Department of Mathematics and Statistics
MCQs Bank of entry test for the Mphil (Mathematics)
Ans
Key
B
MCQ’s
1)
2)
3)
4)
5)
6)
x 2 + 15
A
B
Cx + D
=
+
+ 2
If
, A=
2
2
2
( x + 3) ( x + 3) x + 3 ( x + 3)
x +3
A. 1
B. 1
−
2
2
C. 2
D. none of these
x 2 + 15
A
B
Cx + D
=
+
+ 2
If
, B=
2
2
2
( x + 3) ( x + 3) x + 3 ( x + 3)
x +3
A. 1
B. 1
−
2
2
C. 2
D. none of these
x 2 + 15
A
B
Cx + D
=
+
+ 2
If
, C=
2
2
2
( x + 3) ( x + 3) x + 3 ( x + 3)
x +3
A. −1
B. 1
C. 0
D. none of these
2
x + 15
A
B
Cx + D
=
+
+ 2
If
, D=
2
2
2
( x + 3) ( x + 3) x + 3 ( x + 3)
x +3
A. −1
B. 1
C. 0
D. none of these
1
Derivative of
with respect to x , is
x
A. 1
B. 1
− 2
2
x
x
C. undefined
D. none of these
Derivative of 2 x with respect to x , is
A. 2 x
B.
2x
−
log 2
log 2
x
C. 2
D. none of these
b
C
A
B
B
D
log x
7)
If e x + y = xy ,
C
dy
=
dx
A. y (1 + x)
x( y − 1)
C. y (1 − x )
x ( y − 1)
Discipline: ________________________
B. y (1 − x)
x( y + 1)
D. none of these
Page 1 of 110
8)
 ln x dx =
1
x
C. x ln x + x + C
9)
1
 a 2 − x 2 dx = C + f , f =
A.
x
cos −1 ( )
a
C.
x
sin −1 ( )
a
2
10)
x
dy
If y =
,
=
1 + x dx
A. x 2 − 2 x
( x + 1) 2
C. x 2 + 2 x
A.
( x + 1)
11)
12)
D
1
ln x
D. none of these
B.
C
B. 1 −1 x
sin ( )
a
a
D. none of these
C
B.
x −1+
1
x +1
D. none of these
2
3
1
dt =
4−t
0
A. 2
C. 1
A

B. 0
D. none of these
0

A
3u + 4 du =
−1
14
9
14
C.
3
B. −
A.
13)
14
9
D. none of these
1
C
−x
 e dx =
0
A.
1
e
C.
1
1−
e
14)
1+

 cos
2
B. 0
D. none of these
A
 sin  d =
0
A. 2
3
C. 0
15) e ln x
1 x dx =
Discipline: ________________________
B.
2
3
D. none of these
−
D
Page 2 of 110
A.
2
3
C. 1
−
2
1 −x
16) e + 1
0 e− x dx =
A. e
C. e −1
17) 
−
4
 cos
2
B. 2
3
D. none of these
A
B. 0
D. none of these
C
 d =
0
A.  1
−
8 4
C.  1
+
8 4
18) 2 u
 u 2 − 1 du =
2
B. 
8
D. none of these
A. ln 3
C. 1 − ln 3
19) 1 e x
0 e x + 1 dx =
B. ln 3
A. e + 1
2
C. e − 1
ln
2
A
D. none of these
B
e +1
2
D. none of these
B.
ln
20)
D
The area of the shaded region is
A. 60
C. 32
21)   e− y
0 x y dxdy =
A. 1
C. −1
Discipline: ________________________
B. 30
D. none of these
A
B. 0
D. none of these
Page 3 of 110
22) If D is the region above the x -axis and within a circle centered at the origin of radius 2,
2
2
 ( x + y )dxdy =
D
23)
24)
25)
26)
27)
28)
29)
A. 2
C. 4
4 − x2
lim 2
=
x → x − 1
A. 1
C. 
x
lim =
x →0 x
A. 0
C. undefined
x3 − 8
lim 2
=
x→2 x − 4
A. 1
C. 0
5x − 4
=
2
x −x−2
A. 2
3
−
x − 2 x +1
C. 2
3
+
x + 2 x −1
3x + 11
A
=
+
If 2
x − x−6 x−3
A. 4
C. −4
3x + 11
A
=
+
If 2
x − x−6 x−3
A. 1
C. −1
B. 
D. none of these
B
B. 1
D. none of these
B
B. −3
D. none of these
D
B. −3
D. none of these
D
B.
2
3
+
x − 2 x +1
D. none of these
B
, A=
x+2
B
, B=
x+2
If derivative of f ( x) + C is −
−
A
B. −1
D. none of these
C
B. −4
D. none of these
A
1
, f ( x) =
x2
A. 1
B.
x
C. − ln x2
30) Derivative of 2 − x with respect to x , is
A. 2 − x
log 2
C. −2− x log 2
31)
C
−
1
x
D. none of these
C
B.
−x
2
log x
D. none of these
−
3 x + 11
=
x2 − x − 6
Discipline: ________________________
C
Page 4 of 110
A.
4
1
+
x −3 x + 2
C. 4
1
−
x −3 x + 2
32) (ln x + c) dx =

B.
1
x
C. x ln x + cx + C
33) xe x dx =

B. x ln x − x + cx + C
A. e x ( x − 1) + C
C. xe x − 1 + C
34)
1
 ax + b dx =
A.
1
−
+C
(ax + b) 2
C. 1
ln(ax + b) + C
a
35) e tan −1 x
 1 + x2 dx = C + f , f =
A. e tan −1 x
B. e x ( x + 1) + C
D. none of these
A.
C. ln e tan −1 x
36)
x2
If y =
, y dx = C + f , f =
1+ x 
x2
A.
+ x − ln( x + 1)
2
x2
C.
− x + ln( x + 1)
2
37) x 2
=
1+ x
A. x 2 + x
C.
38)
x −1+
1
1+ x
4
1
+
x−3 x+ 2
D. none of these
−
B
D. none of these
A
C
B. a ln(ax + b) + C
D. none of these
A
B. e− tan−1 x
D. none of these
C
B. x − 1 +
1
1+ x
D. none of these
C
B.
1
1+ x
D. none of these
x +1−
1
B
2
 ( x − x − 1) dx =
−1
A. 4
3
C. 3
−
4
Discipline: ________________________
B.
4
3
D. none of these
−
Page 5 of 110
3x − 1
1 3x dx =
A.
ln 3
1−
2
C.
ln 2
1−
3
3
40)
1
0 4 + t dt =
A. 1
C. −2
41) 0
 3u + 4 du =
39)
2
C
B.
ln 2
3
D. none of these
1+
D
B. 0
D. none of these
C
−1
A. 14
3
C. 14
9
42)
3
B.
14
9
D. none of these
x

