
2 2
( 1) 0
d y y
n n
dx x
   , where n is a constant,
the product of its two independent solutions
is
(A)
1
x
(B) x
(C) n
x (D) 1
1
n
x 
AD [GATE-CH-2016-IISc]
81. What is the solutions for the second order
differential equation
2
2
0
d y
y
dx
  , with the
intial conditions 0
0
| 5and 10
x
x
dy
y
dx


  ?

89. TOPIC 3 – DIFFERENTIAL EQUATIONS
www.targate.org Page 81
(A) 5 10sin
y x
 
(B) 5cos 5sin
y x x
 
(C) 5cos 10
y x x
 
(D) 5cos 10sin
y x x
 
AD [GATE-MT-2016-IISc]
82. The solution of the differential equation
2
2
d y dy
dx dx
 is
(A)
x
y e C
  (B)
x
y e C

 
(C) 1 2
x
y C e C

  (D) 1 2
x
y Ce C
 
[where C, 1
C and 2
C are constants]
AB T3.2 [GATE-PI-2019-IITM]
83. If roots of the auxiliary equation of
2
2
0
d y dy
a by
dx dx
   are real and equal, the
general solution of the differential equation is
(A)
/2 /2
1 2
ax a x
y c e c e

 
(B)
/2
1 2
( ) a x
y c c x e
 
(C)
/2
1 2
( ln ) ax
y c c x e
 
(D)
/2
1 2
( cos sin ) ax
y c x c x e
 
AD T3.2 [GATE-PE-2019-IITM]
84. The general solution of the differential
equation
2
2
2 0
d y dy
y
dx dx
   is (here 1
C and
2
C are arbitrary constants)
(A) 1 2
x x
y C e C e
 
(B)
2
1 2
x x
y C xe C xe
 
(C) 1 2
x x
y C e C xe
 
(D) 1 2
x x
y C e C xe
 
AA [GATE-AE-2016-IISc]
85. Consider a second order linear ordinary
differential equation
2
2
d y dy
4 4y 0
dx dx
   ,
with the boundary conditions
 
x 0
dy
y 0 1; 1
dx 
  . The value of y at x = 1 is
(A) 0 (B) 1
(C) e (D) 2
e
A14.55-14.75 [GATE-BT-2016-IISc]
86.
2
2
0
d y
y
dx
  . The initial conditions for this
second order homogeneous differential
equation are (0 ) 1
y  and 3
dy
dx
 at 0
x 
The value of y when x = 2 is ___________.
AA [GATE-EC-2016-IISc]
87. The particular solution of the initial value
problem given below is
2
2
12 36 0
d y dy
y
dx dx
   with (0) 3
y  and
0
36
x
dy
dx 

(A) (3 – 18x) e−6x
(B) (3 + 25x) e−6x
(C) (3 + 20x) e−6x
(D) (3 − 12x) e−6x
AA [GATE-CE-2016-IISc]
88. The respective expressions for
complimentary function and particular
integral part of the solution of the differential
equation
4 2
2
4 2
3 108
d y d y
x
dx dx
  are
(A) 1 2 3 4
sin 3 cos 3
c c x c x c x
 
  
 
and
4 2
3 12
x x c
 
 
 
(B) 2 3 4
sin 3 cos 3
c x c x c x
 
 
 
and
4 2
5 12
x x c
 
 
 
(C) 1 3 4
sin 3 cos 3
c c x c x
 
 
 
and
4 2
3 12
x x c
 
 
 
(D) 1 2 3 4
sin 3 cos 3
c c x c x c x
 
  
 
and
4 2
5 12
x x c
 
 
 
AB [GATE-EE-2016-IISc]
89. A function y(t), such that y(0) = 1 and y(1) =
3e-1
, is a solution of the differential equation
2
2
2 0
d y dy
y
dt dt
   . Then (2)
y is :
(A) 1
5e
(B) 2
5e
(C) 1
7e
(D) 2
7e
AD [GATE-EC-2007-IITK]
90. The solution for differential equation
2
2
2
2
d y
k y y
dx
  under the boundary conditions
1
y y
 at x = 0
2
y y
 at x = 
Where k, y1 and y2 are constants, is :
(A)  
1 2 2
2
x
y y y exp y
k

 
  
 
 
(B)  
2 1 1
x
y y y exp y
k

 
  
 
 

90. ENGINEERING MATHEMATICS
Page 82 TARGATE EDUCATION GATE-(EE/EC)
(C)  
1 2 1
x
y y y sinh y
k
 
  
 
 
(D) 1 2 2
x
y (y y )sinh y
k

 
  
 
 
C [GATE-IN-2011-IITM]
91. Consider the differential equation
.. .
0
  
y y y
with boundary conditions (0) 1

y
, (1) 0

y .The value of ( 2 )
y is
(A) – 1 (B) - e 1

(C) 2
e 
 (D) 2
e
AC [GATE-MA-2017-IITR]
92. If
2x 2x
y 3e e x

   is the solution of the
initial value problem
 
2
2
d y
y 4 x,y 0 4
dx
     and  
dy
0 1
dx
 ,
where , ,
  then
(A) 3 and 4
   
(B) 1 and 2
   
(C) 3 and 4
    
(D) 1 and 2
    
A93-95 [GATE-ME-2017-IITR]
93. Consider the differential equation
3 "( ) 27 ( ) 0
y x y x
  with initial conditions
(0) 0
y  and '(0) 2000
y  . The value of y at
x = 1 is ______.
AA [GATE-EE-2016-IISc]
94. The solution of the differential equation, for t
> 0, 0, "( ) 2 '( ) ( ) 0
t y t y t y t
    with initial
conditions y(0) = 0 and y’(0) = 1, is (u(t)
denotes the unit step function),
(A) ( )
t
te u t

(B) ( ) ( )
t t
e te u t
 

(C) ( ) ( )
t t
e te u t
 
 
(D) ( )
t
e u t

A7.0-7.5 [GATE-EE-2016-IISc]
95. Let y(x) be the solution
of the differential equation
2
2
4 4 0
d y dy
y
dx dx
  
with initial conditions (0) 0
y 
and
0
1
x
dy
dx 
 . Then the value of y(1) is ____.
ATRUE [GATE-EC-1994-IITKGP]
96. 2x
y e
 is a solution of differential equation
y" y' 2y 0.
   (True/False)
AC [GATE-ME-1996-IISc]
97. The solution of the differential equation
y'' 3y' 2y 0
   is of the form
(A) x 2x
1 2
C e C e

(B) x 3x
1 2
C e C e


(C) x 2x
1 2
C e C e
 

(D) 2x x
1 2
C e C 2
 

[GATE-CE-1998-IITD]
98. Solve
4
4
d y
y 15cos2x
dx
  .
 
x x
1 2 3 4
y=C e +C e + C cosx+C sinx +cos2x
ANS:
AC [GATE-IN-2007-IITK]
99. The boundary-value problem y" y 0,
  
   
y 0 y 0
   will have non-zero solutions
if and only if the values of  are
(A) 0, +1, +2 (B) 1, 2, 3.....
(C) 1, 4, 9 (D) 1, 9, 25
AD [GATE-EC-2010-IITG]
100. A function n(x) satisfies the differential
equation
   
2
2 2
d n x n x
0
dx L
  where L is a
constant. The boundary conditions are: n(0) =
K and  
n 0
  . The solution to this equation
is :
(A)
x
K exp
L
 
 
 
(B)  
x
n x K exp
L

 
  
 
(C)   2 x
n x K exp
L

 
  
 
(D)  
x
n x K exp
L

 
  
 
AA [GATE-ME-2014-IITKGP]
101. Consider two solutions   1
x t x t
   2
x t x t

of the differential equation
 
 
2
2
d x t
x t 0,t 0,
dt
   such that
 
 
1
1
t 0
dx t
x 0 1, 0
dt 
  ,
 
 
2
2
t 0
dx t
x 0 0, 1
dt 
  . The Wronskian
 
   
   
1 2
1 2
x t x t
W t dx t dx t
dt dt
 at t
2

 is
(A) 1 (B) -1
(C) 0 (D)
2


91. TOPIC 3 – DIFFERENTIAL EQUATIONS
www.targate.org Page 83
A34 TO 36 [GATE-ME-2014-IITKGP]
102. If y = f(x) is the solution of
2
2
d y
0
dx
 with the
boundary conditions y = 5 at x = 0, and
dy
2
dx
 at x = 10, f(15) =___________.
AB [GATE-ME-2012-IITD]
103. The solution to the differential equation
2
2
d u du
k 0
dx dx
  where k is a constant , subject
to the boundary conditions  
u 0 0
 and
u(L) =U, is :
(A)
x
u U
L
 (B)
kx
kL
1 e
u U
1 e
 

  

 
(C)
kx
kL
1 e
u U
1 e


 

  

 
(D)
kx
kL
1 e
u U
1 e
 

  

 
AB [GATE-ME-2008-IISc]
104. Given that
" 3 0& (0) 1, '(0) 0
x x x x
    what is x(1)?
(A) – 0.99 (B) -0.16
(C) 0.16 (D) 0.99
AC [GATE-ME-2005-IITB]
105. Which of the following is a solution of the
differential equation
2
d y dy
p qy 0
dx dx
   is
x 3x
1 2
y C e C e
 
  then p and q are
(A) p = 3, q = 3 (B) p =3, q = 4
(C) p = 4, q = 3 (D) p = 4, q = 4
[GATE-ME-2001-IITK]
106. Solve the differential equation
2
2
d y
y x
dx
 
with the following conditions
(i) at x = 0, y = 1 (ii) x = 0, y’ = 1
cosx x

ANS :
[GATE-ME-2000-IITKGP]
107. Find the solution of the differential equation
 
2
2
2
d y
y cos t k
dt
     with initial
conditions y(0) = 0,
 
dy 0
0
dt
 . Here  , 
and k are constants. Use either the method of
undetermined coefficients (or) the operator
d
D
dt
 

 
 
based method.
 
 
2 2 2 2
cos k sin k
y cos( t) sin t

   
      
ANS :
 
 
2 2
1
cos t k
  
  
AD [GATE-ME-2000-IITKGP]
108. The solution of the differential equation
2
2
d y dy
y 0
dx dx
  
(A) x x
Ae Be

(B) x
e (Ax B)

(C) x 3 3
e Acos x Bcos x
2 2

 
   
 

   
 
   
 
   
 
(D)
x
2
3 3
e Acos x Bsin x
2 2
  
   
 

   
 
   
 
   
 
[GATE-ME-1996-IISc]
109. Solve
4
4 2
4
d v
4 v 1 x x
dx
    
 
1 2
cos sin
x x
V e C x C x e
 
    
Ans:
 
2
1 2 4
1
cos sin
4
x x x
e C x C x
  
   

AC [GATE-ME-1995-IITK]
110. The solution of the differential equation
     
f " x 4f ' x 4f x 0
   .
(A)   2 x
1
f x e

` (B)    
2x 2 x
1 2
f x e ,f x e
 
(C)    
2 x 2 x
1 2
f x e , f x xe
 
 
(D)    
2 x x
1 2
f x e , f x e
 
 
A( 1 3
t t
y e te
 
  ) [GATE-ME-1994-IITKGP]
111. Solve for y if
2
2
d y dy
2 y 0
dt dt
   with y(0) =
1 and  
y ' 0 2
 .
AA [GATE-CE-2008-IISc]
112. The general solution of
2
2
d y
y 0
dx
  is
(A) y Pcosx Qsin x
  (B) y Pcosx

(C) y Psin x
 (D) y Psin 2x

AA [GATE-CE-2005-IITB]
113. The solution of
2
2
d y dy
2 17y 0;
dx dx
    
y 0 1,

x
4
dy
0
dx 

 

 
 
in the range 0 x
4

  is given
by
(A) x 1
e cos 4x sin 4x
4
  

 
 
(B) x 1
e cos4x sin 4x
4
 

 
 

92. ENGINEERING MATHEMATICS
Page 84 TARGATE EDUCATION GATE-(EE/EC)
(C) 4x 1
e cos4x sin x
4
  

 
 
(D) 4x 1
e cos 4x sin 4x
4
  

 
 
AC [GATE-CE-2001-IITK]
114. The solution for the following differential
equation with boundary conditions y(0) = 2
and  
y' 1 3
  is, where
2
2
d y
3x 2
dx
 
(A)
3 2
x x
y 3x 6
3 2
   
(B)
2
3 x
y 3x 5x 2
2
   
(C)
3
2
x 5x
y x 2
2 2
   
(D)
2
3 x 3
y x 5x
2 2
   
AD [GATE-IN-2013-IITB]
115. The maximum value of the solution y(t) of
the differential equation    
y t y t 0
 
 with
initial conditions  
y 0 1

 and  
y 0 1
 , for
t 0
 is :
(A) 1 (B) 2
(C)  (D) 2
AC [GATE-IN-2011-IITM]
116. The solution of the differential equation
2
2
d y dy
6 9y 9x 6
dx dx
    with 1
C and 2
C as
constants is :
(A)   3 x
1 2
y C x C e 
 
(B) 3 x 3 x
1 2
y C e C e x

  
(C)   3 x
1 2
y C x C e x

  
(D)   3 x
1 2
y C x C e x
  
AC [GATE-IN-2009-IITR]
117. The solution of the differential equation
2
2
d y
0
dx
 with boundary conditions
(i)
dy
1
dx
 at x = 0
(ii)
dy
1
dx
 at x = 1 is
(A) y = 1
(B) y = x
(C) y = x + C where C is an arbitrary
constants are arbitrary constants
(D) 1 2
y C x C
  where 1
C and 2
C are
arbitrary constants
B [GATE-EC-2014-IITKGP]
118. If a and b are constants the most general
solution of the differential equation
2
2
2 0
d x dx
x
dt dt
   is :
(A)  t
ae (B)  

t t
ae bte
(C) t t
ae bte
 (D) 2 t
ae 
0.53to0.55 [GATE-EC-2014-IITKGP]
119. Which initial value y'(0) = y (0) = 1, the
solution of the differential equation
2
2
4 4 0
d y dy
y
dx dx
   at x = 1 is ------.
C [GATE-EE-2014-IITKGP]
120. The solution for the differential equation
2
2
9 ,
d x
x
dt
  with initial conditions x (0) = 1
and 0 1,
dx
t
dt
  is :
(A) 2
1
t t
 
(B)
1 2
sin 3 cos 3
3 3
t t
 
(C)
1
sin 3 cos 3
3
t t

(D) cos3t t

D [GATE-IN-2005-IITB]
121. The general solution of the differential
equation
2
( 4 4)
D D
  0
y  is of the form
(given D =
d
dx
an C1, C2 are constants)
(A)
2
1
x
C e
(B)
2 2
1 2
x x
C e C e

(C)
2 2
1 2
x x
C e C e

(D)
2 2
1 2
x x
C e C xe

A [GATE-EC-2006-IITKGP]
122. For the differential equation
2
2
2
0,
d y
k y
dx
 
the boundary conditions are
(i) 0
y  for 0
x  and
(ii) 0
y  for x a

The form of non-zero solution of y (where m
varies over all integers) are
(A) sin
m
m
m πx
y A
a
 
  
 

(B) cos
m
m
mπx
y A
a
 
  
 


93. TOPIC 3 – DIFFERENTIAL EQUATIONS
www.targate.org Page 85
(C)
m π
a
m
m
y A x
 
(D)
m πx
a
m
m
y A e

 
A [GATE-PI-2008-IISc]
123. The solutions of the differential equation
2
2
2 2 0
d y dy
y
dx dx
   are :
(A) (1 ) (1 )
,
i x i x
e e
   
(B) (1 ) (1 )
,
i x i x
e e
 
(C) (1 ) (1 )
,
i x i x
e e
  
(D) (1 ) (1 )
,
i x i x
e e
  
B [GATE-EE-2010-IITG]
124. For the differential equation
2
2
6 8 0
d x dx
x
dt
dt
   with initial conditions
x(0) = 1 and
0
0
t
dx
dt 
 

 
 
the solution
(A) 6 2
( ) 2 t t
x t e e
 
 
(B) 2 4
( ) 2 t t
x t e e
 
 
(C) 6 4
( ) 2
t t
x t e e
 
  
(D) 2 4
( ) 2
t t
x t e e
 
  
B [GATE-ME-2006-IITKGP]
125. For
2
2
2
4 3 3 ,
x
d y dy
y e
dx
dx
   the particular
integral is
(A) 2
1
15
x
e (B) 2
1
5
x
e
(C) 2
3 x
e (D) 3
1 2
x x
c e c e
 

A [GATE-PI-2009-IITR]
126. The homogeneous part of the differential
equation
2
2
d y dy
p qy r
dx dx
   (p, q, r are
constants) has real distinct roots if
(A)
2
4 0
p q
  (B)
2
4 0
p q
 
(C)
2
4 0
p q
  (D)
2
4
p q r
 
B [GATE-EC-2005-IITB]
127. A solution of the differential equation
2
2
5 6 0
d y dy
y
dx
dx
   is given by
(A) 2 3
x x
y e e 
 
(B) 2 3
x x
y e e
 
(C) 2 3
x x
y e e

 
(D) None of these.
A [GATE-ME-2008-IISc]
128. It is given that " 2 ' 0,
y y y
   (0) 0
y 
(1) 0
y  what is (0.5)?
y
(A) 0 (B) 0.37
(C) 0.62 (D) 1.13
A [GATE-IN-2006-IITKGP]
129. For initial value problem
'' 2 ' 101 10.4 ,
   x
y y y e y(0)=1.1 and y(0) =
– 0.9. Various solutions are written in the
following groups. Match the type of solution
with the correct expression.
Group-I Group-II
P. General solution
of
Homogeneous
equations
(1) 0.1 x
e
Q. Particular
integral
(2) x
e
[A
cos10 sin10
x B x
 ]
R. Total solution
satisfying
boundary
conditions
(3) cos10 0.1
x x
e x e


Codes:
(A) P – 2, Q – 1, R -3
(B) P -1, Q -3, R – 2
(C) P – 1, Q – 2, R – 3
(D) P -3 , Q – 2, R – 1
AB [GATE-EC-2015-IITK]
130. The solution of the differential equation
2
2
d y 2dy
y 0
dt dt
   with y(0) y '(0) 1
  is
(A)
t
(2 t)e
 (B)
t
(1 2t)e

(C) t
(2 t)e
 (D) None
A–3 [GATE-EE-2015-IITK]
131. A solution of the ordinary differential
equation
2
2
d y dy
5 6y 0
dt dt
   is such that
y(0) = 2 and 3
1 3e
y(1)
e

  . The value of
dy
dt
(t=0) is _____
AC [GATE-ME-2015-IITK]
132. Find the solution of
2
2
d y
y
dx
 which passes
through the origin and the point (ln2,
3
4
).

94. ENGINEERING MATHEMATICS
Page 86 TARGATE EDUCATION GATE-(EE/EC)
(A) x x
1
y e e
2

 
(B)
x x
1
y e e
2

 
(C)  
x x
1
y e e
2

 
(D) None
AA [GATE-EC-2017-IITR]
133. The general solution of the differential
equation
2
2
2 5 0
d y dy
y
dx dx
  
in termsl of arbitrary constants 1
K and 2
K is
(A)
( 1 6) ( 1 6)
1 2
x x
Ke K e
   

(B)
( 1 8) ( 1 8)
1 2
x x
Ke K e
   

(C)
( 2 6) ( 2 6)
1 2
x x
Ke K e
   

(D)
( 2 8) ( 2 8)
1 2
x x
Ke K e
   
 .
A1 [GATE-BT-2017-IITR]
134. For  
y f x ,
 if
2
2
d y dy
0, 0
dx dx
  at x = 0,
and y = 1 at x = 1, the value of y at x = 2 is
_______
AA [GATE-CE-2017-IITR]
135. Consider the following second-order
differential equation:
2
" 4 ' 3 2 3
y y y t t
    
The particular solution of the differential
equation is
(A) 2
2 2t t
   (B) 2
2t t
 
(C) 2
2t 3t
 (D) 2
2 2 t 3t
  
AA [GATE-ME-2017-IITR]
136. The differential equation
2
2
16 0
d y
y
dx
  for
( )
y x with the two boundary conditions
0
1
x
dy
dx 
 and
2
1
x
dy
dx 

  has
(A) no solution
(B) exactly two solutions
(C) exactly one solution
(D) infinitely many solutions
AA [GATE-MT-2017-IITR]
137. For the second order linear ordinary
differential equation,
2
2
0
d y dy
p qy
dx dx
   ,
the following function is a solution :
x
y e

Which one of the following statement is NOT
TRUE ?
(A)  has two values : one complex and one
real
(B)
2
0
p q
    
(C)  has two real values
(D)  has two complex values
AB [GATE-PE-2017-IITR]
138. The roots of the equation
3 2
3 2
6 11 6 0
d y d y dy
y
dx dx dx
    are :
(A) 1, 1, 2 (B) 1, 2, 3
(C) 1, 3, 4 (D) 1, 2, 4
A0.81 to 0.84 [GATE-PH-2018-IITG]
139. Given
2
2
( ) ( )
2 ( ) 0
d f x df x
f x
dx dx
   ,
and boundary conditions (0) 1
f  and
(1) 0
f  , the value of (0.5)
f is ______ (up
to two decimal places).
A4.52 to 4.56 [GATE-PI-2018-IITG]
140. Consider the differential equation
2
2
2 8 0
d y
y
dt
 
with initial conditions :
at 0, 0
t y
  and 10
dy
dt
 .
The value of y (up to two decimal places) at t
= 1 is _______.
A1.45 to 1.48 [GATE-ME-2018-IITG]
141. Given the ordinary differential equation
2
2
6 0
d y dy
y
dx dx
  
with (0) 0
y  and (0) 1
dy
dx
 , the value of
(1)
y is _______ (correct to two decimal
places).
AA [GATE-AG-2018-IITG]
142. The general solution to the second order
linear homogeneous differential equation
" 6 ' 25 0
y y y
   is

95. TOPIC 3 – DIFFERENTIAL EQUATIONS
www.targate.org Page 87
(A) 3
( cos4 sin 4 )
x
e a x b x

(B) 3
( cos 4 sin 4 )
ix
e a x b x

(C) 4
( cos3 sin 3 )
x
e a x b x

(D) 4
( cos3 sin 3 )
ix
e a x b x

A–0.23 to –0.19 [GATE-EC-2018-IITG]
143. The position of a particle ( )
y t is described by
the differential equation :
2
2
5
4
d y dy y
dt dt
   .
The initial conditions are (0) 1
y  and
0
0
t
dy
dt 
 . The position (accurate to two
decimal places) of the particle at t   is
_______.
T 3.2 A0 T3.2 [GATE-XE-2019-IITM]
144. Let 1( )
y x and 2 ( )
y x be two linearly
independent solutions of the differential
equation
2
2
2
4 0, 0
d y dy
x x y x
dx dx
    . If
2
1( )
y x x
 , then 2
lim ( )
x
y x

is ______.
AC T3.2 [GATE-PI-2019-IITM]
145. General solution of the Cauchy-Euler
equation
2
2
2
7 16 0
d y dy
x x y
dx dx
   is
(A)
2 4
1 2
y c x c x
 
(B)
2 4
1 2
y c x c x
 
(C)
4
1 2
( ln )
y c c x x
 
(D)
4 4
1 2 ln
y c x c x x

 
A5.9 to 6.1 T3.2 [GATE-CE-2019-IITM]
146. Consider the ordinary differential equation
2
2
2
2 2 0
d y dy
x x y
dx dx
   . Given the values of
(1) 0
y  and (2) 2
y  , the value of
(3)
y (round off to 1 decimal place), is _____
A5.24 to 5.26 T3.2 [GATE-EC-2019-IITM]
147. Consider the homogeneous ordinary
differential equation
2 2
2
2
3 3 0
d y dy
x x y
dx dx
   , 0
x 
with ( )
y x as a general solution. Given that
(1) 1
y  and (2) 14
y 
the value of (1.5)
y , rounded off to two
decimal places, is ______.
AC [GATE-PI-2015-IITK]
148. The solution to x2
y''+ xy'– y = 0 is :
(A) y=c1x2
+c2x-3
(B) y=c1+c2x-2
(C) 2
1
c
y c x
x
 
