GATE 1994 TO 2021 QUESTIONS (2)

Engineering Mathematics (1994-2021)
By Manish Rajput Sir (MR100)


GATE 1994 : IIT Kharagpur

a 0
Q.1 The inverse of a matrix  
0 b
 ab 0  b 0
(A)   (B) a b
 0 1  
1/ a 0  a 0 
(C)   (D)  0 1/ b 
 0 1/ b   
x
Q.2 The limit of f ( x)  as x  0 is,
sin x
(A) 0 (B) 1
(C) 2 (D) 

dy
Q.3 Integrating factor for the differential equation,  P( x) y  Q ( x)
dx
(A) exp   pdx  (B) exp   pdx 
   
(C)  pdx (D) dP / dx

Q.4 If i, j, k , are the unit vectors in rectangular coordinates, then the curl of the vector.
ix  jy  kz
(A) k (B) k
(C) j  k (D) i  k

d2y dy
Q.5 The solution for the differential equation 2
 5  6y  0
dx dx
(A) C1e2t  C2e3t (B) C1 sin 2t  C2 cos 2t

(C) C1e2t  C2e3t (D) C1e2t  C2e3t
Q.6 Taylor’s series expansion of f ( x) around x  a is _______
Q.7 For a differential function f ( x) to have a maximum, df / dx should be ____and
d 2 f / dx 2 should be _____
Q.8 M dx  N dy is an exact differential when ______
Q.9 The integral of x sin x in ____
Q.10 The Green’s theorem relates ____integrals to surface integrals
Q.11 If ‘a’ is a scalar and ‘b’ is vector, the x (a b ) =_______

1 Manish Rajput Sir : +91-8399972875, Referral Code : MR100
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, Engineering Mathematics (1994-2021)
By Manish Rajput Sir (MR100)

d2y
Q.12 The differential equation y0
dx 2
With the conditions y(0)  0 and y (1)  1 is called is _______value problem.
Q.13 (I) cosh (at)
(II) sin (at)
(A) a / (s 2  a 2 ) (B) a / (s 2  a2 )
(C) s / (s 2  a 2 ) (D) s / (s 2  a 2 )
dy
Q.14 (i)  x2  y 2
dx
dy
(ii)  x2  y
dx
(A) Linear first order O.D.E with constant coefficient
(B) Linear O.D.E. with variable coefficient
(C) First order non linear O.D.E.
(D) Linear second order O.D.E.
0 2
Q.15 Find the eigen value of the matrix A   
 1 1

GATE 1995 : IIT Kanpur

1 0 0 
Q.16 The rank of matrix 0 2 0 
 
 3 0 0 
(A) 0 (B) 1
(C) 2 (D) 3
Q.17 The angle between two vectors 2i  j  k and i  j  2k is
(A) 00 (B) 300
(C) 450 (D) 600
Q.18 A function f ( x)  12 x  x3 has maximum value at x  12
(A) 2 (B) 0
(C) 2 (D) 12
tanh x
Q.19 lim 
x  x
(A)  (B) 1
(C) 0 (D) -1

2 Manish Rajput Sir : +91-8399972875, Referral Code : MR100
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, Engineering Mathematics (1994-2021)
By Manish Rajput Sir (MR100)
Q.20 The second order Taylor series expansion for a function f ( x)  x 2at x  1 is
(A) x 2 (B) 1  x 2
(C) 1  x  x2 (D) 1  x  x 2
Q.21 The average value of function f ( x)  x3 in the interval 0  x  2 is
(A) 1 (B) 2
(C) 4 (D) 8
Q.22 (I) y  x 2
(II) dy / dx  2 x
(A) Linear O.D.E. (B) Nonlinear O.D.E.
(C) Linear algebraic equation (D) Nonlinear algebraic equation
Q.23 (I) dy / dx  5 y  0, y(0)  y0

(II) dy / dx  5  0, y(0)  y0

(A) y  y0  5x

(B) y  y0  5x

(C) y  y0e5t

(D) y  y0e5 x

 4 5
Q.24 Find eigen values and eigen vectors of matrix.  
1 2 

GATE 1996 : IISc Bangalore
x2
dx
 y
( x2  x1 )
x1
Q.25 The ratio
( x2  x1 )
y x  x2

Where 1/y is a monotonically increasing function of x, is:
(A) Less than unity (B) Equal to unity
(C) Greater than unity (D) Less than zero
d2y
Q.26 Solve  m2 y  0
dx 2
Subject to y  1 at x  0 and dy / dx  0 at x  1
 1 2
Q.27 Given the matrix A   
3 4
(i) Write down the characteristic equations.
(ii) Computer [A]4 without direct multiplication.
3 Manish Rajput Sir : +91-8399972875, Referral Code : MR100
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, Engineering Mathematics (1994-2021)
By Manish Rajput Sir (MR100)
Q.28 Solve using
dy
 0.6 y  6e0.5 x
dx
The integrating factor method given y  1 at x  0

GATE 1997 : IIT Madras


Q.29 The sum of the infinite series is
2 n
1 1 1
3      ___   
3 3 3
(A) 9 (B) 9/2
(C) 15/2 (D) Infinity
x3  1
Q.30 lim 2 is
x  2 x  80 x  1


(A) 0 (B) ½
(C) 1 (D) Infinity

Q.31 The value of is  (2  sin x) dx
0

(A) > 0 (B) < 0
(C) 0 (D) Undefined
Q.32 Given f ( x, y)  x2  y 2 , 2 f is
(A) 4 (B) 2
(C) 0 (D) 4( x  y)2
Q.33 A polynomial, f ( x) , is sketched below




f ( x)



x
Its order is
(A) 3 (B) 2
(C) 4 (D) > 5

4 Manish Rajput Sir : +91-8399972875, Referral Code : MR100
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