4 − x2
A. −1
C. 0
−
B
dx =
0
43)
B. 1
D. none of these
1
 (2t −1)
3
B
dt =
0
A. −1
C. 1
44) 9 2 + x
2
B. 0
D. none of these
A
dx =
x
25
A.
3
25
C.
9
3
45)
1
−3 9 + x2 dx =
A. 
−
6
C. 
6
1
46)
−x
 e dx =
4
B.
5
3
D. none of these
C
B. 0
D. none of these
D
0
A. 0
C. 1
Discipline: ________________________
B. −1
D. none of these
Page 6 of 110
47)

e− y
0 x y dxdy =
D
A. −1
B. 0
C. e
D. none of these
48) If D is the region above the x -axis and within a circle centered at the origin of radius 2,
2
2
 ( x + y )dxdy =
C
D
A. 
B. 2
C. 4
D. none of these
49) If  x  is the greatest integer not greater than x , lim  x  =
1
x→
A. 0
C. 2
50)
sin(3 x)
lim
=
x →0 sin(4 x )
A. 4
3
C. 3
4
51) If A is bounded above and s = Sup( A) , for each
A
2
B. 1
D. none of these
C
B. 1
D. none of these
  0 , there is at least one element a0 of A such
B
that
A. a0  s + 
C. a0  s − 
B. a0  s − 
D. none of these
52) If x  0 , x =
C
A. 1
C. 0
53) If limit of a sequence (an )n=1 is l , an → l as
A. n → 
C. n → 1
54)
B. 
D. none of these
A
B. n → 0
D. none of these

B
1
The sequence   is
 n  n =1
A. divergent
C. oscillating
55) For a fixed   0 , if x   for all   0 , x =
B. convergent
D. none of these
A. 1
C. 
56) log3 (243) =
A. 24
C. 8
B. 0
D. none of these
Discipline: ________________________
B
D
B. 6
D. none of these
Page 7 of 110
57) A bounded sequence of real numbers
C
A. converges
C. may converge
58) A Cauchy sequence of real numbers is
B. diverges
D. none of these
A. not bounded
C. divergent

59)
1
The series  is
n =1 n
B. oscillating
D. none of these
A. convergent
C. divergent
60) If a  b and b  c ,
B. =2
D. none of these
A. a = c
C. a  c
61) The set of rational numbers is
D
C
C
a+c
2
D. none of these
B.
b=
A
A. countable
C. finite
62) The interval  0,1 is
B. uncountable
D. none of these
A. infinite
C. finite
63) If a  b , the interval a, b  is
B. bounded
D. none of these
A. countable
C. not countable
64) If A is countable, A
B. unbounded
D. none of these
A. N
C. R
65) The set of natural numbers is equivalent to
B. Q
D. none of these
A.  0,1
A
C
A
B
B. Q
C. R
D. none of these
66) A bounded monotonically decreasing sequence converges to its
A. infimum
C. 1st term
67)  1
=

k =0 k !
A. 
C. e
Discipline: ________________________
A
B. suprimum
D. none of these
C
B. 
D. none of these
Page 8 of 110
68)

The series
1
n
n =1
2
A. convergent
C. undefined
69) A bijection f from N to W is f (n) =
A. n + 1
C. n
70)
A
is
B. divergent
D. none of these
B
B. n −1
D. none of these
C
The function f (n) = 2n − 1 is bijection from Z to the set of
A. natural numbers
C. odd integers
71) The function f ( x) = e x is bijection:
B. even integers
D. none of these
A
A. R → R +
B. R+ → R
C. R → R
D. none of these
72) If A is bounded below and i = Inf ( A) , for each   0 , there is at least one element a1 of A such that
A. a1  i − 
C. a1  i + 
C
B. a1  i + 
D. none of these
73) If d is a metric on X ( ) , d :
B
A. X → R
B. X  X → R
C. X → X
D. none of these
74) If d is a metric on X ( ) , for all x, y  X , d ( x, y ) is
B
A. positive
B. non negative
C. 0
D. none of these
75) If d is a metric on X ( ) , for all x, y  X , d ( x, y ) is
D
A. positive
B. negative
C. 0
D. none of these
76) If b d is a metric on X ( ) and x, y  X be such that y = x , d ( x, y ) is
D
A. positive
B. negative
C. undefined
D. none of these
77) If d is a metric on X ( ) , for all x, y  X , d ( x, y ) =
C
A. −d ( y, x)
B. 0
C. d ( y, x)
D. none of these
78) If d is a metric on X ( ) , for all x, y, z  X , d ( x, y ) is
A
A.  d ( x, z ) + d ( z, y)
C.  d ( x, z ) + d ( z, y )
Discipline: ________________________
B.  d ( x, z ) + d ( z, y)
D. none of these
Page 9 of 110
79) The function f : R → R + defined by f ( x) = e x is
B
A. not one to one
B. one to one
C. decreasing
D. none of these
80) The function f : R + → R defined by f ( x) = ln x is
B
A. decreasing
C. constant
81) The set R + is equivalent to
C
A. Q
C. R
82) The interval  0,1 is equivalent to
A. Q
C.  2,5
B. increasing
D. none of these
B. Q+
D. none of these
C
B. Z
D. none of these
83) The area bounded by the curves y = sin x , y = 0 , x = 0 and x =  is
A
A. 2
B. 0
C. undefined
D. none of these
84) The area bounded by the curves y = sin x , y = 0 , x = 0 and x = 2 is
B
A. 2
C. 4
85)
B. 0
D. none of these
  