(D) y=c1x+c2x4
AD [GATE-PE-2016-IISc]
149. For the differential equation
2
2
2
2 2 0
d y dy
x x y
dx dx
  
the general solution is
(A) 1 2
x
y C x C e
 
(B) 1 2
sin cos
y C x C x
 
(C) 1 2
x x
y Ce C e
 
(D)
2
1 2
y C x C x
 
A(
5
8
) [GATE-ME-1998-IITD]
150. The radial displacement in a rotation disc is
governed by the differential equation
2
2 2
d u 1 du u
8x
dx x dx x
   where u is the
displacement and x is the radius. If u = 0 at x
= 0 and u = 2 at x = 1. Calculate the
displacement at
1
x
2
 .
C [GATE-EE-2014-IITKGP]
151. Consider the differential equation
2
2
2
0
d y dy
x x y
dx dx
   . Which of the
following is a solution to this differential
equation for 0
x  ?
(A) x
e (B) 2
x
(C) 1⁄x (D) ln x
D [GATE-CE-1998-IITD]
152. The general solution of the differential
equation
2
2
2
0
d y dy
x x y
dx
dx
   is :
(A) Ax + Bx2
(A, B are constants)
(B) Ax + B logx (A, B are constants)
(C) Ax + Bx2
logx (A, B are constants)
(D) Ax + Bxlog (A, B are constants)
AA [GATE-ME-2012-IITD]
153. Consider the differential equation
2
2
2
d y dy
x x 4y 0
dx dx
   with the boundary
conditions of y(0) = 0 and y(1) = 1. The
complete solution of the differential equation
is :

96. ENGINEERING MATHEMATICS
Page 88 TARGATE EDUCATION GATE-(EE/EC)
(A)
2
x (B)
x
sin
2

 
 
 
(C)
x x
e sin
2

 
 
 
(D)
x x
e sin
2
 
 
 
 
AC [GATE-AE-2017-IITR]
154. The equation
2
2
2
5 4 0
d y dy
x x y
dx dx
   has a
solution ( )
y x that is :
(A) A polynomial in x
(B) Finite series in terms of non-integer
fractional powers of x
(C) Consists of negative integer powers of x
and logarithmic function of x
(D) Consists of exponential functions of x.
**********
MISCELLANEOUS
T4.1 AB [GATE-MT-2018-IITG]
155. Consider the following Ordinary Differential
Equation:
0
d dc
c
dx dx
 

 
 
In a domain 0 x t
  , with boundary
conditions (0) 0.5
c  and ( ) 1.0
c t  , pick
the appropriate choice for ( )
c x from the
following options :
(A) P (B) Q
(C) R (D) S
T 4.2 AD [GATE-CE-2018-IITG]
156. The solution at 1
x  , 1
t  of the partial
differential equation
2 2
2 2
25
u u
x t
 

 
subject to
initial conditions of (0 3
u x
 and (0) 3
x
t



is _____
(A) 1 (B) 2
(C) 4 (D) 6
AA C [GATE-EE-1998-IITD]
157. Let .
x
f y
 What is
2

 
f
x y
at x = 2, y = 1?
(A) 0 (B) ln 2
(C) 1 (D) 2
1
ln
AC [GATE-CE-2010-IITG]
158. The partial differential equation that can be
formed from z = ax + by + ab has the form
z z
with p and
x y
 
 

 
 
 
(A) z = px + qy (B) z = px + pq
(C) z= px + qy + pq (D) z = qy + pq
AC [GATE-PI-2016-IISc]
159. For the two functions
  3 2
, 3
f x y x xy
  and   2 3
, 3
g x y x y y
 
which one of the following options is correct?
(A)
f g
x x
 

 
(B)
f g
x y
 
 
 
(C)
f g
y x
 
 
 
(D)
f g
y x
 

 
AD [GATE-AE-2016-IISc]
160. The partial differential equation
2
2
u u
,
t x
 
 
 
where  is a positive
constant, is
(A) circular (B) elliptic
(C) hyperbolic (D) parabolic
AA [GATE-TF-2016-IISc]
161. The following partial differential equation
xx yy
U U 0
  is of the type
(A) Elliptic (B) Parabolic
(C) Hyperbolic (D) Mixed type
AA [GATE-IN-2013-IITB]
162. The type of partial differential equation
2
2
f f
dt x
 


is,
(A) Parabolic (B) Elliptic
(C) Hyperbolic (D) Nonlinear
A-0.01 [GATE-TF-2016-IISc]
163. Let  
 
2 2 2
1
f x, y,z
x y z

 
. The value
of
2 2 2
2 2 2
f f f
x y z
  
 
  
is equal to ________

97. TOPIC 3 – DIFFERENTIAL EQUATIONS
www.targate.org Page 89
AC [GATE-CE-2016-IISc]
164. The type of partial differential equation
2 2 2
2 2
3 2 0
P P P P P
x y x y x y
    
    
     
is
(A) elliptic (B) parabolic
(C) hyperbolic (D) none of these
AC [GATE-ME-2005-IITB]
165. If n n 1 n 1 n,
0 1 n 1 n
f a x a x y ..... a x a y
 

    
where i
a (i = 0 to n) are constants, then
f f
x y
x y
 

 
is :
(A)
f
n
(B)
n
f
(C) nf (D) n f
AB [GATE-CE-2016-IISc]
166. The solution of the partial differential
equation
2
2
u u
t x
 
 
 
is of the form
(A) ( / ) ( / )
1 2
cos( ) k x k x
C kt C e C e
  
 

 
(B)
( / ) ( / )
1 2
kt k x k x
Ce C e C e
  
 

 
(C)  
1 2
cos / sin( / )
kt
Ce C k x C k x
 
   
 
(D)
 
1 2
sin( ) cos / sin( / )
C kt C k x C k x
 
   
 
A40 [GATE-EE-2017-IITR]
167. Consider a function f(x, y, z) given by
2 2 2 2 2
( , , ) ( 2 )( )
f x y z x y z y z
   
The partial derivative of this function with
respect to x at the point, x = 2, y = 1 and z = 3
is _______ .
AC [GATE-CE-2017-IITR]
168. Let  
w f x,y
 , where x and y are
functions of t. then, according to the chain
rule,
dw
dt
is equal to
(A)
dw dx dw dt
dx dt dy dt

(B)
w x w y
x t y t
   

   
(C)
w dx w dy
x dt y dt
 

 
(D)
dw x dw y
dx t dy t
 

 
AB [GATE-ME-2017-IITR]
169. Consider the following partial differential
equation for ( , )
u x y with the constant 1
c  :
0
u u
c
y x
 
 
 
Solution of this equation is :
(A) ( , ) ( )
u x y f x cy
 
(B) ( , ) ( )
u x y f x cy
 
(C) ( , ) ( )
u x y f cx y
 
(D) ( , ) ( )
u x y f cx y
 
A10.0 [GATE-MN-2018-IITG]
170. For the given function
( , ) (3 )(4 )
f x y x y
   , the value of
f f
x y
 

 
at 2
x  and 1
y  is ____________.
AD [GATE-IN-2018-IITG]
171. Consider the following equations
2 2
( , )
2
V x y
px y xy
x

  

2 2
( , )
2
V x y
x qy xy
y

  

where p and q constants. ( , )
V x y that
satisfies the above equations is
(A)
3 3
2 6
3 3
x y
p q xy
  
(B)
3 3
5
3 3
x y
p q
 
(C)
3 3
2 2
3 3
x y
p q x y xy xy
   
(D)
3 3
2 2
3 3
x y
p q x y xy
  
A4.4 – 4.6 [GATE-EC-2018-IITG]
172. Let 2
r x y z
   and 3 3
1
z xy yz y
    .
Assume that x and y are independent
variables. At ( , , ) = (2, −1,1), the value
(correct to two decimal places) of
r
x


is
_________ .
T 4.3 AA [GATE-MN-2016-IISc]
173. The differential of the equation,
2 2
1
x y
  ,
with respect to x is
(A) -x/y (B) x/y
(C) -y/x (D) y/x

98. ENGINEERING MATHEMATICS
Page 90 TARGATE EDUCATION GATE-(EE/EC)
C [GATE-PI-2010-IITG]
174. Which one of the following differential
equations has a solution given by the function
5 sin 3
5
π
y x
 
 
 
 
(A)
5
cos(3 ) 0
3
dy
x
dx
 
(B)
5
(cos3 ) 0
3
dy
x
dx
 
(C)
2
2
9 0
d y
y
dx
 
(D)
2
2
9 0
d y
y
dx
 
-------0000-------

99. TOPIC 3 – DIFFERENTIAL EQUATIONS
www.targate.org Page 91
Answer :
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
B B B A B D D B C C
11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
B A 1.0 B A D D B A B
21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
D A * * A C C D D B
31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
D D B D C 0.5 C B A B
41. 42. 43. 44. 45. 46. 47. 48. 49. 50.
D B C A C C D A D A
51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
A A D A D C C 6 D D
61. 62. 63. 64. 65. 66. 67. 68. 69. 70.
2.3 C C A 2.4 A B C D D
71. 72. 73. 74. 75. 76. 77. 78. 79. 80.
C A * A 0 C C * B B
81. 82. 83. 84. 85. 86. 87. 88. 89. 90.
D D B D A * A A B D
91. 92. 93. 94. 95. 96. 97. 98. 99. 100.
C C * A * * C # C D
101. 102. 103. 104. 105. 106. 107. 108. 109. 110.
A * B B C # # D # C
111. 112. 113. 114. 115. 116. 117. 118. 119. 120.
* A A C D C C B * C
121. 122. 123. 124. 125. 126. 127. 128. 129. 130.
D A A B B A B A A B
131. 132. 133. 134. 135. 136. 137. 138. 139. 140.
–3 C A 1 A A A B * *
141. 142. 143. 144. 145. 146. 147. 148. 149. 150.
* A * 0 C * * C D *
151. 152. 153. 154. 155. 156. 157. 158. 159. 160.
C D A C B D C C C D
161. 162. 163. 164. 165. 166. 167. 168. 169. 170.
A A * C C B 40 C B 10.0
171. 172. 173. 174.
D * A C
23. 0.51 – 0.53
24. 10.50 to 12.50
73. TRUE
78. –1.05 to –0.95
86. 14.55 to 14.75
93. 93 to 95
95. 7.0 to 7.5
96. TRUE
102. 34 to 36
111. 1 3
t t
y e te
 
 
119. 0.53 to 0.55
139. 0.81 to 0.84
140. 4.52 to 4.56
141. 1.45 to 1.48
143. –0.23 to –0.19
146. 5.9 to 6.1
147. 5.24 to 5.26
150.
5
8
163. –0.01
172. 4.4 to 4.6

100. Page 92 TARGATE EDUCATION GATE-(EE/EC)
04
Complex Variable
Basic Problems
T 1.1 AD [GATE-CH-2016-IISc]
1. What are the modulus (r) and instrument  

of the complex number 3 + 4i?
(A)
1 4
7, tan
3
r   
    
 
(B)
1 3
7, tan
4
r   
    
 
(C)
1 3
5, tan
4
r   
    
 
(D)
1 4
5, tan
3
r   
    
 
AD T4.2 [GATE-PI-2019-IITM]
2. For a complex number 1 4
z i
  with
1
i   , the value of
3
1
z
z


is
(A) 0 (B) 1/ 2
(C) 1 (D) 2
AA T4.2 [GATE-PE-2019-IITM]
3. Let r and  be the modulus and argument of
the complex number 1
z i
  , respectively.
Then ( , )
r  equals
(A) ( 2, )
4

(B) (2, )
2

(C) (2, )
3

(D) ( 2, )

AB [GATE-MN-2016-IISc]
4. Sinh(x) is
(A)
4
x x
e e

(B)
2
x x
e e

(C)
2
x x
e e

(D)
4
x x
e e

A10 [GATE-EC-2015-IITK]
5. Let
az b
f (z)
cz d



. If 1 2
f (z ) f (z )
 for all
1 2
z z
 , a = 2, b = 4 and c = 5, then d should
be equal to ___.
C [GATE-IN-1994-IITKGP]
6. The real part of the complex number z x iy
 
is given by
(A) Re( ) *
z z z
  (B)
*
Re( )
2
z z
z


(C)
*
Re( )
2
z z
z

 (D) Re( ) *
z z z
 
D [GATE-IN-2009-IITR]
7. If Z = x + jy where x, y are real then the value
of | |
jz
e is
(A) 1 (B)
2 2
x y
e 
(C)
y
e (D)
y
e
D [GATE-PI-2009-IITR]
8. The product of complex numbers (3 – 21) & (3
+ i4) results in
(A) 1 + 6i (B) 9 – 8i
(C) 9 + 8i (D) 17 + i 6
B [GATE-PI-2008-IISc]
9. The value of the expression
5 10
3 4
i
i
 

(A) 1 2i
 (B) 1 2i

(C) 2 i
 (D) 2 i

C [GATE-CE-2005-IITB]
10. Which one of the following is Not true for the
complex numbers z1 and z2?
(A) 1 1 2
2
2 2
| |
z z z
z z

(B) 1 2 1 2
| | | | | |
z z z z
  
(C) 1 2 1 2
| | | | | |
z z z z
  
(D)
2 2 2 2
1 2 1 2 1 2
| | | | 2 | | 2 | |
z z z z z z
    
AA [GATE-ME-2015-IITK]
11. Given two complex numbers 1 5 (5 3)
z i
 
and 2
2
2
3
z i
  , argument of 1
2
z
z
in degrees
is :

101. TOPIC 4 – COMPLEX VARIABLE
www.targate.org Page 93
(A) 0 (B) 30
(C) 60 (D) 90
AC [GATE-ME-2014-IITKGP]
12. The argument of the complex number
1 i
1 i


,
where i 1
  is :
(A)  (B)
2

(C)
2

(D) 
AA [GATE-ME-2011-IITM]
13. The product of two complex numbers 1 +i and
2 – 5i is :
(A) 7-3i (B) 3 - 4i
(C) - 3 – 4i (D) 7 + 3i
AB [GATE-CE-2014-IITKGP]
14.
2 3i
Z
5 i


 
can be expressed as
(A) -0.5 – 0.5i (B) - 0.5 + 0.5i
(C) 0.5 – 0.5i (D) 0.5 + 0.5i
AD [GATE-CE-1994-IITKGP]
15. cos  can be represented as
(A)
i i
e e
2
  

(B)
i i
e e
2i
  

(C)
i i
e e
i
  

(D)
i i
e e
2
  

AA [GATE-PE-2017-IITR]
16. If 5 2 7 2 3
x iy ix y i
     , where 1
i   ,
the values of two real numbers ( , )
x y are,
respectively :
(A) (-1, 1) (B) (1, -1)
(C) (1, 1) (D) (-1, -1)
AC [GATE-PE-2017-IITR]
17. Pick the INCORRECT inequality, where 1 2
,
z z
and 3
z are complex numbers.
(A) 1 2 1 2
| | | | | |
z z z z
  
(B) 1 2 1 2
| | | | | |
z z z z
  
(C) 1 2 1 2
| | | | | |
z z z z
  
(D) 1 2 3 1 2 3
| | | | | | | |
z z z z z z
    
AD [GATE-PE-2017-IITR]
18. Which of the following is NOT true ?
( 1)
i  
(A) cos
2
i i
e e
  

 
(B) cos sin
i
e i

   
(C) sin
2
i i
e e
i
  

 
(D) cos
2
i i
e e
i
  

 
AD [GATE-PE-2017-IITR]
19.
30 19
3
2 1
i i
z
i



, where 1
i   , would simplify
to :
(A) 1 i
 (B) 1
(C) i
 (D) 1 i

T 1.2 AD [GATE-PE-2016-IISc]
20. For a complex number
1 3
2 2
Z i
 
 
 
 
 
, the
value of 6
Z is
(A)
1 3
2 2
i
 
 
 
 
 
(B) -1
(C)
1 3
2 2
i
 

 
 
 
(D) 1
AA T4.2 [GATE-CH-2019-IITM]
21. The value of the complex number 1/2
i
(where
1
i   ) is
(A)
1
(1 )
2
i
 (B)
1
2
i

(C)
1
2
i (D)
1
(1 )
2
i

B [GATE-ME-1996-IISc]
22. ,
i
i where i = 1
 is given by
(A) 0 (B) / 2
π
e 
(C)
2
π
(D) 1
A [GATE-EC-2012-IITD]
23. If 1
x   , then the value of x
x is
(A) / 2
π
e 
(B)
/2
π
e
(C) x (D) 1
D [GATE-IN-2007-IITK]
24. Let j = 1.
 Then one value of j
j is
(A) 3 (B) 1

(C) 1
2 (D) 2
π
e

B [GATE-PI-2007-IITK]
25. If a complex number z =
3 1
2 2
i
 then 4
z is

102. ENGINEERING MATHEMATICS
Page 94 TARGATE EDUCATION GATE-(EE/EC)
(A) 2 2 2i
 (B)
1 3
2 2
i
 
(C)
3 1
2 2
i
 (D)
3 1
8 8
i

AD [GATE-CE-2007-IITK]
26. Let j 1
  . Then the value of j
j is:
(A) j (B) -1
(C)
2

(D) 2
e


T 1.3 B [GATE-EE-2014-IITKGP]
27. All the values of the multi-valued complex
function 1 i
, where 1
i   , are
(A) purely imaginary.
(B) real and non-negative.
(C) on the unit circle.
(D) equal in real and imaginary parts.
B [GATE-CE-1997-IITM]
28.
z
e is a periodic with a period of
(A) 2π (B) 2πi
(C) π (D) iπ
AD [GATE-EC-2008-IISc]
29. The equation sin (z) = 10 has
(A) No real (or) complex solution
(B) Exactly two distinct complex solutions
(C) A unique solution
(D) An infinite number of complex solutions
AB [GATE-EC-2013-IITB]
30. Square roots of i, where i 1
  , are
(A) i, -1
(B) 3
cos isin ,cos
4 4 4
  
     
  
     
     
3
i sin
4

 
  
 
(C) 3 3
cos isin ,cos
4 4 4
  
     

     
     
isin
4

 
  
 
(D)
3 3 3
cos isin ,cos
4 4 4
  
     
  
     
     
3
isin
4

 
  
 
AD [GATE-CE-2013-IITB]
31. The complex function tanh(s) is analytic over
a region of the imaginary axis of the complex s
– plane if the following is TURE everywhere
in the region for all integers n
(A) Re(s) = 0
(B)  
Im s n
 
(C)  
n
Im s
3


(D)  
 
2n 1
Im s
2
 

AB [GATE-IN-2009-IITR]
32. One of the roots of the equation 3
x j
 , where
j is the positive square root of -1, is
(A) j (B)
3 1
j
2 2

(C)
3 1
j
2 2
 (D)
3 1
j
2 2


AC [GATE-CE-2005-IITB]
33. Let 3
z z,
 where z is complex not equal to
zero. Then z is a solution of
(A) 2
z 1
 (B) 3
z 1

(C) 4
z 1
 (D) 9
z 1

AA [GATE-PI-2010-IITG]
34. If a complex number  satisfies the equation
2
1
  , then the value of
1
1  

is :
(A) 0 (B) 1
(C) 2 (D) 4
AD [GATE-EE-2017-IITR]
35. For a complex number z,
 
2
3 2
z i
z 1
lim
z 2z i z 2


  
is
(A) -2i (B) -i
(C) i (D) 2i
AB [GATE-IN-2017-IITR]
36. Let z = x + jy where j 1
  . Then cosz 
(A) cos z (B) cos z
(C) sin z (D) sin z
T1.4 AA [GATE-IN-2016-IISc]
37. In the neighborhood of z = 1, the function f(z)
has a power series expansion of the form
2
( ) 1 (1 ) (1 )
f z z z
      …………
Then f(z) is :
(A)
1
z
(B)
1
2
z


(C)
1
1
z
z


(D)
1
2 1
z 

103. TOPIC 4 – COMPLEX VARIABLE
www.targate.org Page 95
A5.9 to 6.1 T4.1 [GATE-PI-2019-IITM]
38. If z is a complex variable with 1
i   , the
length of the minor axis of an ellipse defined
by | (1 )| | (9 )| 10
z i z i
      is _____
C[GATE-EE-2014-IITKGP]
39. Let S be the set of points in the complex plane
corresponding to the unit circle. (That is, S =
{Z: | Z | = 1}). Consider the function f (z) = z
z*
where z*
denotes the complex conjugate of
z. The f (z) maps S to which one of the
following in the complex plane
(A) Unit circle
(B) Horizontal axis line segment from origin
to (1, 0)
(C) The point (1, 0)
(D) The entire horizontal axis
B [GATE-IN-1997-IITM]
40. The complex number z x jy
  which satisfy
the equation | 1 | 1
z   lie on
(A) a circle with (1, 0) as the centre and
radius 1
(B) a circle with (-1, 0) as the centre and
radius 1
(C) y – axis
(D) x – axis
B [GATE-EC-2006-IITKGP]
41. For the function of a complex variable w = l nz
(where w = u jv
 and z x jy
  ) the u =
constant lines get mapped i the z – plane as
(A) Set of radial straight lines
(B) Set of concentric circles
(C) Set of co focal hyperbolas
(D) Set of co focal ellipses
A [GATE-IN-2002-IISc]
42. The bilinear transformation w =
1
1
z
z


(A) Maps the inside of the unit circle in the z
– plane to the left half of the w - plane
(B) Maps the outside the unit circle in the z –
plane to the left half of the w – plane
(C) maps the inside of the unit circle in the z
– plane to right half of the w – plane
(D) maps the outside of the unit circle in the z
– plane to the right half of the w – plane
**********
Analytic Function
T 2.1 AA T4.2 [GATE-ME-2019-IITM]
43. A harmonic function is analytic if it satisfies
the Laplace equation.
If
2 2
( , ) 2 2 4
u x y x y xy
   is a harmonic
function, then its conjugate harmonic function
( , )
v x y is
(A)
2 2
4 2 2 constant
xy x y
  
(B)
2
4 4 constant
y xy
 
(C)
2 2
2 2 constant
x y xy
  
(D)
2 2
4 2 2 constant
xy y x
   
AB T4.2 [GATE-ME-2019-IITM]
44. An analytic function ( )
f z of complex
variable z x iy
  may be written as
( ) ( , ) ( , )
f z u x y iv x y
  . Then ( , )
u x y and
( , )
v x y must satisfy
(A)
u v
x y
 

 
and
u v
y x
 

 
(B)
u v
x y
 

 
and
u v
y x
 
 
 
(C)
u v
x y
 
 
 
and
u v
y x
 

 
(D)
u v
x y
 
 
 
and
u v
y x
 
 
 
AB [GATE-CE-2010-IITG]
45. If    
3 2
f x iy x 3xy i x, y
     , where
i 1
  and  
f x iy
 is an analytic function,
then  
x,y
 is
(A) 3 2
y 3x y
 (B) 2 3
3x y y

(C) 4 3
x 4x y
 (D) 2
xy y

AD T4.2 [GATE-CE-2019-IITM]
46. Consider two functions : ln
x    and
ln
y    . Which one of the following is the
correct expression for
x


?
(A)
ln
ln ln 1
x 
  
(B)
ln
ln ln 1
x 
  
(C)
ln
ln ln 1

  
(D)
ln
ln ln 1

  
AA [GATE-CE-2005-IITB]
47. The function
 
2 2 1
1 y
w u iv log x y tan
2 x
  
      
 
is not analytic at the point.