The interval  − ,  is equivalent to
 2 2
C
A. Q
C. R
86) f ( x) = cos x is a periodic function with period
B. Z
D. none of these
A. 2
C. 
2
87) If a  b and c  d ,
B. 
D. none of these
A. a + c  b + d
C. a + c  b + d
88) If a  b and c  0 ,
B. a + c  b + d
D. none of these
A. ac  bc
C. ac  bc
89) If a  b and c  0 ,
B. ac = bc
D. none of these
A. ac  bc
C. ac = bc
Discipline: ________________________
A
A
C
A
B. ac  bc
D. none of these
Page 10 of 110
90)
D
sin(2 x)
=
x →0
3x
lim
A. 1
C. 3
2
91) In a metric space, a convergent sequence
B. 0
D. none of these
A. is Cauchy
C. is oscillating
92) A Cauchy sequence of real numbers is
B. is not Cauchy
D. none of these
A. unbounded
C. convergent
93) A convergent sequence of real numbers is
B. divergent
D. none of these
A. unbounded
C. oscillating
94) In a metric space, a Cauchy sequence
B. Cauchy
D. none of these
A. is convergent
C. is oscillating
A
C
B
B
B. may not converge
D. none of these
95)
C
The circumference to diameter ratio of a circle is
A. 1
C. 
96) The sequence
( (−1) )
B. e
D. none of these
n 
A. converges to 1
C. converges to −1
97) Which of the following is true
A.   e
C.  is rational
98) The number  is
A. not real
C. rational
99) The number e belongs to
A. Q
C. Z
100)
D
n =1
B. converges to 0
D. none of these
D
B.
22
7
D. none of these
=
B
B. irrational
D. none of these
B
B. Q
D. none of these
C
n
k =
k =1
A. n 2
Discipline: ________________________
B. n( n − 1)
2
Page 11 of 110
C. n( n + 1)
2

101) b
 1n 
The sequence  n  is
  n =1
A. divergent
C. oscillating
D. none of these
B
B. convergent
D. none of these
102) If a convergent sequence ( a ) consists of infinitely many distinct elements and A = {a , a , a ,...} ,
n n =1
1
2
3
C
the limit of the sequence
A. does not exist
C. is limit point of A
B. is not limit point of A
D. none of these
103) If a convergent sequence ( a ) consists of finitely many distinct elements and A = {a , a , a ,...} , the
n n =1
1
2
3
A
limit of the sequence
A.  A
B.  A
C. is undefined
D. none of these
104) If x0 is an element of a metric space ( X , d ) and r  0 , {x  X : d ( x, x0 )  r} =
B
B. B( x0 ; r )
A. B( x0 ; r )
C. S ( x0 ; r )
D. none of these
105) If x0 is an element of a metric space ( X , d ) and r  0 , {x  X : d ( x, x0 )  r} =
A. X − B( x0 ; r )
B. X − B( x ; r )
0
D. none of these
C. X − S ( x0 ; r )
106)
B

C
 1
The sequence  n n  converges to
  n =1
A. 0
B. −1
C. 1
D. none of these
107) If x0 is an element of a metric space ( X , d ) and r  0 , {x  X : d ( x, x0 ) = r} =
A. B( x0 ; r )
C. S ( x0 ; r )
C
B. B( x ; r )
0
D. none of these
108) If x is an element of a metric space ( X , d ) , r  0 , A = B( x ; r ) and B = B( x ; r ) ,
0
0
0
A
A. A  B
B. B  A
C. A = B
D. none of these
109)If x is an element of a metric space ( X , d ) , r  0 , S = S ( x ; r ) and B = B( x ; r ) ,
0
0
0
B

S
A. S = B
B.
C. S  B
D. none of these
C
Discipline: ________________________
Page 12 of 110
110) If x0 is an element of a metric space ( X , d ) and r  0 , {x  X : d ( x, x0 )  r} =
A. X − B( x0 ; r )
B. X − B( x ; r )
0
D. none of these
C. X  S ( x0 ; r )
111)
D

D
 n 2 − 3n + 1 
The limit of the sequence  2
 is
 2n + 3n − 1  n =1
A. −1
B. 0
2
C. −1
D. none of these
112) If a relation f is such that a = b  f (a) = f (b) then f is
A
A. a function
B. onto
C. one to one
D. none of these
113) If a function f is such that f (a) = f (b)  a = b then f is
C
A. a function
C. one to one
114) If f : A → B is a function, Dom( f )
B. onto
D. none of these
A
A. = A
C.  A
115) If f : A → B is a function, Range( f )
B.  A
D. none of these
B
A. = B
B.  B
C.  A
D. none of these
116) If f : A → B is a function and a  b  f (a)  f (b) , f is
A
A. one to one
B. onto
C. bijection
D. none of these
117) If f : A → B is a function such that different elements of A have different images in B , f is said to
be
C
A. bijection
B. onto
C. one to one
D. none of these
118) If f : A → B is a function such that Range( f )  B , f is said to be
A
A. into
B. onto
C. bijection
D. none of these
119) If f : A → B is a function such that Range( f ) = B , f is said to be
B
A. into
C. bijection
120)
B. onto
D. none of these
C
If f : A → B is a function and such that A1  A , the function f1 : A1 → B defined by
Discipline: ________________________
Page 13 of 110
f1 (a) = f (a) for all a  A1 , is called
A. extension of f on A1
C. restriction of f on A1
B. subset of A
D. none of these
121) For two non-empty sets A and B , the set {(a, b) : a  A, b  B} is called Cartesian product of
A
A. A and B
B. B and A
C. AB
D. none of these
122) For two non-empty sets A and B , the Cartesian product of A and B is denoted by
C
A. AB
C. A B
B. B  A
D. none of these
123) If A B is the Cartesian product of A and B , A B is
124)
A.  A B
B. = A B
C.  A B
D. none of these
n
n
k =0
 
B
B
 k  =
A. 2 − n
C. 2n
B. 2 n
D. none of these
125) Integral of e x2 w.r.t. x , is
A. e x 2
2x
C. x 2 e x2 −1
D
B. 2 xe x2
D. none of these
126) Anb infinite series
B
A. is convergent
C. is divergent
127) An infinite sequence
B. may converge
D. none of these
A. is divergent
C. is convergent
128) a  b a + b
If
,
is
2
B. may converge
D. none of these
A. lesser than a
C. equal to ab
129) A decreasing sequence
B. greater than b
D. none of these
A. is divergent
C. is convergent
130) If a  b , a 2 + b2 is
B. may diverge
D. none of these
A. lesser than 2ab
Discipline: ________________________
B
D
B
B
B. greater than 2ab
Page 14 of 110
C. equal to 2ab
131)
D. none of these
The equation of the line passing through origin at an inclination of
4
B
, is
B. x = y
D. none of these
A. x = 2 y
C. 2x = y
132)

The equation of the line passing through (0,3) at an inclination of
A. x − 3 = y
C. x + 3 = y
133)
x
If f ( x) = , f −1 (1) =
2

4
C
, is
B. −3x = y
D. none of these
C
A. 0
B. 1
C. 2
D. none of these
134) The inverse relation of a function, is a function iff the function is
C
A. onto
C. bijective
135) If a  0 and x  a ,
C
B. one to one
D. none of these
A. x = a
B. a  x  −a
C. −a  x  a
D. none of these
136) If S is the solution set of the relation x = 5 , S =
A. {−5,5}
C. ] − 5,5[
A
B. [−5,5]
D. none of these
137) For any x, y  R , x + y is
A
A.  x + y
B.  x + y
C.  x + y
138)
D. none of these

A. 1
2
C. 0
139)
C
 1 
Infimum of the sequence  2
 , is the number
 n + 2 n =1
Suprimum of the sequence
B. 1
3
D. none of these

2, 2 + 2 , 2 + 2 + 2 ,...
A. 2
A
 , is the number
B.
2
C. 
D. none of these
140) If A and B are two sets such that A  B , inf( A) is
Discipline: ________________________
A
Page 15 of 110
A.  inf( B)
B.  inf( B)
C. = inf( B)
D. none of these
141) If A and B are two sets such that A  B , Sup( A) is
A
A.  Sup( B)
B.  Sup( B)
C. = Sup( B)
D. none of these
142) If A and B are two sets such that A  B , Sup( A) is
D
A. finite
C. 0
143) Which of the followings is not true
D
B. infinite
D. none of these
A. Z  Q
C. W  Z
144)
Domain of the function f ( x) =
B. Q  R
D. none of these
1
4 − x2
B
, is the set
A.  −2, 2
B. −2, 2
C. {2, −2}
D. none of these
145)
C
x
If f ( x) = − 3 , f −1 ( x) =
2
2
x−6
C. 2 x + 6
146) If x = 10 y , y =
A.
A.
B. 2 x − 6
D. none of these
D
1
ln(10)
C. e
147) If x − 3 = 3 − x ,
B.
A. x  3
C. x − 3 = 0
148) ln x is undefined for
B. x = 3
D. none of these
1
ln( x)
D. none of these
D
D
A. x  0
B. x = 10
C. x = e
D. none of these
149) The real line R is a metric space under the metric d0 : R  R → R defined as d0 ( x, y) =
A. x + y
B. x − y
C. x − 2 y
D. none of these
Discipline: ________________________
Page 16 of 110
B
150)
1
In the metric space ( R, d0 ) with usual metric d 0 , if A = N and B =  n − : n  N − {1}
n
=
A. 0
C. 
151)

e
− x2
,
d ( A, B)
A
B. 1
D. none of these
C
dx =
−
A.

B. 
2
C. 
152) The function f ( x) = x is called
D. none of these
A. a linear function
C. a quadratic function
153) The notation y = f ( x) was invented by
B. an identity function
D. none of these
A. Euler
C. Newton
154) The graph of a linear equation is always a
B. Leibnitz
D. none of these
B
A
B
A. Parabola
B. Straight line
C. Cycle
D. none of these
155) The linear function f ( x) = ax + b is identity function if
B
A. a  0, b = 1
C. a = 1, b = 1
156) The notation f ( x) was invented by
B. a = 1, b = 0
D. none of these
C
A. Leibnitz
C. Newton
157) (ax + b) n dx =