104. ENGINEERING MATHEMATICS
Page 96 TARGATE EDUCATION GATE-(EE/EC)
(A)  
0,0 (B)  
0,1
(C)  
1,0 (D)  
2,
AC [GATE-ME-2014-IITKGP]
48. An analytic function of a complex variable
z x iy
  is expressed as
     
f z u x, y iv x, y
  where i 1
  if
 
u x, y 2xy
 , then  
v x,y must be :
(A) 2 2
x y
  constant
(B) 2 2
x y
  constant
(C) 2 2
x y
  constant
(D) 2 2
x y
   constant
AC [GATE-ME-2014-IITKGP]
49. An analytic function of complex variable
z x iy
  is expressed as
( ) ( , ) ( , )
f z u x y j v x y
  where i 1
  . If
  2 2
u x, y x y
  , then expressed for  
v x,y
in terms of x, y and a general constant c would
be :
(A) xy + c (B)
2 2
x y
c
2


(C) 2xy + c (D)
 
2
x y
c
2


AA [GATE-ME-2016-IISc]
50.      
, ,
f z u x y iv x y
  is an analytic
function of complex variable z x iy
  where
1
i   . If  
, 2
u x y xy
 , then  
,
v x y may
be expressed as
(A)
2 2
x y
  constant
(B)
2 2
x y
 constant
(C)
2 2
x y
 constant
(D)  
2 2
x y
   constant
A–1.1 - -0.9 [GATE-ME-2016-IISc]
51. A function f of the complex variable x = x +
iy, is given as      
, , , ,
f x y u x y iv x y
 
where  
, 2
u x y kxy
 and   2 2
,
v x y x y
  .
The value of k, for which the function is
analytic, is______
AA [GATE-PH-2016-IISc]
52. Which of the following is an analytic function
of z everywhere in the complex plane?
(A) 2
z (B)  
2
*
z
(C)
2
| |
z (D) z
A0.0 [GATE-EC-2016-IISc]
53. Consider the complex valued function
3 3
( ) 2 | |
f z z b z
  where z is a complex
variable. The value of b for which the function
f(z) is analytic is ________
AB [GATE-EE-2016-IISc]
54. Consider the function *
( )
f z z z
  where z is
a complex variable and z* denotes its complex
conjugate. Which one of the following is
TRUE?
(A) f(z) is both continuous and analytic
(B) f(z) is continuous but not analytic
(C) f(z) is not continuous but is analytic
(D) f(z) is neither continuous nor analytic
B [GATE-EC-2014-IITKGP]
55. The real part of an analytic function f (z)
where z = x + jy is given by e-y
cos (x). The
imaginary part of f (z) is
(A)  
cos
y
e x (B)  
sin
 y
e x
(C)  
sin
y
e x
 (D)  
sin
y
e x


AB [GATE-MA-2017-IITR]
56. Let    
2 2
f z x y i2xy
   and
   
2 2
g z 2xy i y x
   for z x iy
   .
Then, in the complex plane
(A) f is analytic and g is not analytic
(B) f is not analytic and g is analytic
(C) neither f nor g is analytic
(D) both f and g is analytic
AB [GATE-ME-2017-IITR]
57. If 2 2
( ) ( )
f z x a y i b xy
   is a complex
analytic function of z x iy
  , where
1
i   , then
(A) a = –1, b = –1 (B) a = –1, b = 2
(C) a = 1, b = 2 (D) a = 2, b = 2
AB [GATE-ME-2018-IITG]
58. ( )
F z is a function of the complex variable
z x iy
  given by
( ) Re( ) Im( )
F z iz k z i z
   .
For what value of k will ( )
F z satisfy the
Cauchy-Riemann equations?
(A) 0 (B) 1
(C) –1 (D) y
AB [GATE-IN-2018-IITG]
59. Let 2
1 ( )
f z z
 and 2 ( )
f z z
 be two complex
variable functions. Here z is the complex
conjugate of z. Choose the correct answer.
(A) Both 1( )
f z and 2 ( )
f z are analytic
(B) Only 1( )
f z is analytic

105. TOPIC 4 – COMPLEX VARIABLE
www.targate.org Page 97
(C) Only 2 ( )
f z is analytic
(D) Both 1( )
f z and 2 ( )
f z are not analytic
T2.2 AD [GATE-PI-2016-IISc]
60. The function  
2
2
1
4
z
f z
z



is singular at
(A) 2
z (B) 1
z
(C) z i
 (D) 2
z i
 
AC T4.2 [GATE-XE-2019-IITM]
61. Let
2
| |
( ) z
f z ze
 , where z is the complex
conjugate of z. Then, it is differentiable on
(A) | | 1
z 
(B) | | 1
z 
(C) | | 1
z 
(D) the entire complex plane
D [GATE-CE-2009-IITR]
62. The analytical function has singularities at,
where f(z) = 2
1
1
z
z


(A) 1 and -1 (B) 1 and i
(C) 1 and – i (D) i and – i
AB T4.2 [GATE-EE-2019-IITM]
63. Which one of the following functions is
analytic in the region | | 1

z ?
(A)
2
1

z
z
(B)
2
1
2


z
z
(C)
2
1
0.5


z
z
(D)
2
1
0.5


z
z j
AD T4.2 [GATE-EC-2019-IITM]
64. Which one of the following functions is
analytic over the entire complex plane?
(A) 1/
e z
(B) ln(z)
(C)
1
1 z

(D) cos(z)
T2.3 AB [GATE-CE-2007-IITK]
65. For the function 3
sin z
z
of a complex variable
z, the point z = 0 is:
(A) A pole of order 3
(B) A pole of order 2
(C) A pole of order 1
(D) Not a singularity
**********
Cauchy’s Integral & Residue
Cauchy Integral
T3.1 A81.60 to 81.80 [GATE-PH-2018-IITG]
66. The absolute value of the integral
3 2
2
5 3
,
4
z z
dz
z



over the circle | 1.5| 1
z   in complex plane,
is ______ (up to two decimal places).
AB [GATE-ME-2016-IISc]
67. The value of
  
3 5
1 2
z
z z


 
 dz along a
closed path  is equal to  
4 i
 , where z = x
+iy and 1
i   . The cor12rect path  is
(A)
(B)
(C)
(D)
A–0.0001 to 0.0001 T4.2 [GATE-EC-2019-IITM]
68. The value of the contour integral
2
1 1
2
z dz
j z
 

 
  
 evaluated over the unit
circle | | 1
z  is _____.

106. ENGINEERING MATHEMATICS
Page 98 TARGATE EDUCATION GATE-(EE/EC)
A0.039-0.043 [GATE-MA-2016-IISc]
69. Let  
:| | 2
z z
    be oriented in the
counter-clockwise direction. Let
7
2
1 1
2
I z dz
i z

 
  
  

Then, the value of I is equal to________
AB [GATE-EE-2017-IITR]
70. Consider the line integral 2 2
( )
c
I x jy dz
 

where z x iy
  . the line c is shown in the
figure below.
The value of I is :
(A)
1
i
2
(B)
2
i
3
(C)
3
i
4
(D)
4
i
5
A–136 - –132 [GATE-EC-2016-IISc]
71. In the following integral, the contour C
encloses the points 2 j
 and 2 j
 
3
1 sin
2 ( 2 )

  
C
z
dz
z j
The value of the integral is ________
AA T4.1 [GATE-EE-2019-IITM]
72. The closed loop line integral
3 2
| | 5
8
2
z
z z
dz
z

 

 evaluated counter-
clockwise, is :
(A) 8 j
  (B) 8j
 
(C) 4 j
  (D) 4 j
 
AB [GATE-EC-2016-IISc]
73. The values of the integral
1
2 2
z
c
e
dz
j z
 
 along
a closed contour c in anti-clockwise direction
for
(i) the point z0 = 2 inside the contour c, and
(ii) the point z0 = 2 outside the contour c,
respectively, are
(A) (i) 2.72, (ii) 0 (B) (i) 7.39, (ii) 0
(C) (i) 0, (ii) 2.72 (D) (i) 0, (ii) 7.39
A–1 [GATE-IN-2016-IISc]
74. The value of the integral
2
2
1 1
2 1
C
z
dz
j z

 
 where
z is a complex number and C is a unit circle
with center at 1 0 j
 in the complex plane is
_____.
AB [GATE-EE-2016-IISc]
75. The value of the integral
 
2
2 5
1
4 5
2
C
z
dz
z z z

 
  
 
 

over the contour |z| = 1, taken in the anti-
clockwise direction, would be
(A)
24
13
i

(B)
48
13
i

(C)
24
13
(D)
12
13
C [GATE-EC-2014-IITKGP]
76. C is a closd path in the z-plane given by |z| =3.
The value of the integral
2
4
2
 

C
z z j
dz
z j
is
(A) 4 (1 2)
j
   (B) 4 (3 2)
j
 
(C) 4 (3 2)
j
   (D) 4 (1 2)
j
 
A [GATE-EC-2006-IITKGP]
77. Using Cauchy’s integral theorem, the value of
the integral (integration being taken in contour
clock wise direction)
3
6
3
C
z
dz
z i


 is where C is |z| = 1
(A)
2
4
81
π
πi
 (B) 6
8
π
πi

(C)
4
6
81
π
πi
 (D) 1
B [GATE-EC-2007-IITK]
78. The value of
2
1
(1 )
C
dz
z

 where C is the
contour | / 2 | 1
z i
 
(A) 2 π i (B) 
(C) 1
tan ( )
z
 (D) 1
tan
π i z

A [GATE-EC-2007-IITK]
79. If the semi – circulator contour D of radius 2 is
as shown in the figure. Then the value of the
integral
2
1
1
D
s 
 ds is :

107. TOPIC 4 – COMPLEX VARIABLE
www.targate.org Page 99
(A) iπ (B) iπ

(C) π
 (D) π
C [GATE-IN-2011-IITM]
80. The contour integral
1
z
C
e
 dz with C as the
counter clock – wise unit circle in the z –
plane is equal to
(A) 0 (B) 2π
(C) 2 1
π  (D) 
AD [GATE-EC-2015-IITK]
81. Let Z = x + iy be a complex variable consider
continuous integration is performed along the
unit circle in anti clockwise direction. Which
one of the following statements is NOT
TRUE?
(A) The residue of 2
z
z 1

at z = 1 is
1
2
(B) 2
C
z dz 0


(C)
C
1 1
dz 1
2 i z

 
(D) z (complex conjugate of z) is an
analytical function
AB [GATE-IN-2015-IITK]
82. The value of 2
1
dz
z
 , where the contour is
the unit circle traversed clockwise, is :
(A) 2 i
  (B) 0
(C) 2 i
 (D) 4 i

AB [GATE-EC-2015-IITK]
83. If C is a circle of radius r with centre z0, in the
complex z-plane and if n is a non-zero integer,
then n 1
C
0
dz
(z z ) 

 equals
(A) 2 nj
 (B) 0
(C)
nj
2
(D) 2 n

AC [GATE-EC-2012-IITD]
84. Given  
1 2
f z
z 1 z 3
 
 
. If C is a
counterclockwise path in the z-plane such that
z 1 1
  , the value of  
C
1
f z dz
2 j
  is
(A) -2 (B) -1
(C) 1 (D) 2
AA [GATE-EC-2011-IITM]
85. The value of integral
 
2
C
3z 4
dz
z 4z 5
 
 
 where
C is the circle z 1
 is given by
(A) 0 (B)
1
10
(C)
4
5
(D) 1
AD [GATE-EC-2009-IITR]
86. If   1
0 1
f z c c z
  , then
 
unit
circle
1 f z
dz
z

 is
given by
(A) 1
2 c
 (B)  
0
2 1 c
 
(C) 1
2 jc
 (D)  
0
2 j 1 c
 
AC [GATE-EE-2014-IITKGP]
87. Integration of the complex function
 
2
2
z
f z
z 1


, in the counterclockwise
direction along z 1 1
  is :
(A) i
 (B) 0
(C) i
 (D) 2 i

AA [GATE-EC-2013-IITB]
88.
2
2
z 4
dz
z 4


 evaluated anticlockwise around
the circle z i 2
  , where i 1
  is
(A) 4
  (B) 0
(C) 2   (D) 2 + 2i
AA [GATE-ME-2008-IISc]
89. The integral  
f z dz
 evaluated along the
unit circle on the complex plane for
 
cos z
f z
z
 is:
(A) 2 i
 (B) 4 i

(C) 2 i
  (D) 0
AC [GATE-CE-2008-IISc]
90. The value of the integral  
  
C
co s 2 z
2 z 1 z 3

 

(where C is a closed curve given by z 1
 ) is:
(A) i
 (B)
i
5

(C)
2 i
5

(D) i

AC [GATE-CE-2006-IITKGP]
91. The value of the integral of the complex
function
  
3s 4
f (s)
s 1 s 2


 
. Along the path
| | 3
s  is :
(A) 2 j
 (B) 4 j

(C) 6 j
 (D) 8 j


108. ENGINEERING MATHEMATICS
Page 100 TARGATE EDUCATION GATE-(EE/EC)
D [GATE-EC-2010-IITG]
92. The contour C in the adjoining figure is
described by 2 2
16.
x y
  Then the value of
2
8
(0.5) (1.5)
C
z
dz
z j



(A) 2 π j
 (B) 2 π j
(C) 4 π j (D) 4 π j

AB [GATE-PI-2011-IITM]
93. The value of 4
1
z
dz
z 
 using Cauchy’s
integral around the circle , | 1 | 1
z   where z
= x + iy, is
(A) 2 i
 (B)
i
2

(C)
3 i
2
 
(D) 2
i

AD [GATE-EC-2017-IITR]
94. An integral I over a counter-clockwise circle C
is given by
2
2
1
1
z
C
z
I e dz
z




If C is defined as |z| = 3, thten the value of I is
(A) sin(1)
i
 (B) 2 sin(1)
i
 
(C) 3 sin(1)
i
  (D) 4 sin(1)
i
 
AC [GATE-EE-2017-IITR]
95. The value of the contour integral in the
complex-plane
3
2 3
2
z z
dz
z
 


along the contour |z| = 3, taken counter-
clockwise is
(A) 18 i
  (B) 0
(C) 14 i
 (D) 48 i

A3 [GATE-MA-2017-IITR]
96. Let C be the simple, positively oriented circle
of radius 2 cantered at the origin in the
complex plane. Then
 
  
1/z
C
2 z 1
ze tan dz
i 2 z 1 z 3
 
 
 
 
 
 
  
 
 

equals____________.
AA [GATE-ME-2018-IITG]
97. Let z be a complex variable. For a counter-
clockwise integration around a unit circle C ,
centred at origin,
1
5 4
C
dz A i
z
 

 ,
the value of A is
(A) 2/5 (B) 1/2
(C) 2 (D) 4/5
AB [GATE-EE-2018-IITG]
98. If C is a circle | | 4
z  and
2
2 2
( )
( 3 2)
z
f z
z z

 
, then ( )
C
f z dz
 is
(A) 1 (B) 0
(C) –1 (D) –2
AA [GATE-EE-2018-IITG]
99. The value of the integral 2
1
4
C
z
dz
z


 in
counter clockwise direction around a circle C
of radius 1 with center at the point 2
z   is
(A)
2
i

(B) 2 i

(C)
2
i

 (D) 2 i
 
A6 [GATE-MA-2018-IITG]
100. Let
2
3
( ) z
f z z e
 for z  and let  be the
circle i
z e 
 , where  varies from 0 to 4 .
Then
1 '( )
2 ( )
f z
dz
i f z

  = _______.
AC [GATE-MA-2018-IITG]
101. Let  be the circle given by 4 i
z e 
 , where
 varies from 0 to 2 . Then
2
2
z
e
dz
z z




(A) 2
2 ( 1)
i e
 
(B) 2
(1 )
i e
 
(C) 2
( 1)
i e
 
(D) 2
2 (1 )
i e
 
Residue
T3.2 A1.0 [GATE-EC-2016-IISc]
102. For 2
sin( )
( )
z
f z
z
 , the residue of the pole at z
= 0 is __________
AC [GATE-EC-2010-IITG]
103. The residues of a complex function
 
  
1 2z
X z
z z 1 z 2


 
at its poles are
(A)
1 1
,
2 2
 and 1 (B)
1 1
,
2 2
and -1
(C)
1
,1
2
and
3
2

(D)
1
, 1
2
 and
3
2

109. TOPIC 4 – COMPLEX VARIABLE
www.targate.org Page 101
AD [GATE-EC-2006-IITKGP]
104. The value of the contour integral
2
z j 2
1
dz
z 4
  
 in positive sense is
(A)
j
2

(B)
2

(C)
j
2
 
(D)
2

AD [GATE-EC-2008-IISc]
105. Given  
 2
z
X z
z a


with z a
 , the
residue of   n 1
X z z  at z = a for n 0
 will
be
(A) n 1
a 
(B) n
a
(C) n
na (D) n 1
na 
AB [GATE-ME-2014-IITKGP]
106. If z is a complex variable, the vector of
3i
5
dz
z

is:
(A) – 0.511 – 1.57i
(B) – 0.511 + 1.57i
(C) 0.511 – 1.57i
(D) 0.511 + 1.57i
AB [GATE-EC-2017-IITR]
107. The residues of a function
3
1
( )
( 4)( 1)
f z
z z

 
are
(A)
1
27

and
1
125

(B)
1
125
and
1
125

(C)
1
27

and
1
5
(D)
1
125
and
1
5

A [GATE-EC-2008-IISc]
108. The residue of the function f(z) =
2 2
1
( 2) ( 2)
z z
 
at z = 2 is :
(A) 1
32
 (B) 1
16

(C) 1
16
(D) 1
32
-------0000------

110. ENGINEERING MATHEMATICS
Page 102 TARGATE EDUCATION GATE-(EE/EC)
Answer :
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
D D A B 10 C D D B C
11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
A C A B D A C D D D
21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
A B A D B D B B D B
31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
D B C A D B A * C B
41. 42. 43. 44. 45. 46. 47. 48. 49. 50.
B A A B B D A C C A
51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
* A 0.0 B B B B B B D
61. 62. 63. 64. 65. 66. 67. 68. 69. 70.
C D B D B * B * * B
71. 72. 73. 74. 75. 76. 77. 78. 79. 80.
* A B –1 B C A B A C
81. 82. 83. 84. 85. 86. 87. 88. 89. 90.
D B B C A D C A A C
91. 92. 93. 94. 95. 96. 97. 98. 99. 100.
C D B D C 3 A B A 6
101. 102. 103. 104. 105. 106. 107. 108.
C 1.0 C D D B B A
38. 5.9 to 6.1
51. –1.1 to –0.9
66. 81.60 to 81.80
68. –0.0001 to 0.0001
69. 0.039 to 0.043
71. –136 to –132

111. www.targate.org Page 103
05
Probability and Statistics
Probability Problems
Combined Problems
T1.1 AC [GATE-EC-2015-IITK]
1. Suppose A & B are two independent events
with probabilities P(A)  0 and P(B)  0.
Let A & B be their complements Which of
the following statement is FALSE?
(A) P(A B) P(A)P(B)
 
(B) P(A / B) P(A)

(C) P(A B) P(A) P(B)
  
(D) P(A B) P(A) P(B)
  
A0.75 T5.2 [GATE-BT-2019-IITM]
2. In pea plants, purple colour of flowers is
determined by the dominant allele while
white colour is determined by the recessive
allele. A genetic cross between two purple
flower-bearing plants results in an offspring
with white flowers. The probability that the
third offspring from these parents will have
purple flowers is _____ (rounded off to 2
decimal places).
AC [GATE-ME-2015-IITK]
3. If P(X) = 1/4, P(Y) = 1/3, and
P(XY)=1/12, the value of P(Y/X) is :
(A)
1
4
(B)
4
25
(C)
1
3
(D)
29
50
AA [GATE-CE-2016-IISc]
4. X and Y are two random independent events.
It is known that P(X)=0.40 and
  0.7
C
P X Y
  . Which one of the
following is the value of  
P X Y
 ?
(A) 0.7 (B) 0.5
(C) 0.4 (D) 0.3
D [GATE-EE-2005-IITB]
5. If P and Q are two random events, then the
following is true
(A) Independence of P and Q implies that
probability   0
P Q 

(B) Probability  
P Q 
 probability (P) +
probability (Q)
(C) If P and Q are mutually exclusive then
they must be independent
(D) Probability  
P Q
  probability (P)
AC [GATE-CS-1999-IITB]
6. Consider two events 1
E and 2
E such that
probability of 1
E , P( 1
E ) =
1
2
probability of
2
E , P( 2
E ) =
1
3
and probability of 1
E and
2
E ,P ( 1
E and 2
E ) =
1
5
. Which of the
following statements is/are true?
(A)  
1 2
P E or E is
2
3
(B) Events 1
E and 2
E are independent.
(C) Events 1
E and 2
E are not independent
(D) 1
2
E 4
E 5
 

 
 
AA [GATE-EC-1988-IITKGP]
7. Events A and B are mutually exclusive and
have nonzero probability. Which of the
following statement(s) are true?
(A)      
P A B P A P B
  
(B)    
C
P B P A

(C)      
P A B P A P B
 
(D)    
C
P B P A

D [GATE-CS-2000-IITKGP]
8. E1 and E2 are events in a probability space
satisfying the following constraints
1 2
( ) ( );
P E P E
 1 2
( ) 1
P E Y E  : 1 2
&
E E are
independent then 1
( )
P E 

112. ENGINEERING MATHEMATICS
Page 104 TARGATE EDUCATION GATE-(EE/EC)
(A) 0 (B)
1
4
(C)
1
2
(D) 1
D [GATE--2003-IITM]
9. Let P(E) denote the probability of an event
E. Given P(A) = 1, P(B) =
1
2
the values of
P(A/B) and P(B/A) respectively are
(A)
1 1
,
4 2
(B)
1 1
,
2 4
(C)
1
,1
2
(D)
1
1,
2
A0.06 [GATE-EE-2016-IISc]
10. Candidates were asked to come to an
interview with 3 pens each. Black, blue,
green and red were the permitted pen colours
that the candidate could bring. The
probability that a candidate comes with all 3
pens having the same colour is _____.
AD [GATE-EE-2015-IITK]
11. Two players, A and B, alternately keep
rolling a fair dice. The person to get a six
first wins the game. Given that player A
starts the game, the probability that A wins
the game is
(A) 5/11
(B) 1/2
(C) 7/13
(D) 6/11
A0.66-0.67 [GATE-MN-2016-IISc]
12. Two persons P and Q toss an unbiased coin
alternately on an understanding that whoever
gets the head frist wins. If P starts the game,
then the probability of P winning the game is
_______
AA T5.1 [GATE-AG-2019-IITM]
13. The relay race there are five teams A, B, C,
D and E. Assuming that each team has an
equal chance of securing any position (first,
second, third, fourth or fifth) in the race, the
probability that A, B and C finish first,
second and third, respectively is
(A)
1
60
(B)
1
20
(C)
1
10
(D)
3
10
A0.502 to 0.504 T5.1 [GATE-CS-2019-IITM]
14. Two numbers are chosen independently and
uniformly at random from the set
{1,2,….,13}. The probability (rounded off to
3 decimal places) that their 4-bit (unsigned)
binary representations have the same most
significant bit is _______.
A0.135-0.150 [GATE-MT-2016-IISc]
15. A coin is tossed three times.
It is known that out of three
tosses, one is a HEAD. The
probability of the other two
tosses also being HEADs is :
AC [GATE-EC-2012-IITD]
16. A fair coin is tossed till a head appears for a
first time. The probability that a number of
required tosses is odd, is :
(A)
1
3
(B)
1
2
(C)
2
3
(D)
3
4
D [GATE-CE-1995-IITK]
17. The probability that a number selected at
random between 100 and 999 (both
inclusive) will not contain the digit 7 is
(A)
16
25
(B)
3
9
10
 
 
 
(C)
27
75
(D)
18
25
AA [GATE-IN-2006-IITKGP]
18. Two dices are rolled
simultaneously. The probability
that the sum of digits on
the top surface of the two
dices are even, is:
(A) 0.5 (B) 0.25
(C) 0.167 (D) 0.125
A10 [GATE-CS-2014-IITKGP]
19. Four fair six sided dice are rolled. The
probability that sum of the results being 22 is
X/1296. The value of X is............
A0.18-0.19 [GATE-AG-2016-IISc]
20. The maximum one day rainfall depth at 20
year return period of a city is 150 mm. The
probability of one day rainfall equal to or
greater than 150 mm in the same city
occurring twice in 20 successive years
is________ .
B [GATE--1998-IITD]
21. The probability that two friends share the
same birth-month is
(A) 1/6 (B) 1/12
(C) 1/144 (D) 1/24

113. TOPIC 5 – PROBABILITY & STATISTICS
www.targate.org Page 105
AA [GATE-CS-2011-IITM]
22. If two fair coins are flipped and at least one
of the outcomes is known to be a head, what
is the probability that both outcomes are
heads?
(A)
1
3
(B)
1
4
(C)
1
2
(D)
2
3
B [GATE--2001-IITK]
23. Seven car accidents occurred in a week,
what is the probability that they all occurred
on same day?
(A) 7
1
7
(B) 6
1
7
(C) 7
1
2
(D) 7
7
2
A
24. probability that it will rain today is 0.5, the
probability that it will rain tomorrow is 0.6.
The probability that it will rain either today
or tomorrow is 0.5. What is the probability
that it will rain today and tomorrow?
(A) 0.3 (B) 0.25
(C) 0.35 (D) 0.4
D [GATE-EC-2005-IITB]
25. A fair dice is rolled twice. The probability
that an odd number will follow an even
number is :
(A)
1
2
(B)
1
6
(C)
1
3
(D)
1
4
AA [GATE-IN-2013-IITB]
26. What is the chance that a leap year, selected
at random, will contain 53 Sundays?
(A)
2
7
(B)
3
7
(C)
2
3
(D)
3
4
D [GATE-IN-2009-IITR]
27. If three coins are tossed simultaneously, the
probability of getting at least one head is
(A) 1/8 (B) 3/8
(C) 1/2 (D) 7/8
A0.13 TO 0.15 [GATE-EE-2014-IITKGP]
28. Consider a dice with the property that
probability of a face with n dots showing up
is proportional to n. The probability of the
face with three dots showing up is
_________.
AC [GATE-CE-2010-IITG]
29. Two coins are simultaneously tossed. The
probability of two heads simultaneously
appearing is :
(A)
1
8
(B)
1
6
(C)
1
4
(D)
1
2
AB [GATE-ME-2014-IITKGP]
30. You are given three coins : one has heads on
both faces, the second has tails on both faces
and the third has a head on one face and a
tail on the other. You choose a coin at
random and toss it, and it comes up heads.
The probability that the other face is tails is :
(A)
1
4
(B)
1
3
(C)
1
2
(D)
2
3
AD [GATE-ME-2013-IITB]
31. Out of all the 2-digit integers between 1 and
100, a 2-digit number has to be selected
random. What is the probability that the
selected number is not divisible by 7?
(A)
13
90
(B)
12
90
(C)
78
90
(D)
77
90
AD [GATE-ME-2011-IITM]
32. An unbiased coin is tossed five times. The
outcome of each toss is either a head or a
tail. The probability of getting at least one
head is :
(A)
1
32
(B)
13
32
(C)
16
32
(D)
31
32
AB [GATE-CE-2014-IITKGP]
33. A fair (unbiased) coin was tossed four times
in succession and resulted in the following
outcomes. (i) Head,(ii) Head, (iii) Head, (iv)
Head. The probability of obtaining a ‘Tail’
when the coin is tossed again is :

114. ENGINEERING MATHEMATICS
Page 106 TARGATE EDUCATION GATE-(EE/EC)
(A) 0 (B)
1
2
(C)
4
5
(D)
1
5
AD [GATE-CE-2013-IITB]
34. A 1 hour rainfall of 1cm has return period of
50 year. The probability that 1 hour of
rainfall 10 cm or more will occur in each of
two successive years is :
(A) 0.04 (B) 0.2
(C) 0.02 (C) 0.0004
AA [GATE-CE-2008-IISc]
35. A person on a trip has a choice between
private car and public transport. The
probability of using a private car is 0.45.
While using a public transport, further
choices available are bus and metro, out of
which the probability of commuting by a bus
is 0.55. In such a situation, the probability
(rounded upto two decimals) of using a car,
bus and metro, respectively would be
(A) 0.45, 0.30 and 0.25
(B) 0.45, 0.25 and 0.30
(C) 0.45, 0.55 and 0.00
(D) 0.45, 0.35 and 0.20
AC [GATE-CE-2004-IITD]
36. A hydraulic structure has four gates which
operate independently. The probability of
failure of each gate is 0.2. Given that gate 1
has failed, the probability that both gates 2
and 3 fail is :
(A) 0.240 (B) 0.200
(C) 0.040 (D) 0.008
A0.890 TO 0.899 [GATE-IN-2014-IITKGP]
37. The figure shows the schematic of a
production process with machines A, B and
C. An input job needs to be pre-processed
either by A or by B before it is fed to C,
from which the final finished product comes
out. The probabilities of failure of the
machines are given as:
     
P A 0.15,P B 0.05,P C 0.01
  
Assuming independence of failures of the
machines, the probability that a given job is
successfully processed (up to third decimal
place) is ___________.
AD [GATE-PI-2005-IITB]
38. Two dice are thrown simultaneously. The
probability that the sum of numbers on both
exceeds 8 is :
(A)
4
36
(B)
7
36
(B)
9
36
(D)
10
36
A0.26 [GATE-CS-2014-IITKGP]
39. The probability that a given positive integer
lying between 1 and 100 (both inclusive) is
NOT divisible by 2, 3 and 5 is _________
AB [GATE-CS-2012-IITD]
40. Suppose a fair six-sided dice is rolled once.
If the value on the dice is 1, 2 or 3, the dice
is rolled a second time. What is the
probability that the sum total of values that
turn up is at least 6?
(A)
10
21
(B)
5
12
(C)
2
3
(D)
1
6
AA [GATE-CS-2010-IITG]
41. What is the probability that a divisor of
99
10 is a multiple of 96
10
(A)
1
625
(B)
4
625
(C)
12
625
(D)
16
625
AB [GATE-CS-2009-IITR]
42. An unbiased dice (with 6 faces, numbered
from 1 to 6) is thrown. The probability that
the face value is odd is 90% of the
probability that the face value of even. The
probability of getting any even numbered is
the same.
If the probability that the face is even given
that it is greater than 3 is 0.75, which of the
following options is closest to the
probability that the face value exceeds 3?
(A) 0.453 (B) 0.468
(C) 0.485 (D) 0.492
AC [GATE-CS-2008-IISc]
43. Aishwarya studies either computer science
or mathematics everyday. If she studies
computer science on a day, then the
probability that she studies mathematics next
day is 0.6. If she studies mathematics on a
day, then the probability that she studies
computer science next day is 0.4. Given that
Aishwarya studies computer science on
Monday, what is the probability that she
studies computer science on Wednesday?

115. TOPIC 5 – PROBABILITY & STATISTICS
www.targate.org Page 107
(A) 0.24 (B) 0.36
(C) 0.4 (D) 0.6
AC [GATE-IN-2006-IITKGP]
44. You have gone to a cyber-cafe. You found
that the cyber-café has only three terminals.
All terminals are unoccupied. You and your
friend have to make a random choice of
selecting a terminal? What is the probability
that both of you will not select the same
terminal ?
(A)
1
9
(B)
1
3
(C)
2
3
(D) 1
A0.93 [GATE-CS-1994-IITKGP]
45. Let A, B and C be independent
events which occur with
probabilities 0.8, 0.5 and 0.3
respectively. The probability
of occurrence of at least one of
the event is ____________.
[GATE-CS-1994-IITKGP]
46. The probability of an event B is 1
P . The
probability that events A and B occur
together is 2
P , while the probability that A
and Boccur together is 3
P . The probability
of the event A in terms of 1
P , 2
P and 3
P is
____________.
3
1
P
P(A)
1 P


ANS :
AC [GATE-IT-2007-IITK]
47. Suppose there are two coins. The first coin
gives the heads with probability
5
8
when
tossed, while the second coin gives the heads
with probability
1
4
. One of the two coins is
picked up at random with equal probability
and tossed. What is the probability of
obtaining heads?
(A)
7
8
(B)
1
2
(C)
7
16
(D)
5
32
A7 [GATE-CH-2014-IITKGP]
48. In rolling of two fair dice, the outcome of an
experiment is considered to be the sum of
the numbers appearing on the dice. The
probability is the highest for the outcome
of___________
A0.083 [GATE-CH-2013-IITB]
49. For two rolls of a fair dice, the probability of
getting a 4 in the first roll and a number less
than 4 in the second roll, upto 3 digits after
the decimal point, is___________
A191 TO 199 [GATE-BT-2014-IITKGP]
50. If an unbiased coin is tossed
10 times, the probability that
all outcomes are same will be
__________ 5
10
 .
A0.075 - 0.085 [GATE-BT-2013-IITB]
51. One percent of cars manufactured by a
company are defective. What is the
probability (upto four decimals) that more
than two cars are defective, if 100 cars are
produced?
0.65to0.68 [GATE-EC-2014-IITKGP]
52. In a housing society half of the families have
a single child per family while the remaining
half have two children per family the
probability that a child picked at random has
a sibling is -----
AC [GATE-EC-2014-IITKGP]
53. An unbiased coin is tossed an infinite
number of times. The probability that the
fourth head appears at the tenth loss is
(A) 0.067
(B) 0.073
(C) 0.082
(D) 0.091
0.43to0.45 [GATE-EC-2014-IITKGP]
54. Parcels from sender S to receiver R pass
sequentially through two post offices. Each
post office has a probability
1
5
of losing an
incoming parcel, independently of all other
parcels. Given that a parcel is lost the
probability that it was lost by the second post
office is -----------.
A [GATE-CS-2004-IITD]
55. If a fair coin is tossed 4 times, what is the
probability that two heads and two tails will
result?
(A)
3
8
(B)
1
2
(C)
5
8
(D)
3
4
C [GATE-IT-2004-IITD]
56. In a class of 200 students, 125 students have
taken programming language course, 85
students have taken data structures course,
65 students have taken computer

116. ENGINEERING MATHEMATICS
Page 108 TARGATE EDUCATION GATE-(EE/EC)
organization course, 50 students have taken
both programming languages and data
structures, 35 students Have taken both
programming languages and computer
organization, 30 students have taken both
data structures and computer organization,
15 students have taken all the three courses.
How many students have not taken any of
the three courses?
(A) 15 (B) 20
(C) 25 (D) 35
C [GATE-EC-2007-IITK]
57. An examination consists of two papers,
paper 1 and paper 2. The probability of
failing in probability of failing in paper 1 is
0.3 and that in paper 2 is 0.2.Given that
student has failed in paper 2, probability of
failing in paper 1 is 0.6. The probability of a
student failing in both the papers is :
(A) 0.5 (B) 0.18
(C) 0.12 (D) 0.06
D [GATE-EC-2010-IITG]
58. A fair coin is tossed independently four
times. The probability of the event “The
number of times heads show up is more than
the number of times tails show up” is :
(A) 1/16 (B) 1/8
(C) 1/4 (D) 5/16
D [GATE-ME-2011-IITM]
59. An unbiased coin is tossed five times. The
outcome of each loss is either a head or a
tail.
Probability of getting at least one head is
_______ .
(A)
1
32
(B)
13
32
(C)
16
32
(D)
31
32
AD [GATE-ME-2005-IITB]
60. A single dice is thrown twice. What is the
probability that the sum is neither 8 nor 9?
(A)
1
9
(B)
5
36
(C)
1
4
(D)
3
4
B [GATE-IT-2004-IITD]
61. In a population of N families, 50% of the
families have three children, 30% of families
have two children and the remaining
families have one child. What is the
probability that a randomly picked child
belongs to a family with two children?
(A)
3
23
(B)
6
23
(C)
3
10
(D)
3
5
AB [GATTE-ME-2015-IITK]
62. The chance of a student passing an exam is
20% The chance of a student passing the
exam and getting above 90 % marks in it is
5%. Given that a student passes the
examination, the probability that the student
gets above 90 % marks is :
(A)
1
18
(B)
1
4
(C)
2
9
(D)
5
18
A0.07-0.08 [GATE-EC-2016-IISc]
63. The probability of getting a “head” in a
single toss of a biased coin is 0.3. The coin is
tossed repeatedly till a “head” is obtained. If
the tosses are independent, then the
probability of getting “head” for the first
time in the fifth toss is ______.
A0.027 [GATE-EC-2017-IITR]
64. Three fair cubical dice are thrown
simulataneously. The probability that all
three dice have the same number of dots on
the faces showing up is (up to third decimal
place)_______.
A0.37-0.38 [GATE-AG-2017-IITR]
65. The probability of getting two heads and two
tails from four tosses of the same coin is
_______ .
AC [GATE-AG-2017-IITR]
66. A couple has 2 children.
The probability that both
children are boys if the
older one is a boy is
(A) 1/4 (B) 1/3
(C) 1/2 (D) 1
A90 [GATE-BT-2017-IITR]
67. The angle (in degrees) between the vectors
ˆ ˆ ˆ
x i j 2k
  

and ˆ ˆ ˆ
y 2i j 1.5k
  

is
__________.
A0.5 [GATE-CE-2017-IITR]
68. A two faced fair coin has its faces designed
as head (H) and tail (T). This coin is tossed
three times in succession to record the
following outcomes: H, H, H. If the coin is
tossed one more time, the probability (up to
one decimal place) of obtaining H again,
given the previous realizations of H, H and
H, would be __________.

117. TOPIC 5 – PROBABILITY & STATISTICS
www.targate.org Page 109
AD [GATE-PH-2017-IITR]
69. There are 3 red socks, 4 green socks and 3
blue socks. You choose 2 socks. The
probability that they are of the same colour
is :
(A) 1/5 (B) 7/30
(C) 1/4 (D) 4/15
A0.75 [GATE-ME-2017-IITR]
70. Two coins are tossed simultaneously. The
probabilty (upto two decimal points
accuracy) of getting at least one head is
_____.
AC [GATE-TF-2018-IITG]
71. If A and B are two independent events such
that
1
( )
4
P A  and
2
( )
3
P B  , then
( )
P A B
 is equal to
(A)
11
12
(B)
1
12
(C)
3
4
(D)
5
6
A0.49 to 0.51 [GATE-PI-2018-IITG]
72. The probabilities of occurrence of events F
and G are P(F) = 0.3 and P(G) = 0.4,
respectively. The probability that both events
occur simultaneously is P(F G) 0.2
  . The
probability of occurrence of at least one
event P(F G)
 is _______.
A0.76 to 0.80 [GATE-PE-2018-IITG]
73. A box contains 100 balls of same size, of
which, 25 are black and 75 are white. Out of
25 black balls, 5 have a red dot. A trial
consists of randomly picking a ball and
putting it back in the same box, i.e.,
sampling is done with replacement. Two
such trials are done. The conditional
probability that no black ball with a red dot
is picked given that at least one black ball is
picked, is __________. (in fraction rounded-
off to two decimal places)
A0.49 to 0.51 [GATE-EY-2018-IITG]
74. A plant produces seeds that can be dispersed
by birds or mammals. The probability that a
seed is dispersed by a bird is 0.25, and by a
mammal is 0.5. The bird can disperse a seed
to three patches A, B, or C with a probability
0.5, 0.4 or 0.1, respectively. On the other
hand, the mammal disperses a seed to the
same patches A, B, or C, with a probability
0.15, 0.8 and 0.05, respectively. The
probability that a given seed is dispersed to
patch B is ___ (answer up to 1 decimal
place).
A0.59 to 03.61 T5.1 [GATE-MT-2019-IITM]
75. The probability of solving a problem by
Student A is (1/3), and the probability of
solving the same problem by Student B is
(2/5). The probability (rounded off to two
decimal places) that at least one of the
students solves the problem is _________.
A0.24 to 0.26 T5.1 [GATE-CH-2019-IITM]
76. Two unbiased dice are thrown. Each dice
can show any number between 1 and 6. The
probability that the sum of the outcomes of
the two dice is divisible by 4 is _______
(rounded off to two decimal places).
Problems on Combination
AA [GATE-ME-2014-IITKGP]
77. A box contains 25 parts of which 10 are
defective. Two parts are being drawn
simultaneously random manner from the
box. The probability of both the parts being
good is
(A)
7
20
(B)
42
125
(C)
25
29
(D)
5
9
A0.244 to 0.246 T5.2 [GATE-AG-2019-IITM]
78. Two cards are drawn at random and without
replacement from a pack of 52 playing cards.
The probability that both the cards are black
(rounded off to three decimal places) is
____.
A0.19 to 0.35 T5.1 [GATE-EY-2019-IITM]
79. A beaker contains a large number of
spherical nuts of two types, one with radius
1 cm and the other with 2 cm, in the ratio
2:1. A squirrel picks one nut from a random
point in this beaker. Assuming that the
beaker is well-mixed, the probability of
picking the smaller nut is ___ (round off to 1
decimal place).
A11.9 [GATE-CS-2014-IITKGP]
80. The security system at an IT office is
composed of 10 computers of which exactly
four are working. To check whether the
system is functional, the officials inspect
four of the computers picked at
random(without replacement). The system is
deemed functional if at least three of the four
computers inspected are working. Let the
probability that the system is deemed
functional be denoted by P. Then 100p = ......
AC [GATE-EE-2010-IITG]
81. A box contains 4 white balls and 3 red balls.
In succession, two balls are randomly
selected and removed from the box. Given

118. ENGINEERING MATHEMATICS
Page 110 TARGATE EDUCATION GATE-(EE/EC)
that first removed ball is white, probability
that the second removed ball is red is :
(A)
1
3
(B)
3
7
(C)
1
2
(D)
4
7
AD [GATE-ME-2012-IITD]
82. A box contains 4-red balls and 6-black balls.
Three balls are selected randomly from the
box one after another, without replacement.
The probability that the selected set contains
one red ball and two black balls is :
(A)
1
20
(B)
1
12
(C)
3
10
(D)
1
2
AD [GATE-ME-2006-IITKGP]
83. A box contains 20 defective items and 80
non-defective items. If two items are
selected at random without replacement,
what is the probability that both items are
defective?
(A)
1
5
(B)
1
25
(C)
20
99
(D)
19
495
AD [GATE-ME-2003-IITM]
84. A box contains 5 black and 5 red balls. Two
balls are randomly picked one after another
from the box, without replacement. The
probability for both balls being red is:
(A)
1
90
(B)
1
2
(C)
19
90
(D)
2
9
AC [GATE-ME-1997-IITM]
85. A box contains 5 black balls and 3 red balls.
A total of three balls are picked from the box
one after another, without replacing them
back. The probability of getting two black
balls and one red ball is
(A)
3
8
(B)
2
15
(C)
15
28
(D)
1
2
AB [GATE-CE-2006-IITKGP]
86. There are 25 calculators in a box. Two of
them are defective. Suppose 5 calculators are
randomly picked for inspection(i.e each has
the same change of being selected), what is
the probability that only one of the defective
calculators will be included in the
inspection?
(A)
1
2
(B)
1
3
(C)
1
4
(D)
1
5
AA [GATE-CS-2011-IITM]
87. A deck of 5 cards (each carrying a distinct
number from 1 to 5) is shuffled thoroughly.
Two cards are then removed one at a time
from the deck. What is the probability that
the two cards are selected with the number
on the first card being one higher than the
number on the second card?
(A)
1
5
(B)
4
25
(C)
1
4
(D)
2
5
AC [GATE-CS-1995-IITK]
88. A bag contains 10 white balls and 15 black
balls. Two balls are drawn in succession.
The probability that one of them is black and
other is white is :
(A)
2
3
(B)
4
5
(C)
1
2
(D)
2
1
A0.8145 [GATE-ME-2014-IITKGP]
89. A batch of hundred bulbs is
inspected by testing four
randomly chosen bulbs.
The batch is rejected even
if one of the bulbs is defective.
A batch typically has 5 defective
bulbs. The probability that
the current batch is accepted
is_____________.
AC [GATE-CS-1996-IISc]
90. The probability that top and bottom cards of
a randomly shuffled deck are both aces in
(A)
4 4
52 52
 (B)
4 3
52 52

(C)
4 3
52 51
 (C)
4 4
52 51

AB [GATE-IT-2005-IITB]
91. A bag contains 10 blue marbles, 20 black
marbles and 30 red marbles. A marble is
drawn from the bag, its colour recorded and
it is put back into the bag. This process is
repeated three times. The probability that no
two of the marbles drawn have the same
colour is

119. TOPIC 5 – PROBABILITY & STATISTICS
www.targate.org Page 111
(A)
1
36
(B)
1
6
(C)
1
4
(D)
1
3
D [GATE-EE-2004-IITD]
92. From a pack of regular playing cards, two
cards are drawn at random. What is the
probability that both cards will be kings, if
the card is NOT replaced?
(A) 1/26 (B) 1/52
(C) 1/169 (D) 1/221
C [GATE-EE-2003-IITM]
93. A box contains 10 screws, 3 of which are
defective. Two screws are drawn at random
with replacement. The probability that none
of the two screws is defective will be :
(A) 100% (B) 50%
(C) 49% (D) None of these
C [GATE-ME-2010-IITG]
94. A box contains 2 washers, 3 nuts and 4 bolts.
Items are drawn from the box at random one
at a time without replacement. The
probability of drawing 2 washers first
followed by 3 nuts and subsequently the 4
bots is :
(A) 2/315 (B) 1/630
(C) 1/1260 (D) 1/2520
AA [GATE-PI-2015-IITK]
95. A product is an assemble of 5 different
components. The product can be sequentially
assembled in two possible ways. If the 5
components are placed in a box and these are
drawn at random from the box, then the
probability of getting a correct sequence is :
(A)
2
5!
(B)
2
5
(C)
2
(5 2)!

(D)
2
(5 3)!

AA [GATE-ME-2016-IISc]
96. Three cards were drawn from a pack of 52
cards. The probability that they are a king, a
queen, and a jack is
(A)
16
5525
(B)
64
2197
(C)
3
13
(D)
8
16575
A0.50-0.55 [GATE-PE-2016-IISc]
97. A box has a total of ten identical sized balls.
Seven of these balls are black in colour and
the rest three are red. Three balls are picked
from the box one after another without
replacement.
The probability that two of the balls are
black and one is red is equal to ______.
A0.39-0.43 [GATE-ME-2016-IISc]
98. The probability that a screw manufactured
by a company is defective is 0.1. The
company sells screws in packets containing
5 screws and gives a guarantee of
replacement in one or more screws in the
packet are found to be defective. The
probability that packet would have to be
replaced is _______
AA [GATE-IN-2016-IISc]
99. An urn contains 5 red and 7 green balls. A
ball is drawn at random and its colour is
noted. The ball is placed back into the urn
along with another ball of the same colour.
The probability of getting a red ball in the
next draw is :
(A)
65
156
(B)
67
156
(C)
79
156
(D)
89
156
AB [GATE-AG-2017-IITR]
100. A box contains three white and four red
balls. Two balls are drawn randomly in
sequence. If the first draw resulted in a red
ball, the probability of getting a second red
ball in the next draw is :
(A) 0.33 (B) 0.50
(C) 0.67 (D) 0.75
A0.59 to 0.61 T5.1 [GATE-PE-2019-IITM]
101. A box contains 2 red and 3 black balls.
Three balls are randomly chosen from the
box and are placed in a bag. Then the
probability that there are 1 red and 2 black
balls in the bag, is _____.
AB [GATE-CS-2017-IITR]
102. A test has twenty questions worth 100 marks
in total. There are two types of questions.
Multiple choice questions are worth 3 marks
each and essay questions are worth 11 marks
each. How many multiple choice questions
does the exam have ?
(A) 12 (B) 15
(C) 18 (D) 19

120. ENGINEERING MATHEMATICS
Page 112 TARGATE EDUCATION GATE-(EE/EC)
AB [GATE-MT-2018-IITG]
103. A classroom of 20 students can be
categorized on the basis of blood-types: 5
students each with “A”, “B”, “AB”, and “O”
blood-types. If four students are selected at
random from this class, what is the
probability that each student has a different
blood-type?
(A) 0.2500 (B) 0.1289
(C) 0.0625 (D) 0.0156
A0.0019 to 0.0021 [GATE-EY-2018-IITG]
104. The probability that a bush has a cricket is
0.1. The probability of a spider being present
on a bush is 0.2. When both a spider and a
cricket are present on a bush, the probability
of encountering each other is 0.2. The
probability of a spider consuming a cricket it
encounters is 0.5. Assuming that predation
only occurs on bushes, the probability that a
cricket is preyed on by a spider is ____
(answer up to 3 decimal places).
AC [GATE-ME-2018-IITG]
105. A six-faced fair dice is rolled five times. The
probability (in %) of obtaining“ONE” at
least four times is
(A) 33.3 (B) 3.33
(C) 0.33 (D) 0.0033
AB [GATE-ME-2018-IITG]
106. Four red balls, four green balls and four blue
balls are put in a box. Three balls are pulled
out of the box at random one after another
without replacement. The probability that all
the three balls are red is
(A) 1/72 (B) 1/55
(C) 1/36 (D) 1/27
AB [GATE-IN-2018-IITG]
107. Consider a sequence of tossing of a fair coin
where the outcomes of tosses are
independent. The probability of getting the
head for the third time in the fifth toss is
(A)
5
16
(B)
3
16
(C)
3
5
(D)
9
16
A0.272 to 0.274 [GATE-MA-2018-IITG]
108. Let X be the number of heads in 4 tosses of a
fair coin by Person 1 and let Y be the number
of heads in 4 tosses of a fair coin by Person
2. Assume that all the tosses are
independent. Then the value of ( )
P X Y

correct up to three decimal places is _____.
AB [GATE-MA-2018-IITG]
109. An urn contains four balls, each ball having
equal probability of being white or black.
Three black balls are added to the urn. The
probability that five balls in the urn are black
is
(A) 2/7 (B) 3/8
(C) 1/2 (D) 5/7
AB [GATE-CE-2018-IITG]
110. Each of the letters arranged as below
represents a unique integer from 1 to 9. The
letters are positioned in the figure such that
(A  B  C), (B  G  E) and (D  E  F)
are equal.
Which integer among the following choices
cannot be represented by the letters A, B, C,
D, E, F or G?
A D
B G E
C F
(A) 4 (B) 5
(C) 6 (D) 9
AC [GATE-CE-2018-IITG]
111. Which one of the following matrices is
singular?
(A)
2 5
1 3
 
 
 
(B)
3 2
2 3
 
 
 
(C)
2 4
3 6
 
 
 
(D)
4 3
6 2
 
 
 
AC [GATE-CE-2018-IITG]
112. For the given orthogonal matrix Q,
3/ 7 2 / 7 6 / 7
6 / 7 3/ 7 2 / 7
2 / 7 6 / 7 3 / 7
Q
 
 
 
 
 