B. Lagrange
D. none of these
A. n(a n −1 x + b)
C.
C
B.
1
(ax + b) n +1 + C
n +1
D. none of these
1
(ax + b) n +1 + C
a (n + 1)
158) The change in variable x is called increment of x . It is denoted by  x which is
A. Positive or negative
C. Positive only
159)
d
( f ( x) + C ) = Sec 2 (3 x) , f ( x) =
If
dx
A. tan(3x)
Discipline: ________________________
A
B. Negative only
D. none of these
B
B. 1
tan(3 x)
3
Page 17 of 110
C. 3tan(3x)
160)
f ( x) − f (a)
lim
=
x →a
x−a
D. none of these
A. f (a)
C. f (0)
161)
d
If
( f ( x) + C ) = x n −1 , f ( x) =
dx
B. f ( x)
D. none of these
A. x n
C. x n−2
162) log a a = ?
B. nx n
D. none of these
A
B
A. a
C. 0
163)
D
B. 1
D. none of these
If y = sinh −1 (ax + b) ,
A.
B
dy
=
dx
B.
1
a 1 + (ax + b)2
C.
1
a
1 + (ax + b) 2
D. none of these
1 + (ax + b) 2
164)
If y = e− ax ,
C
dny
=
dx n
B. (−a)− n y
D. none of these
A. a n y
C. (−a)n y
165)
If y = e− ax , y
D
dy
=
dx
A. −ae2ax
e −2ax
a
D. none of these
B.
C. ae−2ax
166)
f ( x) = f (0) + xf (0) +
−
A
x2
f (0) + ... is called
2!
A. Maclaurin’s series expansion
C. Taylor’s theorem
167) 1 − x + x2 − x3 + ... =
A. −1
1+ x
C. 1
1− x
Discipline: ________________________
B. Taylor’s series expansion
D. none of these
B
B.
1
1+ x
D. none of these
Page 18 of 110
168) f is said to be increasing function on a, b if for x1 , x2  a, b
A. f ( x2 )  f ( x1 ) whenever x2  x1
C. f ( x2 )  f ( x1 ) whenever x2  x1
A
B. f ( x2 )  f ( x1 ) whenever x2  x1
D. none of these
169) f is said to be decreasing function on a, b if for x1 , x2  a, b
C
A. f ( x2 )  f ( x1 ) whenever x2  x1
B. f ( x2 )  f ( x1 ) whenever x2  x1
C. f ( x2 )  f ( x1 ) whenever x2  x1
D. none of these
170) A point where first derivative of a function is zero, is called
A
A. Stationary point
C. Point of concurrency
171) f ( x) = sin x is
C
B. Corner point
D. none of these
A. An even function
B. A linear function
C. An odd function
D. none of these
172) The maximum value of the function f ( x) = x 2 − x − 2 is
A. 9
4
C. 9
2
173) d
d2
(cos x) − 2 (sin x) =
dx
dx
B.
A. 0
C. 2cos x
174) If f ( x) = x3 + 2 x + 9 , f ( x) =
B. 2sin x
D. none of these
A. 0
C. 6x
175) d
1 2
( x−
) =
dx
x
B. 3x 2
D. none of these
A.
C.
1+
1
x2
1
x2
176) At x = 0 , the function f ( x) = 1 − x3 has
1−
A. Maximum value
C. Point of inflection
177) If y = sin x , y =
Discipline: ________________________
B
9
4
D. none of these
−
A
C
B
B.
1
x2
D. none of these
1−
C
B. Minimum value
D. none of these
B
Page 19 of 110
A. cos x
B. cos x
x
2 x
D. none of these
C. 2 x cos x
178) y = x x has the value
B
A. Minimum at x = e
B.
C. Maximum at x = e
1
e
D. none of these
179) The degree of the differential equation
180)
Minimum at x =
𝑑2 𝑥
𝑑𝑡 2
B
+ 2𝑥 3 = 0 is
A.
0
B.
C.
2
D. none of these
The order and degree of differential equation
𝑑3 𝑥
𝑑𝑡 3
1
A
𝑑𝑦
+ 4√(𝑑𝑥 )3 + 𝑦 2 =0 are respectively
A.
3 and 2
B.
2 and 3
C.
3 and 3
D. none of these
181) The differential equation 2𝑑𝑦 + 𝑥 2 𝑦 = 2𝑥 + 3, 𝑦(0) = 5 is
A
𝑑𝑥
A.
linear
B.
nonlinear
C.
linear with fixed constants
D. none of these
182) The partial differential equation ∂u + 𝑢 ∂u = ∂2 u is a
∂t
∂x
∂x2
D
A.
Linear equation of order 2
B.
Non linear equation of order 1
C.
Linear equation of order 1
D. none of these
183) Consider the following differential equation 𝑑𝑦 = 5𝑦, initial condition y=2 at t=0, the value pf y at
𝑑𝑡
t=3 is
A.
-5𝑒 −10
Discipline: ________________________
B.
2𝑒 −10
Page 20 of 110
C
2𝑒 −15
C.
184)
185)
D. none of these
𝑑2 𝑦
𝑑𝑦 3
B
The following differential equation 3( 𝑑𝑡 2 )+4( 𝑑𝑡 ) + 𝑦 2 + 2 = 𝑥
A.
Degree=2, order=1
B.
C.
Degree=4, order=3
D. none of these
𝑑2 𝑦
Degree=1, order=2
𝑑𝑦 3
B
The order of the differential equation ( 𝑑𝑡 2 )+ ( 𝑑𝑡 ) + 𝑦 4 = 𝑒 −𝑡 is
A. 1
C. 3
B. 2
D. none of these
186) The differential equation
𝜋
d2 y
dx2
𝑑𝑦
𝑑𝑦
+ 16y=0 for y(x) with two boundary conditions 𝑑𝑥 (𝑥 = 0) = 1 and 𝑑𝑥
A
(x= 2 )=-1 has
A.
No solution
B.
Exactly one solution
C.