 
The inverse is :
(A)
3/ 7 2 / 7 6 / 7
6 / 7 3/ 7 2 / 7
2 / 7 6 / 7 3 / 7
 
 

 
 

 
(B)
3/ 7 2 / 7 6 / 7
6 / 7 3/ 7 2 / 7
2 / 7 6 / 7 3 / 7
  
 
 
 
 
 
 
 
(C)
3/ 7 6 / 7 2 / 7
2 / 7 3/ 7 6 / 7
6 / 7 2 / 7 3 / 7

 
 
 
 

 

121. TOPIC 5 – PROBABILITY & STATISTICS
www.targate.org Page 113
(D)
3/ 7 6 / 7 2 / 7
2 / 7 3/ 7 6 / 7
6 / 7 2 / 7 3 / 7
 
 
 
  
 
 
 
 
AC [GATE-CS-2018-IITG]
113. A six sided unbiased die with four green
faces and two red faces is rolled seven times.
Which of the following combinations is the
most likely outcome of the experiment?
(A) Three green faces and four red faces.
(B) Four green faces and three red faces.
(C) Five green faces and two red faces.
(D) Six green faces and one red face.
Problems from Binomial
AB [GATE-EE-2014-IITKGP]
114. A fair coin tossed n times. The probability
that the difference between the number of
heads and tails is (n - 3) is
(A)
n
2
(B) 0
(C)
n n
n 3
C 2
 (D)
n 3
2 
AA [GATE-TF-2019-IITM]
115. Let X be a binomial random variable with
mean 1 and variance
3
4
. The probability that
X takes the value 3 is :
(A)
3
64
(B)
3
16
(C)
27
64
(D)
3
4
AA [GATE-CS-2007-IITK]
116. There are n stations in a slotted LAN. Each
stations attempts to transmit with a
probability p in each time slot. What is the
probability that ONLY one station transmits
in a given time slot?
(A)  n 1
np 1 p

 (B)  n 1
1 p


(B)  n 1
p 1 p

 (D)  n 1
1 1 p

 
AA [GATE-CS-2010-IITG]
117. Consider a company that assembles
computers. The probability of a faulty
assembly of any computer is p. The
company therefore subjects each computer to
a testing process. This testing process gives
the correct result for any computer with a
probability of q. What is the probability of a
computer being declared faulty?
(A) pq + (1 - p)(1 - q) (B) (1 - q)p
(C) (1 - p)q (D) pq
AC [GATE-PI-2016-IISc]
118. A fair coin is tossed N times. The probability
that head does not turn up in any of the
tosses is
(A)
1
1
2
N 
 
 
 
(B)
1
1
1
2
N 
 
  
 
(C)
1
2
N
 
 
 
(D)
1
1
2
N
 
  
 
AD [GATE-CS-1996-IISc]
119. Two dice are thrown simultaneously. The
probability at least one of them will have 6
facing up is :
(A)
1
36
(B)
1
3
(C)
25
36
(D)
11
36
AD [GATE-ME-1997-IITM]
120. The probability of a defective piece being
produced in a manufacturing process is 0.01.
The probability that out of 5 successive
pieces, only one is defective, is
(A)    
4
0.99 0.01
(B)   4
0.99 0.01
(C)   4
5 0.99 0.01

(D)    
4
5 0.99 0.01

AB [GATE-ME-2015-IITK]
121. The probability of obtaining at least two
‘SIX’ in throwing a fair dice 4 times is
(A) 425/432 (B) 19/144
(C) 13/144 (D) 125/432
AD [GATE-CE-2012-IITD]
122. In an experiment, positive and negative
values are equally likely to occur. The
probability of obtaining atmost one negative
value in five trials is :
(A)
1
32
(B)
2
32
(C)
3
32
(D)
6
32
0.27 [GATE-ME-2014-IITKGP]
123. Consider an unbiased cubic dice with
opposite faces coloured identically and each
face coloured red, blue or green such that
each colour appears only two times on the
dice. If the dice is thrown thrice, the
probability of obtaining red colour on top
face of the dice at least twice is
_____________.

122. ENGINEERING MATHEMATICS
Page 114 TARGATE EDUCATION GATE-(EE/EC)
AB [GATE-EE-2005-IITB]
124. A fair coin is tossed three times in
succession. If the first toss produces a head,
then the probability of getting exactly two
heads in three tosses is :
(A)
1
8
(B)
1
2
(C)
3
8
(D)
3
4
AA [GATE-ME-2008-IISc]
125. A coin is tossed 4 times. What is the
probability of getting heads exactly 3 times?
(A)
1
4
(B)
3
8
(C)
1
2
(D)
3
4
AD [GATE-ME-2001-IITK]
126. An unbiased coin is tossed three times. The
probability that the head turns up in exactly
two cases is
(A)
1
9
(B)
1
8
(C)
2
3
(D)
3
8
AA [GATE-CS-2006-IITKGP]
127. For each element for a set of size 2n, an
unbiased coin is tossed. The 2n coin tosses
are independent. An element is chosen if the
corresponding coin toss is head. The
probability that exactly n elements are
chosen is:
(A)
2n
n
n
C
4
(B)
2n
n
n
C
2
(C) 2n
n
1
C
(D)
1
2
AB [GATE-BT-2012-IITD]
128. A disease is inherited by a child with a
probability of
1
4
. In a family with two
children, the probability that exactly one
sibling is affected by this disease is:
(A)
1
4
(B)
3
8
(C)
7
16
(D)
9
16
B [GATE-CS-1998-IITD]
129. A die is rolled three times. The probability
that exactly one odd number turns up among
the three outcomes is
(A)
1
6
(B)
3
8
(C)
1
8
(D)
1
2
B [GATE-ME-2005-IITB]
130. A lot had 10% defective items. Ten items are
chosen randomly from this lot. The
probability that exactly 2 of the chosen items
are defective is
(A) 0.0036 (B) 0.1937
(C) 0.2234 (D) 0.3874
AA [GATE-ME-1993-IITB
131. If 20 percent managers are technocrats, the
probability that a random committee of 5
managers consists of exactly 2 technocrats
is:
(A) 0.2048 (B) 0.4000
(C) 0.4096 (D) 0.9421
AC [GATE-CH-2018-IITG]
132. A watch uses two electronic circuits (ECs).
Each EC has a failure probability of 0.1 in
one year of operation. Both ECs are required
for functioning of the watch. The probability
of the watch functioning for one year
without failure is
(A) 0.99 (B) 0.90
(C) 0.81 (D) 0.80
Problems on Bay’s
C [GATE-PI-2010-IITG]
133. Two white and two black balls, kept in two
bins, are arranged in four ways as shown
below. In each arrangement, a bin has to be
chosen randomly and only one ball needs to
be picked randomly from the chosen bin.
Which one of the following arrangements
has the highest probability for getting a
white ball picked?
(A)
(B)
(C)
(D)

123. TOPIC 5 – PROBABILITY & STATISTICS
www.targate.org Page 115
A0.25 to 0.27 T5.2 [GATE-EE-2019-IITM]
134. The probability of a resistor being defective
is 0.02. There are 50 such resistors in a
circuit. The probability of two or more
defective resistors in the circuit (round off to
two decimal places) is ____.
0.65 [GATE-ME-2014-IITKGP]
135. A group consists of equal
number of men and women.
Of this group 20% of the
men and 50 % of the
women are unemployed. If a
person is selected at random from the group,
the probability of the selected person being
employed is_____________.
AB [GATE-ME-2012-IITD]
136. An automobile plant contracted to buy shock
absorbers from two suppliers X and Y. X
supplies 60% and y supplies 40% of the
shock absorbers. All the shock absorbers are
subjected to a quality test. The ones that past
the quality test are considered reliable of
‘X’ s shock absorbers, 96% are reliable of
‘Y’ s shock absorbers, 72% are reliable
The probability that a randomly chosen
shock absorber, which is found to reliable, is
made by Y is
(A) 0.288 (B) 0.334
0.48 TO 0.49 [GATE-CE-2014-IITKGP]
137. 10% of the population in a town is HIV+. A
new diagnostic kit for HIV detection is
available, this kit correctly identifies HIV+
individuals 95% of the time, and HIV-
individuals 89% of the time. A particular
patient is tested using this kit and is found to
be positive. The probability that the
individual is actually positive is ________.
AC [GATE-CE-2011-IITM]
138. There are two containers, with one
containing 4 Red and 3 Green balls and the
other containing 3 Blue and 4 Green balls.
One ball is drawn in random from each
container. The probability that one of the
balls is Red and other is Blue will be
(A)
1
7
(B)
9
49
(C)
12
49
(D)
3
7
AD [GATE-PI-2014-IITKGP]
139. In a given day in the rainy season, it may
rain 70% of the time. If it rains, chance that
village fair will make a loss on that day is
80%. However, if it does not rain, chance,
that the fair will make a loss on that day is
only 10%. If the fair has not made loss on a
given day in the rainy season, what is the
probability it has not rained on the day?
(A)
3
10
(B)
9
11
(C)
14
17
(D)
27
41
AD [GATE-ME-2013-IITB]
140. The probability that a student knows the
correct answer to a multiple choice question
is is 2/3. If the student does not know the
answer, then the student guesses the answer.
The probability of the guessed answer being
correct is
1
4
. Given that the student has
answered the question correctly, the
conditional probability that the student
knows the correct answer is
(A)
2
3
(B)
3
4
(C)
5
6
(D)
8
9
AC [GATE-CH-2013-IITB]
141. In a factory, two machines M1 and M2
manufactures 60% and 40% of the auto
components respectively. Out of the total
production, 2% of M1 and 3% of M2 are
found to be defective. If a randomly drawn
auto component from the combined lot is
found defective, what is the probability that
it was manufactured by M2?
(A) 0.35 (B) 0.45
(C) 0.5 (D) 0.4
AA [GATE-EE-2017-IITR]
142. An urn contains 5 red balls and 5 black balls.
In the first draw, one ball is picked at
random and discarded without noticing its
colour. The probability to get a red ball in
the second draw is :
(A)
1
2
(B)
4
9
(C)
5
9
(D)
6
9
A0.65 to 0.68 [GATE-MA-2017-IITR]
143. Let E and F be any two events with
P[E]=0.4,P(F)=0.3 and c
P(F|E)=3P(F|E ) .
Then P(E|F) equals (rounded to 2 decimal
places)_______.
AB [GATE-IN-2018-IITG]
144. Two bags A and B have equal number of
balls. Bag A has 20% red balls and 80%
green balls. Bag B has 30% red balls, 60%
green balls and 10% yellow balls. Contents
of Bags A and B are mixed thoroughly and a
ball is randomly picked from the mixture.

124. ENGINEERING MATHEMATICS
Page 116 TARGATE EDUCATION GATE-(EE/EC)
What is the chance that the ball picked is
red?
(A) 20% (B) 25%
(C) 30% (D) 40%
AC [GATE-EC-2018-IITG]
145. A cab was involved in a hit and run accident
at night. You are given the following data
about the cabs in the city and the accident.
(i) 85% of cabs in the city are green and
the remaining cabs are blue.
(ii) A witness identified the cab involved in
the accident as blue.
(iii) It is known that a witness can correctly
identify the cab colour only 80% of the
time.
Which of the following options is closest
to the probability that the accident was
caused by a blue cab?
(A) 12%
(B) 15%
(C) 41%
(D) 80%
A0.60 to 0.62 [GATE-CS-2018-IITG]
146. Consider Guwahati (G) and Delhi (D) whose
temperatures can be classified as high (H),
medium (M) and low (L). Let ( )
G
P H denote
the probability that Guwahati has high
temperature. Similarly, ( )
G
P M and ( )
G
P L
denotes the probability of Guwahati having
medium and low temperatures respectively.
Similarly, we use ( ), ( )
D D
P H P M and
( )
D
P L for Delhi.
The following table gives the conditional
probabilities for Delhi’s temperature given
Guwahati’s temperature.
D
H D
M D
L
G
H 0.40 0.48 0.12
G
M 0.10 0.65 0.25
G
L 0.01 0.50 0.49
Consider the first row in the table above. The
first entry denotes that if Guwahati has high
temperature ( )
G
H then the probability of
Delhi also having a high temperature ( )
D
H
is 0.40; i.e., Similarly, the next two entries
are ( | )
D G
P M H = 0.48
and ( | ) 0.12
D G
P L H  . Similarly for the
other rows.
If it is known that ( ) 0.2
G
P H  ,
( ) 0.5
G
P M  , and ( ) 0.3
G
P L  , then the
probability (correct to two decimal places)
that Guwahati has high temperature given
that Delhi has high temperature is _____.
**********
Probability Distribution
Statistics
AD [GATE-ME-2014-IITKGP]
147. In the following table, X is a discrete
random variable and p(x) is the probability
density. The standard deviation of x is
X 1 2 3
P(x) 0.3 0.6 0.1
(A) 0.18 (B) 0.36
(C) 0.54 (D) 0.6
A2.5 [GATE-IN-2016-IISc]
148. A voltage V1 is measured 100 times and its
average and standard deviation are 100 V
and 1.5 V respectively. A second voltage V2,
which is independent of V1, is measured 200
times and its average and standard deviation
are 150 V and 2 V respectively. V3 is
computed as: V3 = V1 + V2. Then the
standard deviation of V3 in volt is ____.
AD [GATE-EC-2006-IITKGP]
149. Three companies X, Y and Z supply
computers to a university. The percentage of
computers supplied by them and probability
of those being defective are tabulated below
Com-
pany
% of
computers
supplied
Probability of
being
defective
X 60% 0.01
Y 30% 0.02
Z 10% 0.03
Given that a computer is defective, the
probability that it was supplied by Y is:
(A) 0.1 (B) 0.2
(C) 0.3 (D) 0.4
AC [GATE-EE-2007-IITK]
150. A loaded dice has following probability
distribution of occurances
Dice Value Probability
1 1/4
2 1/8
3 1/8
4 1/8
5 1/8
6 1/4

125. TOPIC 5 – PROBABILITY & STATISTICS
www.targate.org Page 117
If three identical dice as the above are
thrown, the probability of occurrence of
values 1, 5 and 6 on the three dice is
(A) Same as that of occurrence of 3, 4, 5
(B) Same as that of occurrence of 1, 2, 5
(C)
1
128
(D)
5
8
A128 to 130 T5 statistics [GATE-BT-2019-
IITM]
151. A new game is being introduced in a casino.
A player can lose Rs. 100, break even, win
Rs. 100, or win Rs. 500. The probabilities
(P(X)) of each of these outcomes (X) are
given in the following table :
X (in Rs.) -100 0 100 500
P(X) 0.25 0.5 0.2 0.05
The standard deviation ( )
 for the casino
payout is Rs. ______ (rounded off to the
nearest integer).
AC [GATE-PI-2014-IITKGP]
152. Marks obtained by 100 students in an
examination are given in the table
No. Mark obtained
Number of
students
1. 25 20
2. 30 20
3. 35 40
4. 40 20
What would be the mean, median and mode
of the marks obtained by the students?
(A) Mean: 33; Median: 35; Mode: 40
(B) Mean: 35; Median: 32.5; Mode: 40
(C) Mean: 33; Median: 35; Mode: 35
(D) Mean: 35; Median: 32.5; Mode: 35
A54.49-54.51 [GATE-CE-2016-IISc]
153. The spot speeds (expressed in km/hr)
observed at a road section are 66, 62, 45, 79,
32, 51, 56, 60, 53 and 49. The median speed
(expressed in km/hr) is ________.
(Note: answer with one decimal accuracy)
AB [GATE-EC-2009-IITR]
154. A discrete variable X takes values from 1 to
5 with probabilities as shown in the table. A
student calculates the mean X as 3.5 and her
teacher calculates the variance of X as 1.5.
Which of the following statements is true?
k 1 2 3 4 5
P(X = k) 0.1 0.2 0.4 0.2 0.1
(A) Both the student and the teacher are
right
(B) Both the student and the teacher are
wrong
(C) The student is wrong but the teacher is
right
(D) The student is right but the teacher is
wrong
AC [GATE-ME-2004-IITD]
155. The following data about the flow of liquid
was observed in a continuous chemical
process plant
Flow
rate(litre/sec)
7.5
to
7.7
7.7
to
7.9
7.9
to
8.1
8.1
to
8.3
8.3
to
8.5
8.5
to
8.7
Frequency 1 5 35 17 12 10
Mean flow rate of liquid is:
(A) 8.00 litres/sec (B) 8.06 litres/sec
(B) 8.16 litres/sec (D) 8.26 litres/sec
A3 T5 [GATE-ME-2019-IITM]
156. If x is the mean of data 3, x, 2 and 4, then the
mode is ______.
A20 T5 Statistic[GATE-BT-2019-IITM]
157. The median value for the dataset (12, 10, 16,
8, 90, 50, 30, 24) is ______.
A5.26 to 5.28 T5 Statistics [GATE-AG-2019-
IITM]
158. The mean absolute deviation about the
median for the data 3, 9, 5, 3, 12, 10, 18, 4,
7, 19, 21 (rounded off to two decimal places)
is _____.
AA [GATE-ME-2014-IITKGP]
159. A machine produces 0, 1 and 2 defective
pieces in a day with associated probability of
1 2
,
6 3
and
1
6
respectively. The mean value
and the variance of the number of defective
pieces produced by the machine in a day,
respectively are
(A)
1
1 and
3
(B)
1
3
and 1
(C)
4
1 and
3
(D) None
A [GATE--2000-IITKGP]
160. In a manufacturing plant, the probability of
making a defective bolt is 0.1. The mean and
standard deviation of defective bolts in a
total of 900 bolts are respectively

126. ENGINEERING MATHEMATICS
Page 118 TARGATE EDUCATION GATE-(EE/EC)
(A) 90 and 9 (B) 9 and 90
(C) 81 and 9 (D) 9 and 81
C [GATE-CE-2007-IITK]
161. If the standard deviation of the spot speed of
vehicles in a highway is 8.8 kemps and the
mean speed of the vehicles is 33 kmph, the
coefficient of variation in speed is
(A) 0.1517 (B) 0.1867
(C) 0.2666 (D) 0.3646
74 TO 75 [GATE-ME-2014-IITKGP]
162. Demand during lead time with associated
probability shown below:
Demand 50 70 75 80 85
Probability 0.15 0.14 0.21 0.20 0.30
Expected demand during lead time
is_________________.
AC [GATE-ME-2017-IITR]
163. A sample of 15 data is as follows : 17, 18,
17, 17, 13, 18, 5, 5, 6, 7, 8, 9, 20, 17, 3. The
mode of the data is
(A) 4 (B) 13
(C) 17 (D) 20
A6.4 to 6.5 [GATE-MN-2018-IITG]
164. The sample standard deviation for the
following set of observations is _______. 40,
45, 50 and 55
AC [GATE-PI-2018-IITG]
165. In a mass production firm, measurements are
carried out on 10000 pairs of shaft and hole.
The mean diameters of the shaft and the hole
are 37.53 mm and 37.59 mm, respectively.
The corresponding standard deviations are
0.03 mm and 0.04 mm. The mean clearance
and its standard deviation (both in mm),
respectively, are
(A) 0.06 and 0.07
(B) 0.06 and 0.06
(C) 0.06 and 0.05
(D) 0.07 and 0.01
A3.49 to 3.51 [GATE-PI-2018-IITG]
166. Weights (in kg) of six products are 3, 7, 6, 2,
3 and 4. The median weight (in kg, up to one
decimal place) is _______.
A99-101 [GATE-EY-2018-IITG]
167. If the mean of a sample is 5, and the
variance is 25, the PERCENT coefficient of
variation is ___.
A5.9 to 6.1 [GATE-EY-2018-IITG]
168. The frequency distribution of beak sizes of a
bird species is symmetric but not normally
distributed. If the mean value of beak size is
6 mm, standard deviation is 25 mm and
kurtosis is 10, then the median is ____ mm.
AD [GATE-IN-2018-IITG]
169. X and Y are two independent random
variables with variances 1 and 2,
respectively. Let Z X Y
  . The variance of
Z is
(A) 0 (B) 1
(C) 2 (D) 3
Expectation
A25 TO 25 [GATE-CE-2014-IITKGP]
170. In any given year, the probability of an
earthquake greater than Magnitude of 6
occurring in the Garhwal Himalayas is 0.04.
The average time between successive
occurrences of such earthquakes is
__________ years.
A11 T5.2 [GATE-EC-2019-IITM]
171. If X and Y are random variables such that
E[2X+Y] = 0 and E[X+2Y] = 33, then
E[X]+E[Y] = _____ .
A3.88 [GATE-CS-2014-IITKGP]
172. Each of nine words in the sentence “The
quick brown fox jumps over the lazy dog” is
written on a separate piece of paper. The
nine pieces of paper are kept in a box. One
of the pieces is drawn at random from the
box. The expected length of word drawn
is................
(The answer should be rounded to one
decimal place.)
AD [GATE-CS-2004-IITD]
173. A point is randomly selected with uniform
probability in the x, y plane within the
rectangle with corners at (0, 0) , (1, 0), (1, 2)
and (0, 2). If p is the length of the position
vector of the point, the expected value of
2
p
is :
(A)
2
3
(B) 1
(C)
4
3
(D)
5
3
AD [GATE-CS-2014-IITKGP]
174. An examination paper has 150 multiple-
choice questions of one mark each, with
each question having four choices. Each
incorrect answer fetches -0.25 mark.
Suppose 1000 students choose all their
answers randomly with uniform probability.
The sum total of the expected marks
obtained by all these students is:

127. TOPIC 5 – PROBABILITY & STATISTICS
www.targate.org Page 119
(A) 0 (B) 2550
(C) 7525 (D) 9375
2.9to3.1 [GATE-EC-2014-IITKGP]
175. A fair coin is tossed reqeatedly
till both head and tail appear at
least once. The average number
of tosses required is ----------.
A6 [GATE-EC-2015-IITK]
176. A fair die with faces {1, 2, 3, 4, 5, 6} is
thrown repeatedly till ‘3’ is observed for the
first time. Let X denote the number of times
the dies is thrown. The expected value of X
is _____.
D [GATE-ME-2007-IITK]
177. Let X and Y be two independent random
variables. Which one of the relations b/w
expectation (E), variance (Var) and
covariance (Cov) given below is FALSE?
(A) E(XY) = E(X) E(Y)
(B) cov (X, Y) = 0
(C) Var(X + Y) = Var(X) + Var(Y)
(D) E(X2
Y2
) = (E(X))2
(E(y))2
AB [GATE-EC-2014-IITKGP]
178. Let X be a real-valued random variable with
E[X] and E[X2
] denoting the mean values of
X and X2
, respectively. The relation which
always holds
(A) (E[X])2
> E[X2
]
(B) (E[X2
])  (E[X2
])
(C) E[X2
] = (E[X]) 2
(D) E[X2
] > (E[X]) 2
AA [GATE-PI-2007-IITK]
179. The random variable X takes on the values
1, 2 or 3 with probabilities
   
2 5P 1 3P
,
5 5
 
and
 
1.5 2P
5

, respectively. The values of
P and E[X] are respectively
(A) 0.05, 1.87 (B) 1.90, 5.87
(C) 0.05, 1.10 (D) 0.25, 1.40
A1.5 [GATE-EC-2015-IITK]
180. Let the random variable X
represent the number of times
a fair coin needs to be tossed
till two consecutive heads appear
for the first time. The expectation
of X is _____.
49.9 TO 50.1 [GATE-EC-2014-IITKGP]
181. Let X be a random variable
which is uniformly chosen
from the set of positive odd
numbers less than 100. The
expectation, E[X], is ____.
A0.25 [GATE-CS-2014-IITKGP]
182. Suppose you break a stick of unit length at a
point chosen uniformly at random, then the
expected length of shorter stick is...........
A2.4-2.6 [GATE-EC-2017-IITR]
183. Passengers try repeatedly to get a seat
reservation in any train running between two
stations until they are successful. If there is
40% chance of getting reservation in any
attempt by a passenger, then the average
number of attempts that passengers need to
make to get a seat reserved is _____ .
A3.5 [GATE-ME-2017-IITR]
184. A six-face fair dice is rolled a
large number of times. The
mean value of the outcomes
is _______ .
A2.1 to 2.1 [GATE-BT-2018-IITG]
185. The probability distribution
for a discrete random variable
X is given below.
X 1 2 3 4
P(X) 0.3 0.4 0.2 0.1
The exceptation value
of X is (up to one decimal place) ______.
A0.25 [GATE-EC-2018-IITG]
186. Let 1 2 3
, ,
X X X and 4
X be independent
normal random variables with zero mean and
unit variance. The probability that 4
X is the
smallest among the four is _______.
Normal Distribution
49 TO 51 [GATE-ME-2014-IITKGP]
187. A nationalized bank has found that the daily
balance available in its savings accounts
follows a normal distribution with a mean of
Rs. 500 and a standard deviation of Rs.50.
The percentage of savings account holders,
who maintain an average daily balance more
than Rs.500 is_______________.
0.79-3.01 [GATE-EC-2014-IITKGP]
188. Let X be a zero mean unit variance Gaussian
random variable. E[|X|] is equal to _______
B [GATE-IN-2008-IISc]
189. Consider a Gaussian distributed random
variable with zero mean and standard
deviation  . The value of its cumulative
distribution function at the origin will be
(A) 0 (B) 0.5
(C) 1 (D) 4

128. ENGINEERING MATHEMATICS
Page 120 TARGATE EDUCATION GATE-(EE/EC)
AC [GATE-IN-2010-IITG]
190. The diameters of 10000 ball bearings were
measured. The mean diameter and standard
deviation were found to be 10mm and 0.05
mm respectively. Assuming Gaussian
distribution of measurements, it can be
expected that the number of measurements
more than 10.15 mm will be :
(A) 230 (B) 115
(C) 15 (D) 2
AA [GATE-CE-2012-IITD]
191. The annual precipitation data of a city is
normally distributed with mean and standard
deviation as 1000mm and 200mm,
respectively. The probability that the annual
precipitation will be more than 1200 mm is:
(A) <50% (B) 50%
(C) 75% (D) 100%
AD [GATE-CE-2006-IITKGP]
192. A class of first year B.Tech. students is
composed of four batches A, B, C and D,
each consisting of 30 students. It is found
that the sessional marks of students in
Engineering Drawing in batch C have a
mean of 6.6 and standard deviation of 2.3.
The mean and standard deviation of the
marks for the entire class are 5.5 and 4.2
respectively. It is decided by the course
instructor to normalize the marks of the
students of all batches to have the same
mean and standard deviation as that of the
entire class. Due to this, the marks of a
student in batch C are changed from 8.5 to
(A) 6.0 (B) 7.0
(C) 8.0 (D) 9.0
AB [GATE-XE-2016-IISc]
193. A company records heights of all employees.
Let X and Y denote the errors in the average
height of male and female employees
respectively. Assume that  
X ~ N 0,4 and
 
Y ~ N 0,9 and they are independent. Then
the distribution of  
Z X Y / 2
  is
(A) N(0, 6.5) (B) N(0, 3.25)
(C) N(0, 2) (D) N(0, 1)
AA [GATE-PI-2008-IISc]
194. For a random variable X 
x
   
following normal distribution, the mean is
100
  . If the probability is P   for
X 110.
 Then the probability of X lying
between 90 and 110, i.e,  
P 90 X 110
 
will be equal to
(A) 1 2
  (B) 1  
(C) 1
2

 (D) 2
AD [GATE-ME-2015-IITK]
195. Among the four normal distribution with
probability density functions as shown
below, which one has the lowest variance?
(A) I (B) II
(C) III (D) IV
AB [GATE-ME-2013-IITB]
196. Let X be the normal random variable with
mean 1 and variance 4. The probability P{X
< 0} is :
(A) 0.5
(B) greater than zero and less than 0.5
(C) greater than 0.5 and less than 0.1
(D) 1.0
AC [GATE-EC-2001-IITK]
197. The PDF of a Gaussian random variable X is
given by
2
1
( ) exp[ ( 4) /18].
3 2
x
p x x
  

The
probability of the event {X = 4} is
(A) 1/2 (B) 1/ (3 / 2 )

(C) 0 (D) 1/4
A99.6-99.8 [GATE-ME-2016-IISc]
198. The area (in percentage) under standard
normal distribution curve of random variable
Z within limits from -3 to +3 is ______
AB [GATE-PI-2016-IISc]
199. A normal random variable X has the
following density function
 
 2
1
8
1
8
x
X
f x e
 

 
 
 
 


, x
   
 
1 X
f x dx



(A) 0 (B)
1
2
(C)
1
1
e
 (D) 1

129. TOPIC 5 – PROBABILITY & STATISTICS
www.targate.org Page 121
A0 [GATE-TF-2016-IISc]
200. Let X be normally distributed random
variable with mean 2 and 4. Then, the mean
of
x 2
2

is equal to________
AD [GATE-AG-2016-IISc]
201. The function f(x) represents a normal
distribution whose standard deviation and
mean are 1 and 5, respectively. The value of
f(x) at x = 5 is
(A) 0.0 (B) 0.159
(C) 0.282 (D) 0.398
AB [GATE-CE-2014-IITKGP]
202. If X is a continuous, real valued random
variable defined over the interval  
,
 
and its occurrence is defined by the density
function given as:
2
1 x a
2 b
1
f(x) e
2 *b

 
  
 


where ‘a’ and ‘b’ are the statistical attributes
of the random variable X. The value of the
integral
2
1 x a
a
2 b
1
e dx
2 *b

 
  
 
 
 is :
(A) 1 (B) 0.5
(C)  (D)
2

AA [GATE-EC-1997-IITM]
203. A probability density function is given by
p(x) = K 2
exp( / 2),
x x
    . The
value of K should be
(A) 1/ 2 (B) 2 / 
(C) 1/ 2  (D) 1/ 2

AB [GATE-CE-2017-IITR]
204. The number of parameters in the univariate
exponential and Gaussian distributions,
respectively, are
(A) 2 and 2 (B) 1 and 2
(C) 2 and 1 (D) 1 and 1
Uniform Distribution
A [GATE-ME-2009-IITR]
205. The standard deviation of a uniformly
distributed random variable b/w 0 and 1 is
(A)
1
12
(B)
1
3
(C)
5
12
(D)
7
12
A1.0 to 1.4 T5.2 [GATE-MT-2019-IITM]
206. The standard deviation (rounded off to one
decimal place) of the following set of five
numbers is ______.
6, 8, 8, 9, 9
A0.AA [GATE-EE-2011-IITM]
207. A zero mean random signal is uniformly
distributed between limits –a and + a and its
mean square value is equal to its variance.
Then the r.m.s value of the signal is
(A)
a
3
(B)
a
2
(C) a 2 (D) a 3
AC [GATE-EE-2008-IISc]
208. X is uniformly distributed random variable
that takes values between 0 and 1. The value
of E[
3
X ] will be :
(A) 0 (B)
1
8
(C)
1
4
(D)
1
2
AB [GATE-EC-1992-IITD]
209. For a random variable ‘X’ following the
probability density function, p(x), shown in
figure, the mean and the variance are,
respectively.
(A) 1/2 and 2/3 (B) 1 and 4/3
(C) 1 and 2/3 (D) 2 and 4/3
AD [GATE-CS-2007-IITK]
210. Suppose we uniformly and randomly select a
permutation from the 20! Permutations of 1,
2, 3,............,20. What is the probability that
2 appears at an earlier position than any
other even number in the selected
permutation?
(A)
1
2
(B)
1
10
(C)
9!
20!
(D) None of these
AA [GATE-IN-2008-IISc]
211. A random variable is uniformly distributed
over the interval 2 to 10. Its variance will be
(A)
16
3
(B) 6
(C)
256
9
(D) 36

130. ENGINEERING MATHEMATICS
Page 122 TARGATE EDUCATION GATE-(EE/EC)
AB [GATE-MN-2017-IITR]
212. F(y) and f(y) are the probability distribution
function and density function respectively of
a continuous variable Y in the interval
(0,  ). Which one of the following is
TRUE ?
(A) ( ) ( )
y
F y f x dx

 
(B)
0
( ) ( )
y
F y f x dx
 
(C)
( )
( )
df y
F y
dy

(D) ( ) 1 ( )
F y f y
 
A0.325 to 0.365 [GATE-TF-2018-IITG]
213. Let X be a random variable following the
binomial distribution. If E(X) = 2 and
Var(X)= 1.2 , then P(X = 2) , accurate to
three decimal places, is equal to ________.
AB [GATE-PI-2018-IITG]
214. In a service centre, cars arrive according to
Poisson distribution with a mean of 2 cars
per hour. The time for servicing a car is
exponential with a mean of 15 minutes. The
expected waiting time (in minute) in the
queue is
(A) 10 (B) 15
(C) 25 (D) 30
AB [GATE-PE-2018-IITG]
215. The probability density for three binomial
distributions (D1, D2, and D3) is plotted
against number of successful trials in the
given figure.
Each of the plotted distributions
corresponds to a unique pair of (n, p)
values, where, n is the number of trials and
p is the probability of success in a trial.
Three sets of (n, p) values are provided in
the table.
Set (n, p)
I (60, 0.3)
II (60, 0.2)
III (24, 0.5)
Pick the correct match between the (n, p) set
and the plotted distribution.
(A) Set I – D1, Set II – D2, Set III – D3
(B) Set I – D3, Set II – D1, Set III – D2
(C) Set I – D2, Set II – D3, Set III – D1
(D) Set I – D2, Set II – D1, Set III – D3
A [GATE-ME-2018-IITG]
216. Let 1
X and 2
X be two independent
exponentially distributed random variables
with means 0.5 and 0.25, respectively. Then
1 2
min( , )
Y X X
 is
(A) exponentially distributed with mean 1⁄6
(B) exponentially distributed with mean 2
(C) normally distributed with mean 3⁄4
(D) normally distributed with mean 1⁄6
AB [GATE-ME-2018-IITG]
217. Let 1
X , 2
X be two independent normal
random variables with means 1
 , 2
 and
standard deviations 1 2
,
  , respectively.
Consider 1 2
Y X X
  ; 1 2 1
    , 1 1
  ,
2 2
  . Then,
(A) Y is normally distributed with mean 0
and variance 1
(B) Y is normally distributed with mean 0
and variance 5
(C) Y has mean 0 and variance 5, but is
NOT normally distributed
(D) Y has mean 0 and variance 1, but is
NOT normally distributed
A0.1 [GATE-MA-2018-IITG]
218. Let 1 2 3 4
, , ,
X X X X be independent
exponential random variables with mean
1,1/ 2,1/ 3,1/ 4, respectively. Then
1 2 3 4
min( , , , )
Y X X X X
 has exponential
distribution with mean equal to ______.
AC [GATE-MA-2018-IITG]
219. Let  
i
X be a sequence of independent
Poisson (  ) variables and let
1
1 n
n i
i
W X
n 
  . Then the limiting
distribution of  
n
n W   is the normal
distribution with zero mean and variance
given by
(A) 1 (B) 
(C)  (D) 2


131. TOPIC 5 – PROBABILITY & STATISTICS
www.targate.org Page 123
AC [GATE-MA-2018-IITG]
220. Let 1 2
, ,...., n
X X X be independent and
identically distributed random variables with
probability density function given by
( 1)
, 1,
( ; )
0 otherwise
x
X
e x
f x
 
 
  

Also, let 1
1 n
i
i
X X
n 
  . Then the
maximum likelihood estimator of  is
(A) 1/ X (B) (1 / ) 1
X 
(C) 1 / ( 1)
X  (D) X
AA [GATE-CE-2018-IITG]
221. The graph of a function f(x) is shown in the
figure.
For f(x) to be a valid probability density
function, the value of h is
(A) 1/3 (B) 2/3
(C) 1 (D) 3
AC [GATE-CE-2018-IITG]
222. A probability distribution with right skew is
shown in the figure.
The correct statement for the probability
distribution is
(A) Mean is equal to mode
(B) Mean is greater than median but less
than mode
(C) Mean is greater than median and mode
(D) Mode is greater than median
A0.32 to 0.32 [GATE-BT-2018-IITG]
223. The variable z has a standard normal
distribution. If (0 1) 0.34
P z
   , then
2
( 1)
P z  is equal to (up to two decimal
places) ______.
AB T5.2 [GATE-CE-2019-IITM]
224. The probability density function of a
continuous random variable distributed
uiniformly between x and y (for y > x) is
(A)
1
x y

(B)
1
y x

(C) x y
 (D) y x

Combined Continuous Dist.
A0.65-0.71 [GATE-MA-2016-IISc]
225. Let X be a random variable with the
following cumulative distribution function:
 
2
0 0
1
0
2
3 1
1
4 2
1
1
x
x
x
F x
x
x

 

  

 
  





Then 1
1
4
P X
 
 
 
 
is equal to__________
A0.5 [GATE-CS-2016-IISc]
226. A probability density function
on the interval [a, 1] is given by
2
1/ x and outside this interval
the value of the function is zero.
The value of a is __________.
AB [GATE-CE-2016-IISc]
227. If f(x) and g(x) are two probability density
functions,
 
1 : 0
1 : 0
0 :
x
a x
a
x
f x x a
a
otherwise

   



    





 
: 0
: 0
0 :
x
a x
a
x
g x x a
a
otherwise

   



  





Which one of the following statements is
true?

132. ENGINEERING MATHEMATICS
Page 124 TARGATE EDUCATION GATE-(EE/EC)
(A) Mean of f(x) and g(x) are same;
Variance of f(x) and g(x) are same
(B) Mean of f(x) and g(x) are same;
Variance of f(x) and g(x) are different
(C) Mean of f(x) and g(x) are different;
Variance of f(x) and g(x) are same
(D) Mean of f(x) and g(x) are different;
Variance of f(x) and g(x) are different
A1.99-2.01 [GATE-TF-2016-IISc]
228. Let X be a continuous type random variable
with probability density function
 
1
1 x 3
f x 4
0 otherwise

  




. When
 
P X x 0.75
  , the value of x is equal to
_________
AA [GATE-CE-2016-IISc]
229. Probability density function of a random
variable X is given below
0.25 if1 5
( )
0 otherwise
x
f x
 

 

( 4)
P X  is
(A)
3
4
(B)
1
2
(C)
1
4
(D)
1
8
AA T5.2 [GATE-IN-2019-IITM]
230. The function ( )
p x is given by ( ) /
p x A x

where A and  are constants with 1
  and
1 x
   and ( ) 0
p x  for 1
x
   . For
( )
p x to be a probability density function,
the value of A should be equal to
(A) 1
  (B) 1
 
(C) 1/ ( 1)
  (D) 1/ ( 1)
 
AB T5.2 [GATE-EE-2019-IITM]
231. The mean-square of a zero-mean random
process is
kT
c
,where k is Boltzmann’s
constant, T is the absolute temperature, and c
is a capacitance. The standard deviation of
the random process is
(A)
kT
c
(B)
kT
c
(C)
c
kT
(D)
kT
c
A0.5 to 0.7 T5.2 [GATE-MN-2019-IITM]
232. The random variable X has probability
density function as given by
2
3 , 0 1
( )
0, otherwise
x x
f x
  
 

The value
2
( )
E X (rounded off to one
decimal place) is
AA [GATE-EE-2016-IISc]
233. Let the probability density function of a
random variable, X, be given as :
3 4
3
( ) ( ) ( )
2
x x
X
f x e u x ae u x

 
where ( )
u x is the unit step function.
Then the value of ‘a’ and { 0}
Prob X  ,
respectively, are
(A)
1
2,
2
(B)
1
4,
2
(C)
1
2,
4
(D)
1
4,
4
A6 [GATE-EC-2015-IITK]
234. The variance of the random variable X with
probability density function
| |
1
( ) | |
2
x
f x x e
 is ________.
A5.2-5.3 [GATE-MA-2016-IISc]
235. Let the probability density function of a
random variable X be
   
2
1
0
2
1
1
2 1 2
0 otherwise
x
x
x
f x c x

 



  
 






Then, the value of c is equal to _______.
AA [GATE-EC-2008-IISC]
236. Px(x) = M exp(–2|x|) + N exp(–3|x|) is the
probability density function for the real
random variable X, over the entire x axis. M
and N are both positive real numbers. The
equation relating M and N is
(A)
2
1
3
M N
 
(B)
1
2 1
3
M N
 
(C) 1
M N
 
(D) 3
M N
 

133. TOPIC 5 – PROBABILITY & STATISTICS
www.targate.org Page 125
AA [GATE-PI-2005-IITB]
237. The life of a bulb (in hours) is a random
variable with an exponential
distribution   at
f t e
  , 0 t
   . The
probability that its values lies between 100
and 200 hours is,
(A) 100 200
e e
   

(B) 100 200
e e
 

(C) 100 200
e e
   

(D) 200 100
e e
   

AB [GATE-CE-2008-IISc]
238. If the probability density function of a
random variable x is :
2
x , 1 x 1
0 elsewhere
   


Then, the percentage probability
1 1
P x
3 3

 
 
 
 
is :
(A) 0.274 (B) 2.47
(C) 24.7 (D) 247
A6 [GATE-CE-2013-IITB]
239. Find the value of  such that function f(x)
is valid probability density function is
    
f x x 1 2 x for 1 x 2
     
0 otherwise

A0.4 [GATE-CE-2014-IITKGP]
240. The probability density function of
evaporation E on any day during a year in a
watershed is given by
 
1
0 E 5mm / day
f E 5
0 otherwise

 

 


The probability that E lies in between 2 and
4 mm/day in the watershed is(in
decimal)__________
AA [GATE-EE-2013-IITB]
241. A continuous random variable X has a
probability density function   x
f x e
 . Then
P{ X > 1 } is
(A) 0.368 (B) 0.5
(C) 0.632 (D) 1.0
AC [GATE-EC-2006-IITKGP]
242. A probability density function is of the form
  a x
p x Ke
 ,  
x ,
   . The value f K
is:
(A) 0.5 (B) 1
(C) 0.5a (C) a
D [GATE-PI-2007-IITK]
243. If X is a continuous random variable whose
probability density function is given by
2
(5 2 ), 0 2
( )
0,
k x x x
f x
otherwise
   

 


Then P(x
> 1) is :
(A) 3/14 (B) 4/5
(C) 14/17 (D) 17/28
0.35 to 0.45 [GATE-EE-2014-IITKGP]
244. Let X be a random variable with probability
density function
0.2,
( ) 0.1,
0,
f x


 


for| | 1
for1 | | 4
otherwise
x
x

 
The probability (0.5 5)
P X
  is ____.
2 to 2 [GATE-IN-2014-IITKGP]
245. Given that x is a random variable in the
range[0,∞]with a probability densityfunction
2
x
e
K

, the value of the constant K
is___________.
A [GATE-IN-2007-IITK]
246. Assume that the duration in minutes of a
telephone conversation follows the
exponential distribution f(x) =
/5
1
, .
5
x
e x o


The probability that the conversation will
exceed five minutes is
(A)
1
e
(B)
1
1
e

(C) 2
1
e
(D) 2
1
1
e

0.4 TO 0.5 [GATE-EE-2014-IITKGP]
247. Lifetime of an electric bulb is a random
variable with density   2
f x kx ,
 where x is
measured in years. If the minimum and
maximum lifetimes of both are 1 and 2 years
respectively, then the value of k is
________.
AB [GATE-ME-2006-IITKGP]
248. Consider the continuous random variable
with probability density function
 
f t 1 t for 1 t 0
    
1 t for 0 t 1
   
The standard deviation for the random
variable is

134. ENGINEERING MATHEMATICS
Page 126 TARGATE EDUCATION GATE-(EE/EC)
(A)
1
3
(B)
1
6
(C)
1
3
(D)
1
6
AA [GATE-EC-2008-IISc]
249.   2 x 3 x
x
P x Me Ne
 
  is the probability
density function for the real random variable
X over the entire x axis. M and N both
positive real numbers. The equation relating
M and N is :
(A)
2
M N 1
3
  (B)
1
2M N 1
3
 
(C) M N 1
  (D) M N 3
 
A0.25 [GATE-EE-2015-IITK]
250. A random variable X has probability density
function f(x) as given below:
a bx for0 x 1
f (x)
0 otherwise
  

 

If the expected value E[X] = 2/3, then Pr[X
< 0.5] is ___________
A4 [GATE-IN-2015-IITK]
251. The probability density function of a random
variable X is px(x) = e–x
for x  0 and 0
otherwise. The expected value of the
function gx(x) = e3x/4
is ________.
AA [GATE-EC-2008-IISC]
252. The Probability Density Function (PDF) of a
radon variable X is as shown below
The corresponding Cumulative Distribution
Function (CDF) has the form
(A)
(B)
(C)
(D)
A1/8 [GATE-EC-1993-IITB
253. The function shown in the figure can
represent a probability density function for A
____________.
AA [GATE-IN-2006-IITKGP]
254. Probability density function differential
function p(x) of a random variable X is
shown below. The value of  is:
(A)
2
c
(B)
1
c
(C)
 
2
b c

(D)
 
1
b c

AD [GATE-EC-2004-IITD]
255. The distribution function ( )
x
F x of a random
variable X is shown in the figure. The
probability that X = 1 is :

135. TOPIC 5 – PROBABILITY & STATISTICS
www.targate.org Page 127
(A) zero (B) 0.25
(C) 0.55 (D) 0.3
A1 [GATE-BT-2017-IITR]
256. For the probability density   0.5x
P x 0.5e
 ,
the integral  
0
P x dx ________



AB [GATE-CE-2017-IITR]
257. For the function  
f x a bx,0 x 1
    , to
be a valid probability density function,
which one of the following statements is
correct?
(A) a = 1, b = 4
(B) a = 0.5, b = 1
(C) a = 0, b = 1
(D) a = 1 and b = -1
AB [GATE-PE-2017-IITR]
258. The value of constant a for which :
2
0 5
( )
0,
ax x
f x
otherwise
 

 

is a valid
probability density function, is (given,
0
a  ) :
(A)
1
125
(B)
3
125
(C)
6
125
(D)
9
125
A0.25 [GATE-MA-2018-IITG]
259. Let the cumulative distribution function of
the random variable X be given by
0, 0
, 0 1/ 2
( )
(1 ) / 2 1/ 2 1
1, 1.
X
x
x x
F x
x x
x


  

 
  

 

Then ( 1/ 2)
P X  = _________.
A0.5 [GATE-CE-2018-IITG]
260. Probability (up to one decimal place) of
consecutively picking 3 red balls without
replacement from a box containing 5 red
balls and 1 white ball is ______.
A6.80 to 7.20 [GATE-AG-2018-IITG]
261. In a box, there are 2 red, 3 black and 4 blue
coloured balls. The probability of drawing 2
blue balls in sequence without replacing, and
then drawing 1 black ball from this box is
______%.
AD [GATE-CH/AR/CY-2018-IITG]
262. To pass a test, a candidate needs to answer at
least 2 out of 3 questions correctly. A total of
6,30,000 candidates appeared for the test.
Question A was correctly answered by
3,30,000 candidates. Question B was
answered correctly by 2,50,000 candidates.
Question C was answered correctly by
2,60,000 candidates. Both questions A and B
were answered correctly by 1,00,000
candidates. Both questions B and C were
answered correctly by 90,000 candidates.
Both questions A and C were answered
correctly by 80,000 candidates. If the
number of students answering all questions
correctly is the same as the number
answering none, how many candidates failed
to clear the test?
(A) 30,000 (B) 2,70,000
(C) 3,90,000 (D) 4,20,000
A0.021 to 0.024 [GATE-CS-2018-IITG]
263. Two people, P and Q, decide to
independently roll two identical dice, each
with 6 faces, numbered 1 to 6. The person
with the lower number wins. In case of a tie,
they roll the dice repeatedly until there is no
tie. Define a trial as a throw of the dice by P
and Q. Assume that all 6 numbers on each
dice are equi-probable and that all trials are
independent. The probability (rounded to 3
decimal places) that one of them wins on the
third trial is _____.
AB T5.2 [GATE-ST-2019-IITM]
264. A fair die is rolled two times independently.
Given that the outcome on the first roll is 1,
the expected value of the sum of the two
outcomes is
(A) 4 (B) 4.5
(C) 3 (D) 5.5
Poisson Distribution
0.265 [GATE-CE-2014-IITKGP]
265. A traffic officer imposes on an average 5
number of penalties daily on traffic
violators. Assume that the number of
penalties on different days is independent
and follows a Poisson distribution. The
probability that there will be less than 4
penalties in a day is__________.