Exactly two solutions
D. none of these
187) A differential equation is considered to be ordinary if it has
C
A.
one dependent variable
B.
more than one dependent variable
C.
one independent variable
D. none of these
188) PDE has independent variable
A
A.
More than 1
B.
Less than 1
C.
1
D. none of these
189) In homogeneous first order linear constant coefficient ordinary DE is
A.
𝜕𝑢
𝜕𝑥
=0
Discipline: ________________________
B.
𝜕𝑢
𝜕𝑥
B
=cu+𝑥 2
Page 21 of 110
C. 𝜕𝑢 𝑐
= + 𝑥2
𝜕𝑥 𝑢
190) The P.D.E
𝜕2 𝑈
𝜕𝑥 2
D. none of these
𝜕2𝑈
A
+ 𝜕𝑦 2 = 𝑓(𝑥, 𝑦); is known as
A.
Laplace equation
B.
Poisson equation
C.
Wave equation
D. none of these
191) The solution of a differential equation which is not obtained from the general solution is known as:
A.
Particular solution
B.
C.
Auxiliary solution
D. none of these
Singular solution
dy
192)
A
The differential equation dx = y2 is:
A.
Non-linear
B.
C.
Quasilinear
D. none of these
Linear
193) The DE formed by y= acosx+bcosx+4 where a and b are arbitrary constants is:
𝑑2 𝑦
B.
𝑑2 𝑦
D. none of these
A.
(𝑑𝑥 2 )+y=0
C.
(𝑑𝑥 2 )+y=4
C
𝑑2 𝑦
(𝑑𝑥 2 )-y=0
194) The equation a0 x2+𝑑2 𝑦+a1 xdy +a2y=𝜑(𝑥) is called:
𝑑𝑥 2
dx
B
A.
Legendre’s linear equation
B.
C.
Simultaneous equation
D. none of these
Cauchy’s linear equation
195) Solution of the DE 𝑑𝑦=sin(x+y)+ cos(x+y), is
𝑑𝑥
Discipline: ________________________
B
D
Page 22 of 110
A.
C.
(𝑥+𝑦)
Log|1+tan
2
|=y+c
Log |1+tan(x+y)|=y+c
B.
(𝑥+𝑦)
Log|2+sec
|=x+c
2
D. none of these
196) If y=a cos (log x)+ b sin (log x), then
B
𝑑2 𝑦
𝑑𝑦
B.
𝑑2 𝑦
𝑑𝑦
D. none of these
A.
x2 𝑑𝑥 2 + x𝑑𝑥 + y=0
C.
x2 𝑑𝑥 2 + x𝑑𝑥 - y=0
𝑑2 𝑦
𝑑𝑦
x2 𝑑𝑥 2 - x𝑑𝑥 + y=0
197) If y=sin(a 𝑠𝑖𝑛−1 𝑥), then
A
𝑑2 𝑦
𝑑𝑦
B.
𝑑2 𝑦
𝑑𝑦
D. none of these
A.
(1-x2 ) 𝑑𝑥 2 - x𝑑𝑥 +a2 y=0
C.
(1-x2 ) 𝑑𝑥 2 - x𝑑𝑥 -a2 y=0
𝑑2 𝑦
𝑑𝑦
(1-x2 ) 𝑑𝑥 2 + x𝑑𝑥 -a2 y=0
198) The DE of the family of curves y2= 4a(x+ is
A.
C.
𝑑𝑦
y2=4
𝑑𝑥
(𝑥 +
𝑑𝑦
𝑑𝑥
B
B.
)
𝑑𝑦
𝑑𝑦
𝑑𝑦
𝑑𝑥
𝑑𝑥
y2( )2+2xy
- y2=0
D. none of these
y2𝑑𝑥 + 4y=0
199) Inverse derivative of sin x is:
A.
A
1
B.
√1 + 𝑥 2
√1 − 𝑥 2
C.
−1
1
D. none of these
√1 − 𝑥 2
200)
Inverse derivative of cos x is:
A.
1
√1 − 𝑥 2
Discipline: ________________________
C
B.
1
√1 + 𝑥 2
Page 23 of 110
−1
C.
D. none of these
√1 − 𝑥 2
201) If y=𝑡𝑎𝑛−1 𝑥 3⁄2 , then 𝑑𝑦 =
𝑑𝑥
A
A.
3√𝑥
2(1 + 𝑥 3 )
B.
C.
3√𝑥
2(1 − 𝑥 3 )
D. none of these
2√𝑥
(1 + 𝑥 3 )
202) The equation of the curves, satisfying the DE 𝑑2 𝑦 (𝑥 2 + 1) = 2𝑥 𝑑𝑦 passing through the point (0,1)
𝑑𝑥 2
𝑑𝑥
B
and having the slope of tangent at x=0 as 6 is
𝑦 2 = 2𝑥 3 + 6𝑥 + 1
A.
𝑦 2 = 𝑥 3 + 6𝑥 + 1
C.
𝑦 = 2𝑥 3 + 6𝑥 + 1
B.
D. none of these
203) A particle, initially at origin moves along x-axis according to the rule 𝑑𝑥 = x+4. The time taken by the
C
𝑑𝑡
particle to traverse a distance of 96 units is:
ln 5
A.
C.
B.
2 ln 5
log 5 𝑒
D. none of these
204) If y=𝑐𝑜𝑠 −1 (𝑙𝑛 𝑥), then the value of 𝑑𝑦 is
𝑑𝑥
A.
1
B
B.
𝑥√1 − (ln 𝑥)2
C.
−1
−1
𝑥√1 − (ln 𝑥)2
D. none of these
𝑥√1 + (ln 𝑥)2
Discipline: ________________________
Page 24 of 110
205) If x=2 𝑙𝑛 𝑐𝑜𝑡( 𝑡) and y= tan(t) + cot(t), the value of 𝑑𝑦 is
𝑑𝑥
A
A.
cot(2t)
B.
tan(2t)
C.
cos(2t)
D. none of these
206) Solution of the DE 𝑙𝑛(𝑑𝑦)= ax + by is
𝑑𝑥
A.
C.
1
- 𝑏 𝑒 −𝑏𝑦 =
1
𝑏
𝑒 −𝑏𝑦 = -
1
𝑎
1
𝑎
A
1
𝑒 𝑎𝑥 + 𝑐
B.
𝑒 𝑎𝑥 + 𝑐
D. none of these
𝑏
𝑒 −𝑏𝑦 =
1
𝑎
𝑒 𝑎𝑥 + 𝑐
207) If the general solution of a differential equation is (y+c)2=cx, where c is an arbitrary constant, then the
order and degree of differential equation is
A.
1, 2
B.
C.
1, 3
D. none of these
2, 1
208) Solution of (𝑥 2 sin3y – y2cos x) dx + (x3 cos y sin2y – 2y sin x) dy = 0 is
A.
(x3sin3y/3 ) = c
C.
(x3sin3y/3 )= y2 sin x +c
B.
C
𝑥 3 sin3y = y2 sin x + c
D. none of these
209) Solution of 𝑥 𝑑𝑦 = ( 𝑦 − 1)dx is
𝑥 2 +𝑦 2
𝑥 2 +𝑦 2
A.
C.
𝑥 − tan−1
𝑥 tan−1
𝑦
𝑥
𝑦
=𝑐
𝑥
C
B.
tan−1
𝑦
=𝑐
𝑥
D. none of these
210) Solution of (𝑦 + 𝑥 √𝑥𝑦 (𝑥 + 𝑦))𝑑𝑥 + (𝑦 √𝑥𝑦 (𝑥 + 𝑦) − 𝑥)𝑑𝑦 = 0 is
A.
A
𝑦
𝑥 2 + 𝑦 2 = 2 tan−1 √ + 𝑐
𝑥
Discipline: ________________________
B.
D
𝑦
𝑥 2 + 𝑦 2 = 4 tan−1 √ + 𝑐
𝑥