136. ENGINEERING MATHEMATICS
Page 128 TARGATE EDUCATION GATE-(EE/EC)
A0.36 to 0.38 T5.2 [GATE-EC-2019-IITM]
266. Let Z be an exponential random variable
with mean 1. That is, the cumulative
distribution function of Z is given by
1 if 0
( )
0 if 0
x
z
e x
F x
x

  
 


Then Pr (Z > 2 | Z > 1), rounded off to two
decimal places, is equal to _____.
AB [GATE-ME-2014-IITKGP]
267. The number of accidents occurring in a plant
in a month follows Poisson distribution with
mean as 5.2. The probability of occurrence
of less than 2 accidents in a plant during a
randomly selected month is :
(A) 0.029 (B) 0.034
(C) 0.039 (D) 0.044
A0.27 [GATE-CE-2014-IITKGP]
268. An observer counts 240veh/h at a specific
highway location. Assume that the vehicle
arrival at the location is Poisson distributed,
the probability of having one vehicle
arriving over a30-second time interval
is____________.
AD [GATE-PI-2010-IITG]
269. If a random variable X satisfies the
Poisson’s distribution with a mean value of
2, then the probability that X 2
 is
(A) 2
2e
(B) 2
1 2e

(C) 2
3e
(D) 2
1 3e

AA [GATE-EC-2014-IITKGP]
270. If calls arrive at a telephone exchange such
that the time of arrival of any call is
independent of the time of arrival of earlier
or future calls, the probability distribution
function of the total number of calls in a
fixed time interval will be :
(A) Poisson (B) Gaussian
(C) Exponential (D) Gamma
A0.9-1.1 [GATE-EC-2016-IISc]
271. The second moment of a Poisson-distributed
random variable is 2. The mean of the
random variable is _____ .
AC [GATE-CS-2013-IITB]
272. Suppose p is the number of cars per minute
passing through a certain road junction
between 5 PM, and p has a Poison’s
distribution with mean 3. What is the
probability of observing fewer than 3 cars
during any given minute in this interval?
(A)
 
3
8
2e
(B)
 
3
9
2e
(C)
 
3
17
2e
(D)
 
3
26
2e
A0.17 to 0.19 T5.2 [GATE-CE-2019-IITM]
273. Traffic on a highway is moving at a rate of
360 vehicles per hour at a location. If the
number of vehicles arriving on this highway
follows Poisson distribution, the probability
(round off to 2 decimal places) that the
headway between successive vehicles lies
between 6 and 10 seconds is ______.
A54 [GATE-CS-2017-IITR]
274. If a random variable X has a Poisson
distribution with mean 5, then the expression
 
2
E X 2
 

 
equals________.
AB T5.2 [GATE-XE-2019-IITM]
275. Let X be the Poisson random variable with
parameter 1
  . Then, the probability
(2 4)
P X
  equals
(A)
19
24e
(B)
17
24e
(C)
13
24e
(D)
11
24e
Miscellaneous
A0.32-0.34 [GATE-EC-2016-IISc]
276. Two random variables x and y are distributed
according to
,
( ), 0 1 0 1
( , )
0, otherwise.
X Y
x y x y
f x y
    



The probability ( 1)
P X Y
  is ________
AD [GATE-EC-2003-IITM]
277. Let X and Y be two statistically independent
random variables uniformly distributed in
the ranges (–1, 1) and (–2, 1) respectively.
Let Z = X + Y. Then the probability that
(Z 2)
  is :
(A) zero (B)
1
6
(C)
1
3
(D)
1
12
A0333 [GATE-EC-2014-IITKGP]
278. Let 1 2
X ,X and 3
X be independent and
identically distributed random variables with
the uniform distribution [0, 1]. The
probability P 
1
X is the largest is_______.

137. TOPIC 5 – PROBABILITY & STATISTICS
www.targate.org Page 129
A0.16 [GATE-EC-2014-IITKGP]
279. Let 1 2
,
X X and 3
X be independent and
identically distributed random variables with
the uniform distribution on [0, 1]. The
probability  
1 2 3
,
P X X X
  is ………..
0.15to0.18 [GATE-EC-2014-IITKGP]
280. Let X1, X2, and X3 be in dependent and
identically distributed random variables with
the uniform distribution on [0, 1]. The
probability P {X1, + X2 < X3} is ----------.
AB [GATE-EC-2013-IITB]
281. Let U and V be two independent zero mean
Gaussian random variables of variances
1
4
and
1
9
respectively. The probability
P(3V 2U)
 is :
(A) 4/9 (B) 1/2
(C) 2/3 (D) 5/9
AC [GATE-EC-2009-IITR]
282. Consider two independent random variables
X and Y with identical distributions. The
variables X and Y take values 0, 1 and 2
with probabilities
1 1 1
, ,
2 4 4
and respectively.
What is the conditional probability
( 2 | 0)?
P X Y X Y
   
(A) 0 (B)
1
16
(C)
1
6
(D) 1
B [GATE-EC-2012-IITD]
283. Two independent random variables X and Y
are uniformly distributed in the interval [-1,
1]. The probability that max[X, Y] is less
than ½ is :
(A) 3/4 (B) 9/16
(C) 1/4 (D) 2/3
AB [GATE-MT-2017-IITR]
284. The mean of a numerical data-set is X and
the standard deviation is S. If a number K is
added to each term in the data-set then the
mean and standard deviation become :
(A) ,
X S (B) ,
X K S

(C) ,
X S K
 (D) ,
X K S K
 
B [GATE-ME-1999-IITB]
285. Four arbitrary points 1 1
( , )
x y ,
2 2 3 3
( , ),( , )
x y x y , 4 4
( , )
x y , are given in the
xy – plane using the method of least squares,
if, regressing y upon x gives the fitted line y
= ax + b; and regressing x upon y gives the
fitted line x = cy + d, then
(A) The two fitted lines must coincide
(B) the two fitted lines need not coincide
(C) It is possible that ac = 0
(D) a must be 1/c
B [GATE-IN-2009-IITR]
286. Using given data points tabulated below, a
straight line passing through the origin is
fitted using least squares method. The slope
of the line
x 1 2 3
y 1.5 2.2 2.7
(A) 0.9 (B) 1
(C) 1.1 (D) 1.5
A [GATE-ME-2008-IISc]
287. Three values of x and y are to be fitted in a
straight line in the form y a bx
  by the
method of least squares. Given
6, 21,
x y
    2
14, 46,
x xy
    the
values of a and b are respectively
(A) 2, 3 (B) 1, 2
(C) 2, 1 (D) 3, 2
AD [GATE-EC-1987-IITB]
288. The variance of a random variable x is
2
x
 .
Then the variance of –kx (where k is a
positive constant) is
(A)
2
x
 (B) -k
2
x

(C) k
2
x
 (D)
2
k 2
x

AC [GATE-ME-2002-IISc]
289. A regression model is used to express a
variable Y as a function of another variable
X. This implies that
(A) There is a causal relationship between
X and Y.
(B) A value of X may be used to estimate a
value of Y.
(C) Value of X exactly determine values of
Y.
(D) There is no causal relationship between
Y and X.
AD [GATE-CE-2005-IITB]
290. Which one of the following statements is
NOT true?
(A) The measure of skewness is dependent
upon the amount of dispersion.

138. ENGINEERING MATHEMATICS
Page 130 TARGATE EDUCATION GATE-(EE/EC)
(B) In a symmetric distribution, the values
of mean, mode and median are the
same.
(C) In a positively skewed distribution:
mean > median > mode.
(D) In a negatively skewed distribution:
Mode > mean > median
AC [GATE-CS-2012-IITD]
291. Consider a random variable X that takes
values + 1 and – 1 with probability 0.5 each.
The values of the cumulative distributive
function F(x) at x = -1 and +1 are
(A) 0 and 0.5 (B) 0 and 1
(C) 0.5 and 1 (D) 0.25 and 0.75
A 0.332 to 0.A1 [GATE-MA-2018-IITG]
292. Let X and Y have joint probability density
function given by
,
2, 0 1 , 0 1
( , )
0 otherwise.
X Y
x y y
f x y
    

 

If Y
f denotes the marginal probability
density function of Y , then (1/ 2)
Y
f  ___.
A–0.5 [GATE-EC-2018-IITG]
293. A random variable X takes values −0.5 and
0.5 with probabilities
1
4
and
3
4
,
respectively. The noisy observation of is
= + , where has uniform probability
density over the interval (−1, 1). and are
independent. If the MAP rule based detector
outputs X̂ as
0.5,
ˆ
0.5,
X


 
 ,
Y
Y
 
 
then the value of  (accurate to two decimal
places) is _______.
-----00000-----

139. TOPIC 5 – PROBABILITY & STATISTICS
www.targate.org Page 131
Answer :
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
C 0.75 C A D C A D D 0.06
11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
D * A * * C D A 10 *
21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
B A B A D A D * C B
31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
D D B D A C * D 0.26 B
41. 42. 43. 44. 45. 46. 47. 48. 49. 50.
A B C C 0.93 # C 7 * *
51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
* * C * A C C D D D
61. 62. 63. 64. 65. 66. 67. 68. 69. 70.
B B * * * C 90 0.5 D 0.75
71. 72. 73. 74. 75. 76. 77. 78. 79. 80.
C * * * * * A * * *
81. 82. 83. 84. 85. 86. 87. 88. 89. 90.
C D D D C B A C * C
91. 92. 93. 94. 95. 96. 97. 98. 99. 100.
B D C C A A * * A B
101. 102. 103. 104. 105. 106. 107. 108. 109. 110.
* B B * C B B * B B
111. 112. 113. 114. 115. 116. 117. 118. 119. 120.
C C C B A A A C D D
121. 122. 123. 124. 125. 126. 127. 128. 129. 130.
B D * B A D A B B B
131. 132. 133. 134. 135. 136. 137. 138. 139. 140.
A C C * * B * C D D
141. 142. 143. 144. 145. 146. 147. 148. 149. 150.
C A * B C * D 2.5 D C
151. 152. 153. 154. 155. 156. 157. 158. 159. 160.
* C * B C 3 20 * A A
161. 162. 163. 164. 165. 166. 167. 168. 169. 170.
C * C * C * * * D 25
171. 172. 173. 174. 175. 176. 177. 178. 179. 180.
11 3.88 D D * 6 D B A 1.5
181. 182. 183. 184. 185. 186. 187. 188. 189. 190.
* 0.25 * 3.5 2.1 0.25 * * B C
191. 192. 193. 194. 195. 196. 197. 198. 199. 200.
A D B A D B C * B 0
201. 202. 203. 204. 205. 206. 207. 208. 209. 210.
D B A B A * A C B D
211. 212. 213. 214. 215. 216. 217. 218. 219. 220.
A B * B B A B 0.1 C C
221. 222. 223. 224. 225. 226. 227. 228. 229. 230.
A C * B * 0.5 B * A A
231. 232. 233. 234. 235. 236. 237. 238. 239. 240.
B * A 6 * A A B 6 0.4
241. 242. 243. 244. 245. 246. 247. 248. 249. 250.
A C D * 2 A * B A 0.25
251. 252. 253. 254. 255. 256. 257. 258. 259. 260.
4 A 1/8 A D 1 B B 0.25 0.5
261. 262. 263. 264. 265. 266. 267. 268. 269. 270.
* D * B * * B 0.27 D A
271. 272. 273. 274. 275. 276. 277. 278. 279. 280.
* C * 54 B * D * 0.16 *
281. 282. 283. 284. 285. 286. 287. 288. 289. 290.
B C B B B B A D C D
291. 292. 293. 294. 295. 296. 297. 298. 299. 300.
C 1 *
12. 0.66 to 0.67
14. 0.502 to 0.504
15. 0.135 to 0.150
20. 0.18 to 0.19
28. 0.13 to 0.15
37. 0.890 to 0.899
49. 0.083
50. 191 to 199
51. 0.075 to 0.085
52. 0.65 to 0.68
54. 0.43 to 0.45
63. 0.07 to 0.08
64. 0.027
65. 0.37 to 0.38
72. 0.49 to 0.51
73. 0.76 to 0.80
74. 0.49 to 0.51
75. 0.59 to 03.61
76. 0.24 to 0.26
78. 0.244 to 0.246
79. 0.19 to 0.35
80. 11.9
89. 0.8145
97. 0.50 to 0.55
98. 0.39 to 0.43
101. 0.59 to 0.61
104. 0.0019 to 0.0021
108. 0.272 to 0.274
123. 0.27
134. 0.25 to 0.27
135. 0.65
137. 0.48 to 0.49
143. 0.65 to 0.68
146. 0.60 to 0.62
151. 128 to 130
153. 54.49 to 54.51
158. 5.26 to 5.28
162. 74 to 75
164. 6.4 to 6.5
166. 3.49 to 3.51
167. 99 to 101
168. 5.9 to 6.1
175. 2.9 to 3.1
181. 49.9 to 50.1
183. 2.4 to 2.6
187. 49 to 51
188. 0.79 to 3.01
198. 99.6 to 99.8
206. 1.0 to 1.4
213. 0.325 to 0.365
223. 0.32 to 0.32
225. 0.65 to 0.71
228. 1.99 to 2.01
232. 0.5 to 0.7
235. 5.2 to 5.3
244. 0.35 to 0.45
247. 0.4 to 0.5
261. 6.80 to 7.20
263. 0.021 to 0.024
265. 0.265
266. 0.36 to 0.38
271. 0.9 to 1.1
273. 0.17 to 0.19
276. 0.32 to 0.34
278. 0.33
280. 0.15 to 0.18
293. –0.5

140. Page 132 TARGATE EDUCATION GATE-(EE/EC)
06
Numerical Methods
Basic Problems
B [GATE-EC-2014-IITKGP]
1. Match the application to appropriate
numerical method.
Application
P1: Numerical integration
P2: Solution to a transoandental eqution
P3: Solution to a system of linear equations
P4: Solution to a differential equation
Numerical Method
M1: Newton-Raphson Method
M2: Rurge katta Mathod
M3: Simpson’s 1/3-rule
M4: Gauss Elimination Method
(A) 1 3, 2 2, 3 4, 4 1
P M P M P M P M
   
(B) 1 3, 2 1, 3 4, 4 2
P M P M P M P M
   
(C) 1 4, 2 1, 3 3, 4 2
P M P M P M P M
   
(D) 1 2, 2 1, 3 3, 4 4
P M P M P M P M
   
B [GATE-ME-2013-IITB]
2. For solving algebraic and transcendental
equation which one of the following is used?
(A) Coulomb’s theorem
(B) Newton-Raphson method
(C) Euler’s method
(D) Stoke’s theorem
AD [GATE-ME-2013-IITB]
3. Match the CORRECT pairs
Numerical Integration
Scheme
Order of Fitting
Polynomial
P
Simpson’s
3
8
Rule
1 First
Q Trapezoidal Rule 2 Second
R
Simpson’s
1
3
rule
3 Third
(A) P-2, Q-1, R-3 (B) P-3, Q-2, R-1
(C) P-1, Q-2, R-3 (D) P-3, Q-1, R-2
C [GATE-EC-2005-IITB]
4. Match the following and
choose the correct combination
E. Newton –
Raphson
method
(1) Solving non-
linear equations
F. Runge-Kutta
method
(2) Solving linear
simultaneous
equations
G. Simpson’s
Rule
(3) Solving ordinary
differential
equations
H. Gauss
elimination
(4) Numerical
intergration
method
(5) Interpolation
(A) E – 6, F – 1, G
– 5, H – 3
(B) E – 1, F – 6, G –
4, H – 3
(C) E – 1, F – 3, G
– 4, H – 2
(D) E – 5, F – 3, G –
4, H – 1
C
5. Matching exercise choose the correct one out
of the alternatives A, B, C, D
Group-I Group-II
P. 2nd
order differential
equations
(1) Runge – Kutta
method
Q. Non-linear algebraic
equations
(2) Newton –
Raphson
method
R. Linear algebraic
equations
(3) Gauss
Elimination
S. Numerical
integration
(4) Simpson’s Rule
(A) P-3, Q-2, R-4, S-1
(B) P-2, Q-4, R-3, S-1

141. TOPIC 6 – NUMERICAL METHODS
www.targate.org Page 133
(C) P-1, Q-2, R-3, S-4
(D) P-1, Q-3, R-2, S-4
AD [GATE-ME-2006-IITKGP]
6. Match the items in columns I and II
Column I Column II
P Gauss-Seidel
method
1 Interpolation
Q Forward Newton-
Gauss method
2 Non-linear
differential
equations
R Runge-Kutta
method
3 Numerical
integration
S Trapezoidal Rule 4 Linear algebraic
equation
Codes:
P Q R S
(A) 1 4 3 2
(B) 1 4 2 3
(C) 1 3 2 4
(D) 4 1 2 3
AB [GATE-CS-1998-IITD]
7. Which of the following statements applies to
the bisection method used for finding roots
of functions:
(A) Converges within a few iterations.
(B) Guaranteed to work for all continuous
functions.
(C) Is faster than the Newton-Raphson’s
method.
(D) Required that there be no error in
determining the sign of the function.
AB [GATE-IT-2004-IITD]
8. Consider the following iterative root finding
methods and convergence properties:
Iterative root finding
methods
Convergence properties
Q False position I Order of convergence =
1.62
R Newton-
Raphson’s
II Order of convergence =
2
S Secant III Order of convergence =
1 with guarantee of
convergence
T Successive
approximation
IV Order of convergence =
1 with no guarantee of
convergence
The correct matching of the methods and
properties is :
(A) Q-II, R-IV, S-III, T-I
(B) Q-III, R-II, S-I, T-IV
(C) Q-II, R-I, S-IV, T-III
(D) Q-I, R-IV, S-II, T-III
AC [GATE-CH-2016-IISc]
9. Which one of the following is an iterative
technique for solving a system of linear
algebraic equations?
(A) Gauss elimination
(B) Gauss-Jordan
(C) Gauss-Seidel
(D) LU decomposition
AD [GATE-TF-2016-IISc]
10. Which of the following is a multi-step
numerical method for solving the ordinary
differential equation?
(A) Euler method
(B) Improved Euler Method
(C) Runge-Kutta method
(D) Adams-Multon method
AC [GATE-PE-2017-IITR]
11. The numerical method used to find the root
of a non-linear algebraic equation, that
converges quadratically, is :
(A) Bisection method
(B) Regula-Falsi method (Method of False
Position).
(C) Newton-Raphson method.
(D) None of the above
B [GATE-EE-1998-IITD]
12. In the interval [0, ]
 the equation cos
x x

has
(A) No solution
(B) Exactly one solution
(C) Exactly 2 solutions
(D) An infinite number of solutions
C [GATE-EC-2009-IITR]
13. It is known that two roots of the non-linear
equation 3 2
6 11 6 0
x x x
    are 1 and 3.
The third root will be
(A) j (B) j

(C) 2 (D) 4

142. ENGINEERING MATHEMATICS
Page 134 TARGATE EDUCATION GATE-(EE/EC)
AA [GATE-CE-2007-IITK]
14. Given that one root of the equation
3 2
x 10x 31x 30 0
    is 5. The other two
roots are:
(A) 2 and 3 (B) 2 and 4
(C) 3 and 4 (D) -2 and -3
AA [GATE-CS-2014-IITKGP]
15. A non-zero polynomial f(x) of degree 3 has
roots at x = 1, x = 2 and x = 3. Which one of
the following must be TRUE?
(A)    
f 0 f 4 0

(B)    
f 0 f 4 0

(C)    
f 0 f 4 0
 
(D)    
f 0 f 4 0
 
A2 [GATE-MA-2016-IISc]
16. The number of roots of the equation
 
2
cos 0
x x
  in the interval ,
2 2
 
 

 
 
is
equal to ___________ .
AC [GATE-EC-2016-IISc]
17. How many distinct values of satisfy the
equation sin( ) = /2, where is in radians?
(A) 1 (B) 2
(C) 3 (D) 4 or more
AC [GATE-BT-2018-IITG]
18. Which one of the following is the solution
for 2
cos 2cos 1 0
x x
   , for values of x in
the range of 0 0
0 360
x
  ?
(A) 0
45 (B) 0
90
(C) 0
180 (D) 0
270
AC [GATE-AG-2018-IITG]
19. Solution of
4 3 2
( ) 2 4 3 1 0
f x x x x x
      is
(A) 0.333 (B) 0.646
(C) 0.658 (D) 1.000
**********
Roots Finding Methods
Newton Raphson Method
AC [GATE-EC-2008-IISc]
20. The recursion relation to solve x
x e
 using
Newton-Raphson’s method is:
(A) n
x
n 1
x e
 
(B) n
x
n 1 n
x x e
  
(C)  
n
n
x
n 1 n x
e
x 1 x
1 e

 
 

(D)  
n
n
x
2
n n
n 1 x
n
x e 1 x 1
x
x e

 
  


A [GATE-CE-2009-IITR]
21. The following equation needs to be
numerically solved using the Newton –
Raphson method 3
4 9 0.
x x
   The iterative
equation for this purpose is ( k indicates the
iteration level)
(A)
3
1 2
2 9
3 4
k
k
k
x
x
x




(B)
3
1 2
3 9
2 9
k
k
k
x
x
x




(C) 2
1 3 4
k k k
x x
    (D)
2
1 2
4 3
9 2
k
k
k
x
x
x




AA [GATE-CE-2011-IITM]
22. The square root of number N is to be
obtained by applying the Newton-Raphson’s
iterations to the equation 2
x N 0
  . If i
denotes the iteration index, the correct
iterative method will be :
(A) i+1 i
i
1 N
x = x +
2 x
 
 
 
(B) 2
i 1 i 2
i
1 N
x x
2 x

 
 
 
 
(C)
2
i 1 i
i
1 N
x x
2 x

 
 
 
 
(D) i 1 i
i
1 N
x x
2 x

 
 
 
 
AB [GATE-IN-2014-IITKGP]
23. The iteration step in order to solve for the
cube roots of a given number ‘N’ using the
Newton-Raphson’s method is
(A)  
3
k 1 k k
1
x x N x
3
   
(B) k 1 k 2
k
1 N
x 2x
3 x

 
 
 
 
(C)  
3
k 1 k
1
x N x
3
  
(D) k 1 k 2
k
1 N
x 2x
3 x

 
 
 
 
A8.50 to 9.00 T6.2 [GATE-MT-2019-IITM]
24. The estimated value of the cube root of 37
(rounded off to two decimal places) obtained
from the Newton-Raphson method after two
interactions 2
( )
x is ______.
[Start with an initial guess value of 0 1
x  ].

143. TOPIC 6 – NUMERICAL METHODS
www.targate.org Page 135
AA [GATE-IN-2007-IITK]
25. Identify the Newton-Raphson’s iteration
scheme for finding the square root of 2.
(A) n 1 n
n
1 2
x x
2 x

 
 
 
 
(B) n 1 n
n
1 2
x x
2 x

 
 
 
 
(C) n 1
x  
n
2
x
(D) n 1 n
x 2 x
  
AA [GATE-CS-2007-IITK]
26. Consider the series
n
n 1 0
n
x 9
x ,x 0.5
2 8x
    obtained from the
Newton-Raphson’s method. The series
converges to
(A) 1.5 (B) 2
(C) 1.6 (D) 1.4
AA [GATE-CS-2002-IISc]
27. The Newton-Raphson’s iteration
n
n 1
n
x 1
x 3
2 2x

 
 
   
 
   
can be used to solve
the equation :
(A) 2
x 3
 (B) 3
x 3

(C)
2
x 2
 (D)
3
x 2

C [GATE-CE-2005-IITB]
28. Given a > 0, we wish to calculate it
reciprocal value
1
a
by using Newton –
Raphson method for ( ) 0.
f x  The Newton-
Raphson algorithm for the function will be :
(A) 1
1
2
k k
k
a
x x
x

 
 
 
 
(B) 2
1
2
k k k
a
x x x
  
(C) 2
1 2
k k k
x x ax
  
(D) 2
1
2
k k k
a
x x x
  
C [GATE-CS-2008-IISc]
29. The Newton-Raphson iteration
1
1
2
n n
n
R
x x
x

 
 
 
 
can be used to compute
the
(A) square or R
(B) reciprocal of R
(C) square root of R
(D) logarithm of R
A [GATE-EE-2009-IITR]
30. Let
2
117 0.
x   The iterative steps for the
solution using Newton – Raphson’s method
given by
(A) 1
1 117
2
k k
k
x x
x

 
 
 
 
(B) 1
117
k k
k
x x
x
  
(C) 1
117
k
k k
x
x x
  
(D) 1
1 117
2
k k k
k
x x x
x

 
  
 
 
A [GATE-EC-2009-IITR]
31. Newton-Raphson formula to find the roots of
an equation ( ) 0
f x  is given by
(A) 1 1
( )
( )
n
n n
n
f x
x x
f x
  
(B) 1 1
( )
( )
n
n n
n
f x
x x
f x
  
(C) 1 1
( )
( )
n
n
n n
f x
x
x f x
 
(D) none of the above
C [GATE-PI-2009-IITR]
32. The Newton-Raphson iteration formula for
finding 3
,
c where c > 0 is
(A)
3 3
1 2
2
3
n
n
n
x c
x
x



(B)
3 3
1 2
2
3
n
n
n
x c
x
x



(C)
3
1 2
2
3
n
x
n
x c
x
x



(D)
3
1 2
2
3
n
n
n
x c
x
x



AC [GATE-ME-2016-IISc]
33. The root of the function   3
1
f x x x
  
obtained after first iterations on application
of Newton-Raphson scheme using an initial
guess of 0 1
x  is :
(A) 0.682 (B) 0.686
(C) 0.750 (D) 1.000

144. ENGINEERING MATHEMATICS
Page 136 TARGATE EDUCATION GATE-(EE/EC)
A0.49-0.51 [GATE-MT-2016-IISc]
34. Solve the equation x
x e
 using Newton-
Raphson method. Starting with an intial
guess value 0 0
x  , the value of x after the
first iteration is ________
A0.355-0.365 [GATE-CE-2016-IISc]
35. Newton-Raphson method is to be used to
find root of equation 3 sin 0
x
x e x
   . If
the initial trial value for the root is taken as
0.333, the next approximation for the root
would be _________ (note: answer up to
three decimal)
AB [GATE-PI-2016-IISc]
36. To solve the equation
2sinx = x
by Newton-Raphson method,
the initial guess was chosen to
be x = 2.0. Consider x in radian only. The
value of x(in radian) obtained after one
iteration will be closest to
(A) -8.101 (B) 1.901
(C) 2.099 (D) 12.101
A0.56-0.58 [GATE-AE-2016-IISc]
37. Use Newton-Raphson method
to solve the equation: x
xe 1
 .
Begin with the intial guess
0
x 0.5
 . The solution after
one step is x = _______.
A0.06 [GATE-EE-2014-IITKGP]
38. The function f(x) = x
e 1
 is to be solved
using Newton-Raphson’s method. If the
initial value of 0
x is taken 1.0, then the
absolute error observed at 2nd
iteration is __.
AC [GATE-EE-2013-IITB]
39. When the Newton-Raphson’s method is
applied to solve the equation
  3
f x x 2x 1 0
    , the solution at the
end of the first iteration with the initial guess
value as 0
x 1.2
 is :
(A) -0.82 (B) 0.49
(C) 0.705 (D) 1.69
A0.543 [GATE-ME-2014-IITKGP]
40. The real root of the equation
5x – 2cosx –1 = 0
(upto two decimal accuracy)
is_______________.
AC [GATE-ME-2005-IITB]
41. Starting from 0
x 1
 , one step of Newton-
Raphson’s method in solving the equation
3
x 3x 7 0
   gives the next value 
1
x as
(A) 1
x 0.5
 (B) 1
x 1.406

(C) 1
x 1.5
 (D) 1
x 2

AB [GATE-ME-1999-IITB]
42. We wish to solve 2
2 0
x   by Newton-
Raphson’s technique. Let the initial guess be
0
x 1.0
 . Subsequent estimated 1
x will be :
(A) 1.414 (B)`1.5
(C) 2.0 (D) 3.0
AC [GATE-IN-2011-IITM]
43. The extremum (minimum or maximum)
point of a function f(t) is to be determined by
solving
 
df x
0
dx
 using the Newton-
Raphson’s method. Let   3
f x x 6x
  and
0
x 1
 be the initial guess of x. The value f x
after two iterations  
2
x is
(A) 0.0141 (B) 0.4142
(C) 1.4167 (D) 1.5000
AB [GATE-CE-2005-IITB]
44. Given a > 0, we wish to calculate its
reciprocal value 1/a by using Newton-
Raphson’s method for f(x) = 0
For a = 7 and starting with 0
x 0.2
 , the first
two iterations will be :
(A) 0.11, 0.1299 (B) 0.12, 0.1392
(C) 0.12, 0.1416 (D) 0.13, 0.1428
AA [GATE-CS-2014-IITKGP]
45. In the Newton-Raphson’s method, an initial
guess of 0
x 2
 is made and the sequence
0 1, 2
x ,x x ,.......... is obtained for the function
  3 2
f x 0.75x 2x 2x 4 0
    
Consider of the statements
(I) 3
x 0

(II) The method converges to a solution in a
finite number of iterations.
Which of the following is TRUE?
(A) Only I (B) Only II
(C) Both I and II (D) Neither I nor II
A0.73 [GATE-CS-2014-IITKGP]
46. If the equation   2
sin x x
 is solved by
Newton-Raphson’s method with the initial
guess of x =1, then the value of x after 2
iterations would be____________
D [GATE-EE-2008-IISC]
47. The Newton-Raphson method is to be used
to find the root of the equation and '( )
f x is
the derivative of f the method converges

145. TOPIC 6 – NUMERICAL METHODS
www.targate.org Page 137
(A) Always
(B) Only is f is a polynomial
(C) Only if 0
( ) 0
f x 
(D) None of the above
A4.3 [GATE-EC-2015-IITK]
48. The Newton-Raphson method is used to
solve the equation 3 2
f(x) x 5x 6x 8 0
     .
Taking the initial guess as x = 5, the solution
obtained at the end of the first iteration is
_____.
AA [GATE-IN-2006-IITKGP]
49. For k = 0, 1, 2,............, the steps of Newton-
Raphson’s method for solving a non-linear
equation is given as 2
k 1 k k
2 5
x x x
3 3

   .
Starting from a suitable initial choice as k
tends to , the iterate k
x tends to
(A) 1.7099 (B) 2.2361
(C) 3.1251 (D) 5.0000
A0.65-0.72 [GATE-EC-2017-IITR]
50. Starting with x = 1, the solution of the
equation 3
x x 1
  , after two iterations of
Newton-Raphson’s method (up to two
decimal places) is ________.
A5.95 to 6.25 [GATE-TF-2018-IITG]
51. Starting from the initial point 0 10
x  , if the
sequence  
n
x is generated using Newton
Raphson method to compute the root of the
equation 4
600 0
x   , then 2
x , accurate to
two decimal places, is equal to ________.
A1 [GATE-CE-2018-IITG]
52. The quadratic equation 2
2 3 3 0
x x
   is to
be solved numerically starting with an initial
guess as 0 2
x  . The new estimate of x after
the first iteration using Newton-Raphson
method is ______.
Other Methods
A10 TO 10 [GATE-EE-2017-IITR]
53. Only one of the real roots of
  6
f x x x 1
   lies in the interval
1 x 2
  and bisection method is used to
find its value. For achieving an accuracy of
0.001, the required minimum number of
iterations is _____.(Give the answer up to
two decimal places.)
A0.74-0.76 [GATE-MT-2017-IITR]
54. Using the bisection method,
the root of the equation
3
1 0
x x
   after three
iterations is _____
(answer up to two decimal places)
(Assume starting values of x = –1 and +1)
Solution of Differential
Equation
Eulers Method
AB [GATE-CE-2006-IITKGP]
55. The differential equation
2
dy
0.25y
dx
 is to
be solved using the backward (implicit)
Euler’s method with the boundary condition
y = 1 at x = 0 and with a step size of 1. What
would be value of y at x = 1?
(A) 1.33 (B) 1.67
(C) 2.00 (D) 2.33
A0.6-0.7 [GATE-EC-2016-IISc]
56. The ordinary differential equation
3 2
dx
x
dt
  , with (0) 1
x 
is to be solved using the forward Euler
method. The largest time step that can be
used to solve the equation without making
the numerical solution unstable is ________
AA [GATE-IN-2010-IITG]
57. The velocity v(in m/s) of a moving mass,
starting from rest, is given as
dv
v t
dt
  .
Using Euler forward difference method (also
known as Cauchy-Euler method) with a step
size of 0.1 s, the velocity of 0.2 s evaluates
to
(A) 0.01 m/s
(B) 0.1 m/s
(C) 0.2 m/s
(D) 1 m/s
AD [GATE-IN-2006-IITKGP]
58. A linear ordinary differential equation is
given as  
2
2
d y dy
3 2y t
dt dt
    . Where
 
t
 is an impulse input. The solution is
found by Euler’s forward-difference method
that uses an integration step h. What is a
suitable value of k?
(A) 2.0 (B) 1.5
(C) 1.0 (D) 0.2
A [GATE-PI-2011-IITM]
59. Consider a differential equation
( )
( )
dy x
y x x
dx
  with initial condition
(0) 0.
y  Using Euler’s first order method
with a step size of 0.1 then the value of
y(0.3) is :
(A) 0.01 (B) 0.031
(C) 0.0631 (D) 0.1

146. ENGINEERING MATHEMATICS
Page 138 TARGATE EDUCATION GATE-(EE/EC)
AA [GATE-IN-2013-IITB]
60. While numerically solving the differential
equation  
2
dy
2xy 0,y 0 1
dx
   using
Euler’s predictor corrector (improved Euler-
Cauchy) method with a step size of 0.2, the
value of y after the first step is
(A) 1.00
(B) 1.03
(C) 0.97
(D) 0.96
A10.5-11.5 [GATE-PE-2017-IITR]
61. Solve
dy
y
dx
  numerically from x = 0 to 1
using explicit, forward, first order Euler
method with initial condition of (0) 1
y  and
step size (h) of 0.2. The absolute value of
error in y(1) calculated using analytical and
numerical solution is _____ % (calculate the
error using analytical solution as the basis
and use three decimal places).
A0.78 to 0.80 [GATE-CE-2018-IITG]
62. Variation of water depth ( )
y in a gradually
varied open channel flow is given by the
first order differential equation
10
ln( )
3
3ln( )
1
250 45
y
y
dy e
dx e





Given initial condition : ( 0) 0.8
y x m
  .
The depth (in m, up to three decimal places)
of flow at a downstream section at x = 1 m
from one calculation step of Single Step
Euler Method is _____.
A
63. During the numerical solution of a first order
differential equation using the Euler (also
known as Euler Cauchy) method with step
size h, the local truncation error is of the
order of
(A)
2
h (B)
3
h
(C)
4
h (D)
5
h
Runge Kutta Method
AD [GATE-ME-2014-IITKGP]
64. Consider an ordinary differential equation
dx
4t 4
dt
  . If 0 0
x x
  at t = 0, the
increment in x is calculated using Runge-
Kutta fourth order multi step method with a
step size of 0.2
t
  is :
(A) 0.22 (B) 0.44
(C) 0.66 (D) 0.88
0.060-0.063 [GATE-EC-2016-IISc]
65. Consider the first order initial value problem
2
' 2 , (0) 1,(0 )
y y x x y x
     
with exact solution
2
( ) x
y x x e
  . For x =
0.1, the percentage difference between the
exact solution and the solution obtained
using a single iteration of the second-order
Runge-Kutta method with step-size h = 0.1
is ________ .
A3.12 to 3.26 [GATE-PE-2018-IITG]
66. Solve the given differential equation using
the 2nd
order Runge-Kutta (RK2) method:
dy
t y
dt
  ; Initial condition: ( 0) 4
y t  
Use the following form of RK2 method with
an integration step-size, h = 0.5:
1 2 1
( , ); ( 0.5 , 0.5 )
i i i i
k f t y k f t h y k h
   
1 2
i i
y y k h
  
The value of ( 0.5)
y t   =_____________.
(rounded-off to two decimal places)
AD [GATE-CH-2018-IITG]
67. The fourth order Runge-Kutte (RK4) method
to solve an ordinarydifferential equation
( , )
dy
f x y
dx
 is given as
1 2 3 4
1
( ) ( ) ( 2 2 )
6
y x h y x k k k k
     
1 ( , )
k h f x y

1
2 ,
2 2
k
h
k h f x y
 
  
 
 
2
3 ,
2 2
k
h
k h f x y
 
  
 
 
 
4 3
,
k h f x h y k
  
For a special case when the function f
depends solely on x, the above RK4 method
reduces to
(A) Euler’s explicit method
(B) Trapezoidal rule
(C) Euler’s implicit method
(D) Simpson’s 1/3 rule
**********
Numerical Integration
Trapezoidal Rule
A0.175-0.195 [GATE-ME-2016-IISc]
68. The error in numerically computing the
integral  
0
sin cos
x x dx


 using the
trapezoidal rule with three intervals of equal
length between 0 and  is _________.

147. TOPIC 6 – NUMERICAL METHODS
www.targate.org Page 139
A0.640 to 0.650 T6.4 [GATE-PI-2019-IITM]
69. The numerical value of the definite integral
1
0
x
e dx

 using trapezoidal rule with function
evaluations at points 0
x  , 0.5 and 1 is
____ (round off to 3 decimal places)
A1.32 to 1.34 T6.4 [GATE-PE-2019-IITM]
70. The values of a function ( )
f x over the
interval [0,4] are given in the table below :
x 0 1 2 3 4
f(x) 1 0.5 0.2 0.1 0.06
Then, according to the trapezoidal rule, the
value of the integral
4
0
( )
f x dx
 is ______
(round off to 2 decimal places).
A2.20 to 2.50 T6.4 [GATE-BT-2019-IITM]
71.
1
1
( )
f x dx

 calculated using trapezoidal rule
for the values given in the table is
_____(rounded off to 2 decimal places).
x -1 -2/3 -1/3 0 1/3 2/3 1
f(x) 0.37 0.51 0.71 1.0 1.40 1.95 2.71
A1.820 to 1.830 T6.4 [GATE-AG-2019-IITM]
72. Using trapezoidal rule, the value of
5.2
4.0
ln( )
I x dx
  (rounded off to three
decimal places) is ______.
x 4.0 4.2 4.4 4.6 4.8 5.0 5.2
y = ln(x) 1.386 1.435 1.482 1.526 1.569 1.609 1.648
A1.8-1.9 [GATE-PE-2016-IISc]
73. For a function f(x), the values of the function
in the interval [0, 1] are given in the table
below.
x f(x)
0.0 1.0
0.2 1.24
0.4 1.56
0.6 1.96
0.8 2.44
1.0 3.0
The value of the integral  
1
0
f x dx

according to the trapezoidal rule is
______________.
AD [GATE-AG-2016-IISc]
74. Integration by trapezoidal method of 10
log ( )
x
with lower limit of 1 to upper limit of 3
using seven distinct values (equally covering
the whole range) is _______________ .
C [GATE-EE-2008-IISc]
75. A differential equation
2
( )
t
dx
e u t
dt

 has to
be solved using trapezoidal rule of
integration with a step size h = 0.01s.
Function u(t) indicates a unit step function.
If  
x 0 0

 , then value of x at t = 0.01s will
be given by
(A) 0.00099 (B) 0.00495
(C) 0.0099 (D) 0.0198
A1.1 [GATE-ME-2014-IITKGP]
76. Using the trapezoidal rule and dividing the
interval of integration into the three equal
subintervals, the definite integral
1
1
x dx



is_________________.
A1.7532 [GATE-ME-2014-IITKGP]
77. The value of  
4
2.5
In x dx
 calculated using
the trapezoidal rule with five subintervals
is_______________.
A1.16 [GATE-ME-2013-IITB]
78. The definite integral
3
1
1
dx
x

is evaluated using Trapezoidal
rule with a step size of 1.
The correct answer is _______.
AA [GATE-ME-2007-IITK]
79. A calculator has accuracy upto 8 digits after
decimal place. The value of
2
0
sinxdx

 ,
when evaluated using this calculator by
trapezoidal method with 8 equal intervals to
5 significant digits
(A) 0.00000 (B) 1.0000
(C) 0.00500 (D) 0.00025
AA [GATE-CE-2006-IITKGP]
80. A 2nd
degree polynomial, f(x) has value of 1,
4 and 15 at x = 0, 1 and 2 respectively. The
integral  
2
0
f x dx
 is to be estimated by
applying the trapezoidal rule to this data.
What is the error(defined as “true value –
approximate value”) in the estimate?
(A)
4
3
 (B)
2
3

(C) 0 (D)
2
3
AD [GATE-CS-2013-IITB]
81. Function f is known at the following points.

148. ENGINEERING MATHEMATICS
Page 140 TARGATE EDUCATION GATE-(EE/EC)
x F(x)
0 0
0.3 0.09
0.6 0.36
0.9 0.81
1.2 1.44
1.5 2.25
1.8 3.24
2.1 4.41
2.4 5.76
2.7 7.29
3.0 9.00
The values of  
3
0
f x dx
 computed using the
trapezoidal rule is :
(A) 8.983 (B) 9.003
(C) 9.017 (D) 9.045
AD [GATE-CS-2011-IITM]
82. The value of
2
1
x
0
e dx

 , using trapezoidal
rule for 10- trapezoids, is equal to
(A) 0.6778 (B) 0.7165
(C) 0.6985 (D) 0.7462
C
83. The trapezoidal rule for integration give
exact result when the integrand is a
polynomial of degree
(A) but not 1 (B) 1 but not 0
(C) 0 (or) 1 (D) 2
A0.70-0.80 [GATE-MT-2017-IITR]
84. The definite integral,
2
1
0
x
e dx

 is to be
evaluated numerically. Devide the
integration interval into exactly 2
subintervals of equal length. Applying the
trapezoidal rule, the approximate value of
the integral is ___________ (answer up to
two decimal places)
A1.95 to 2.05 [GATE-MT-2018-IITG]
85. Using the trapezoidal rule with two equal
intervals ( 2
n  , 1
x
  ), the definite
integral
4
2
1
ln( )
dx
x

 ________ (to two
decimal places).
Simpsons Rule
A14760 [GATE-AG-2017-IITR]
86. The areas of seven horizontal cross-sections
of a water reservoir at intervals of 9 m are
210, 250, 320, 350, 290, 230 and 170 m2
.
The estimated volume of the reservoir in m3
using Simpson’s rule is _______ .
A0.24 to 0.28 T6.4 [GATE-TF-2019-IITM]
87. The value of the integral
/2
0
6 cos2
1 sin
x
dx
x

 

obtained using Simpson’s
1
3
rule (rounded
off to 2 decimal places) is ________.
A94.5-94.8 [GATE-CH-2016-IISc]
88. Values of f(x) in the interval [0, 4] are given
below.
X 0 1 2 3 4
f(x) 3 10 21 36 55
Using Simpson’s 1/3 rule with a step size of
1, the numerical approximation (rounded off
to the second decimal place) of  
4
0
f x dx

is________.
C [GATE-CE-2013-IITB]
89. The integral
3
1
1
dx
x
 when evaluated by
using simpson’s 1/ 3rd
rule on two equal
sub intervals each of length 1, equal to
(A) 1.000 (B) 1.008
(C) 1.1111 (D) 1.120
A0.53 [GATE-CE-2013-IITB]
90. The magnitude as the error (correct to two
decimal places) in the estimation of integral
 
4
4
0
x 10 dx

 using Simphson 1/3 rule
is____________.[Take the step length as 1]
AD [GATE-CE-2012-IITD]
91. The estimate of
1.5
0.5
dx
x
 obtained using
Simphson’s rule with three-point function
evaluation exceeds the exact value by
(h=0.5)
(A) 0.235 (B) 0.068
(C) 0.024 (D) 0.012
AA [GATE-CE-2010-IITG]
92. The table given below gives values of a
function F(x) obtained for values of x at
intervals of 0.25
x 0 0.25 0.5 0.75 1.0
F(x) 1 0.9412 0.8 0.64 0.50
The value of the integral of the function
between the limits 0 to 1 using Simphson’s
rule is:
(A) 0.7854 (B) 2.3562
(C) 3.1416 (D) 7.5000

149. TOPIC 6 – NUMERICAL METHODS
www.targate.org Page 141
AB [GATE-CS-1992-IITD]
93. Simpson’s rule for integration gives exact
result when f(x) is a polynomial degree:
(A) 1 (B) 2
(C) 3 (D) 4
AD [GATE-CS-2008-IISc]
94. If interval of integration is divided into two
equal intervals of width 1.0, the value of the
definite integral
3
e
1
log xdx
 , using Simpson’s
one-third rule, will be
(A) 0.50 (B) 0.80
(C) 1.00 (D) 1.29
AC [GATE-PI-2018-IITG]
95. In order to evaluate the integral
1
0
x
e dx
 with
Simpson’s 1/3rd
rule, values of the function
x
e are used at 0.0,0.5
x  and 1.0. The
absolute value of the error of numerical
integration is
(A) 0.000171 (B) 0.000440
(C) 0.000579 (D) 0.002718
A25.80 to 25.90 [GATE-AG-2018-IITG]
96. The velocity (v) of a tractor, which starts
from rest, is given at fixed intervals of time
(t) as follows :
t (min) v (m min-1
)
0 0
2 0.8
4 1.5
6 2.1
8 2.4
10 2.7
12 1.7
14 0.9
16 0.4
18 0.2
20 0
Using Simpson’s 1/3rd
rule, the distance
covered by the tractor in 20 minutes will be
____m.
Mixed
AC [GATE-ME-2003-IITM]
97. The accuracy of Simphson’s rule quadrature
for a step size h is
(A)  
2
O h (B)  
3
O h
(C)  
4
O h (D)  
5
O h
AD [GATE-ME-1997-IITM]
98. The order of error is Simphson’s rule for
numerical integration with a step size h is:
(A) h (B)
2
h
(C)
3
h (D)
4
h
AC [GATE-CS-2014-IITKGP]
99. With respect to numerical evaluation of the
definite integral,
b
2
a
K x dx
  ,, where a and b
are given, which of the following statements
is/are TRUE?
(I) The value of K obtained using the
trapezoidal rule is always greater than
or equal to the exact value of the
definite integral.
(II) The value of K obtained using the
Simpson’s rule is always equal to the
exact value of the definite integral.
(A) Only I
(B) Only II
(C) Both I and II
(D) Neither I nor II
AA [GATE-ME-2017-IITR]
100. P(0, 3), Q(0.5, 4) and R(1, 5) are three points
on the curve defined by f(x). Numerical
integration is carried out using both
Trapezoidal rule and Simpson’s rule within
limits x = 0 and x = 1 for the curve. The
difference between the two results will be :
(A) 0 (B) 0.25
(C) 0.5 (D) 1
**********
Miscellaneous
T5.1AA [GATE-CE-2009-IITR]
101. In the solution of the following set of linear
equation by Gauss elimination using partial
pivoting 5x + y + 2z = 34, 4y – 3z = 12, amd
10x – 2y + z = -4.
The pivots for elimination of x and y are:
(A) 10 and 4 (B) 10 and 2
(C) 5 and 4 (D) 5 and -2
A–6 [GATE-ME-2016-IISc]
102. Gauss-Siedel method is used to solve the
following equations (as per the given order):
1 2 3
2 3 5
x x x
  
1 2 3
2 3 1
x x x
  
1 2 3
3 2 3
x x x
  
Assuming initial guess as 1 2 3 0
x x x
   ,
the value of 3
x after the first iteration is
_______

150. ENGINEERING MATHEMATICS
Page 142 TARGATE EDUCATION GATE-(EE/EC)
AT5.2, 2.25-2.72 [GATE-CH-2016-IISc]
103. The model
2
y mx
 is to be fit to the data
given below.
x 1 2 3
y 2 5 8
Using linear regression, the value (rounded
off to the second decimal place) of m is
______ .
AD [GATE-ME-2004-IITD]
104. The value of the function f(x) are tabulated
below
X 0 1 2 3
f(x) 1 2 1 10
Using Newton’s forward difference formula,
the cubic polynomial that can be fitted to the
above data is
(A) 3 2
2x 7x 6x 2
  
(B) 3 2
2x 7x 6x 2
  
(C) 3 2 2
x 7x 6x
 
(D) 3 2 2
2x 7x 6x 1
  
-------0000-------

151. TOPIC 6 – NUMERICAL METHODS
www.targate.org Page 143
Answer :
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
B B D C C D B B C D
11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
C B C A A 2 C C C C
21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
A A B * A A A C C A
31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
A C C * * B * 0.06 C *
41. 42. 43. 44. 45. 46. 47. 48. 49. 50.
C B C B A * D 4.3 A *
51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
* 1 10 * B * A D A A
61. 62. 63. 64. 65. 66. 67. 68. 69. 70.
* * A D * * D * * *
71. 72. 73. 74. 75. 76. 77. 78. 79. 80.
* * * D C 1.1 * * A A
81. 82. 83. 84. 85. 86. 87. 88. 89. 90.
D D C * * * * * C 0.53
91. 92. 93. 94. 95. 96. 97. 98. 99. 100.
D A B D C * C D C A
101. 102. 103. 104.
A –6 * D
24. 8.50 to 9.00
34. 0.49 to 0.51
35. 0.355 to 0.365
37. 0.56 to 0.58
40. 0.543
46. 0.73
50. 0.65 to 0.72
51. 5.95 to 6.25
54. 0.74 to 0.76
56. 0.6 to 0.7
61. 10.5 to 11.5
62. 0.78 to 0.80
65. 0.060 to 0.063
66. 3.12 to 3.26
68. 0.175 to 0.195
69. 0.640 to 0.650
70. 1.32 to 1.34
71. 2.20 to 2.50
72. 1.820 to 1.830
73. 1.8 to 1.9
77. 1.7532
78. 1.16
84. 0.70 to 0.80
85. 1.95 to 2.05
86. 14760
87. 0.24 to 0.28
88. 94.5 to 94.8
96. 25.80 to 25.90
103. 2.25 to 2.72
**** END OF THE BOOKLET ****