Page 25 of 110
𝑦
𝑥 2 + 𝑦 2 = tan−1 √ + 𝑐
𝑥
C.
D. none of these
211) Solution of the DE 𝑑𝑦 + 2𝑥𝑦 = 𝑦 is
𝑑𝑥
A.
𝑦 = 𝑐𝑒 𝑥−𝑥
C.
𝑦 = 𝑐𝑒 𝑥
2
A
2
𝑦 = 𝑐𝑒 𝑥 − 𝑥
B.
D. none of these
212) Solution of the differential equation 𝑑𝑦 = sin(𝑥 + 𝑦) + cos(𝑥 + 𝑦), is
𝑑𝑥
A.
C.
log |1 + 𝑡𝑎𝑛
(𝑥+𝑦)
|=y+c
2
log |1 + 𝑡𝑎𝑛 (𝑥 + 𝑦)| = y + c
B.
log |2 + 𝑠𝑒𝑐
D
(𝑥+𝑦)
2
|=x+c
D. none of these
213) The Blasius equation 𝑑3 𝑓 + 𝑓 𝑑2 𝑓 = 0 is a
𝑑η3
2 𝑑η2
B
A.
Second order non linear differential
equation
C.
Third order linear ordinary differential D. none of these
equation
214) The general solution of DE
𝑑𝑥
𝑥+𝑦
cos(
2
Third order non linear ordinary
differential equation
D
= cos(𝑥 + 𝑦) with c as a constant is
𝑦 + 𝑠𝑖𝑛 (𝑥 + 𝑦) = 𝑥 + 𝑐
A.
C.
𝑑𝑦
B.
B.
𝑥+𝑦
tan(
2
)=𝑦+𝑐
D. none of these
)=𝑥+𝑐
215) The solution of the initial value problem 𝑑𝑦 = −2𝑥𝑦; 𝑦(0) = 2 is
𝑑𝑥
A.
1 + 𝑒 −𝑥
2
Discipline: ________________________
B.
B
2𝑒 −𝑥
2
Page 26 of 110
1 + 𝑒𝑥
C.
2
D. none of these
216) The solution of ODE 𝑑2 𝑦 + 𝑑𝑦 − 6𝑦 = 0 is
𝑑𝑥 2
𝑑𝑥
A
A.
𝑦 = 𝑐1 𝑒 −3𝑥 + 𝑐2 𝑒 2𝑥
B.
C.
𝑦 = 𝑐1 𝑒 3𝑥 + 𝑐2 𝑒 −2𝑥
D. none of these
217) The solution for the differential equation
A.
𝑥2
𝑑𝑥
= 𝑥 2 𝑦 with two condition that y=1 at x=0
B.
2𝑒 2
𝑥2
C.
𝑒2
218) The solution of
𝑑𝑦
𝑑𝑦
𝑑𝑥
𝑦 = 𝑐1 𝑒 3𝑥 + 𝑐2 𝑒 2𝑥
𝑥
3𝑒 2
D. none of these
𝑥
C
= − 𝑦 with initial condition y(1)=√3 is
A.
𝑥3 + 𝑦3 = 4
B.
C.
𝑥2 + 𝑦2 = 4
D. none of these
𝑦 = 4𝑎𝑥
219) Which of these is the solution of differential equation 𝑑𝑥 + 3𝑥 = 0
𝑑𝑡
A.
2𝑒 −3𝑡
B.
C.
2𝑒 2𝑡
D. none of these
A
𝑒 −3𝑡
220) The general solution of DE 𝑑𝑦 = 𝑦 is
𝑑𝑥
𝑥
A.
log y= kx
C.
y=𝑥
𝑘
Discipline: ________________________
C
B
B.
y=kx
D. none of these
Page 27 of 110
221) Integrating factor of DE cos𝑑𝑦 + 𝑦𝑠𝑖𝑛𝑥 = 1 is
𝑑𝑥
B
A.
sin x
B.
sec x
C.
tan x
D. none of these
222) If 2xy dx + P(x, y)dy = 0 is exact then P(x, y) is
D
A.
x–y
B.
C.
x – y2
D. none of these
x+y
223) A differential equation of first degree
B
A.
Is of first order
B.
May or may not be linear
C.
Is always linear
D. none of these
224) A general solution of an nth order differential equation contains
A.
n – 1 arbitrary constants
B.
C.
n+1 arbitrary constants
D. none of these
B
n arbitrary constants
225) The differential equation Mdx + Ndy = 0 is defined as an exact differential equation of
A.
𝜕𝑀 𝜕𝑁
=
𝜕𝑥
𝜕𝑦
C.
𝜕𝑀
𝜕𝑁
=−
𝜕𝑦
𝜕𝑥
B.
𝜕𝑀 𝜕𝑁
≠
𝜕𝑦
𝜕𝑥
D. none of these
226) The order of the differential equation 𝜕2 𝑦 + 𝑦 2 = 𝑥 + 𝑒 𝑥 is
𝜕𝑥 2
A. 2
C. 0
227)
A
B. 3
D. none of these
‘’Infinitely many differential equation have the same integrating factor’’. This statement is
Discipline: ________________________
D
Page 28 of 110
C
A.
Never true
B.
C.
Always true
D. none of these
228) If
𝑑𝑦
𝑑𝑥
May be true
𝑓(𝑥,𝑦)
= 𝜑(𝑥,𝑦) is a homogeneous DE then it can be made in the form ‘’separable in variables’’ by
C
putting
A.
y2 =vx
B.
x2=vy
C.
y=vx
D. none of these
229) The differential equation 𝑑𝑦 + 𝑃𝑦 = 𝑄𝑦 𝑛 , 𝑛 ≥ 2 can be reduced to linear form by substituting
𝑑𝑥
A.
z= yn-1
B.
C.
x=y1-n
D. none of these
x=yn+1
230) The differential equation (𝑦 − 2𝑥 3 )𝑑𝑥 − 𝑥(1 − 𝑥𝑦)𝑑𝑦 = 0 becomes exact on multiplication by
A.
1
𝑥
B.
C.
1
𝑥3
D. none of these
B
1
𝑥2
231) If the DE 𝑓(𝑥, 𝑦)𝑑𝑥 + 𝑥𝑠𝑖𝑛𝑦𝑑𝑦 = 0 is exact then 𝑓(𝑥, 𝑦) equals
B
–cos(y) +x2
A.
cos y
B.
C.
–sin y
D. none of these
232) The differential equation 𝑥 𝑑𝑦 =𝑥 − 𝑦 is
𝑑𝑥
C
A.
Exact
B.
C.
Both A and B
D. none of these
Discipline: ________________________
D
Linear
Page 29 of 110
233) The general solution of the equation 𝑥 ′ + 5𝑥 = 3 is
A.
𝑥(𝑡) =
C.
𝑥(𝑡) =
3
+ 𝑒 −5𝑡
5
3
+ 𝐶𝑒 −5𝑡
5
C
𝑥(𝑡) = 3 + 𝐶 sin 5𝑡
B.
D. none of these
234) A solution of the initial value problem 𝑦 ′ + 8𝑦 = 1 + 𝑒 −6𝑡 is
A.
𝑥(𝑡) =
1 1 6𝑡 5 8𝑡
+ 𝑒 − 𝑒
8 2
8
𝑥(𝑡) = 4 − 𝑒 2𝑡 + 3𝑒 8𝑡
C.
B
B.
𝑥(𝑡) =
1 1 −6𝑡 5 −8𝑡
+ 𝑒
− 𝑒
8 2
8
D. none of these
235) Two linearly independent solutions of the equation 𝑦 ′′ + 𝑦 ′ − 6𝑦 = 0 are
236)
A.
𝑒 −3𝑥 and 𝑒 2𝑥
C.
𝑒 −𝑥 and 𝑒 6𝑥
0
e
− x2
B.
A
𝑒 −2𝑥 and 𝑒 3𝑥
D. none of these
A
dx =
−
A.
B. 

2
C. 
D. none of these
237) A particular solution for the differential equation 𝑦 ′′ + 2𝑦 ′ + 𝑦 = 3 − 2 sin 𝑥is
A.
𝐴 + 𝐵𝑥 2 + 𝐶 cos 𝑥 + 𝐷 sin 𝑥
B.
C.
𝐴 + 𝐵𝑥 cos 𝑥 + 𝐶𝑥 sin 𝑥
D. none of these
B
𝐴 + 𝐵 cos 𝑥 + 𝐶 sin 𝑥
238) The solution of the initial value problem 𝑥 2 𝑦 ′′ − 𝑥𝑦 ′ − 3𝑦 = 0, 𝑦(1) = 1, 𝑦 ′ (1) = −2 is y=
A.
5
4
1
𝑥 −1 − 4 𝑥 3
Discipline: ________________________
B.
1
4
3
𝑥 + 4 𝑥 −3
Page 30 of 110
A
5
C.
1
D. none of these
𝑒 −𝑥 − 4 𝑒 3𝑥
4
239) The differential equation 𝑥 ′′ + 2𝑥 ′ − 5𝑥 = sin 𝑡 is equivalent to the system
A.
𝑥 ′ = 𝑦 , 𝑦 ′ = 5𝑥 − 2𝑦 + sin 𝑡
B.
C.
𝑥 ′ = 𝑦 , 𝑦 ′ = 2𝑥 − 5𝑦 + sin 𝑡
D. none of these
A
𝑥 ′ = 2𝑥 − 5𝑦 , 𝑦 ′ = sin 𝑡
240) The system of differential equation 𝑥 ′′ = − 2 𝑦 2 , 𝑦 ′′ = − 2 𝑦 2 is equivalent to a 1st order system
𝑥 +𝑦
𝑥 +𝑦
C
consisting of
A.
Two equations
B.
Three equations
C.
Four equations
D. none of these
𝑛
241) The series solution for the differential equation 𝑦 ′′ + 𝑥𝑦 ′ + 𝑦 = 0 is of the form ∑∞
𝑛=0 𝐶𝑛 𝑋 has
recursion relation
A.
𝐶𝑛+2 + 𝐶𝑛+1 + 𝐶𝑛 = 0, 𝑛 ≥ 0
C.
𝑛𝑐𝑛+1 − 𝑐𝑛 = 0, 𝑛 ≥ 1
B.
(𝑐 + 2)𝐶𝑛+2 + 𝐶𝑛 = 0, 𝑛 ≥ 2
D. none of these
242) The differential equation 𝑥𝑦 ′′ + (𝑥 − 2)𝑦 ′ + 𝑦 = 0 has a solution of the form
∞
A.
2
𝑛
B
B.
𝑦 = 𝑥 ∑ 𝑐𝑛 𝑥 , 𝑐0 ≠ 0
𝑦 = 𝑥1/2 ∑
∞
C
𝑐𝑛 𝑥 𝑛 , 𝑐0 ≠ 0
𝑛=0
𝑛=0
C.
3
𝑦=𝑥 ∑
∞
𝑐𝑛 𝑥 𝑛 , 𝑐0 ≠ 0
D. none of these
𝑛=0
243) The solution of the differential equation(2𝑥 − 1)𝑦 ′ + 2𝑦 = 0, can be represented as a power series
𝑛
∑∞
𝑛 = 0 𝐶𝑛 𝑋 with radius of convergence equal to
A.
C.
0
1
Discipline: ________________________
B.
1/2
D. none of these
Page 31 of 110
B
𝑠+4
244) The partial fraction decomposition of
is
(𝑠−1)2 (𝑠2 +4)
𝐴
A.
𝐵 +𝐶
(𝑠−1)2
𝐴
C.
𝑠−1
𝐴
B.
+ 𝑠2𝑠 +4
𝐵
B
𝐶
+ (𝑠−1)2 + 𝑠2 +4
𝑠−1
𝐵
𝐶 +𝐷
+ (𝑠−1)2 + 𝑠𝑠2 +4
D. none of these
245) Which of the following equations can be rearranged into separable equations?
A.
(𝑥 + 𝑦)𝑦 ′ = 𝑥 − 𝑦
B.
C.
𝑦 ′ = 𝑙𝑛(𝑥𝑦)
D. none of these
B
𝑦 ′ − 𝑒 𝑦 = 𝑒 𝑥=𝑦
246) A body with mass 𝑚 = 1 𝑘𝑔 is attached to the end of a spring that is stretched 2𝑚 by a force of 100N.
2
A
It is set in motion with initial position 𝑥0 = 1 m and initial velocity 𝑣0 =−5m/s. The position function
of the body is given by
A.
1
𝑥(𝑡) = cos 10𝑡 − 2 𝑠𝑖𝑛10𝑡
C. 𝑥(𝑡) = 𝑒 −𝑡 (𝑐𝑜𝑠9𝑡 − 3𝑠𝑖𝑛9𝑡)
B. 𝑥(𝑡) = cos 5𝑡 − 𝑠𝑖𝑛5𝑡
D. none of these
247) The appropriate form of a particular solution 𝑦𝑝 of the equation 𝑦 ′′ + 2𝑦 ′ + 𝑦 = 𝑒 −𝑡 is
A. 𝐴𝑒 −𝑡
B. 𝐴𝑡𝑒 −𝑡
C. 𝐴𝑡 2 𝑒 −𝑡
D. none of these
248) Particular integral of (𝐷2 + 4)𝑦 = 𝑠𝑖𝑛3𝑥 is
B
A.
1
− 𝑠𝑖𝑛3𝑥
5
B.
C.
1
− 𝑐𝑜𝑠3𝑥
5
D. none of these
1
− 𝑡𝑎𝑛3𝑥
5
249) The DE 𝑑𝑦 = 𝑥 2 +𝑥𝑦+2𝑦 is
𝑑𝑥
2𝑥 2 +𝑦 2
Discipline: ________________________
C
B
Page 32 of 110
250)
A. Exact
B. Homogeneous
C. Cauchy
D. none of these

e
− x2
B
dx =
0
251)
A. 
B.
C. 
2
D. none of these
The order and degree of the DE (

𝑑𝑦 2 3
)
𝑑𝑥
2
𝑑2 𝑦
𝑑𝑥2
1+(
C
) =b are respectively
A. 1 and 3
B. 1 and 2
C. 2 and 3
D. none of these
252) A general solution of 3rd order DE contains
C
A. One constant
B. Two constants
C. Three constants
D. none of these
253) Solving 𝑦 ′ + 𝑦 ′ + 2𝑦 = 0 with 𝑦(0), 𝑦(1) = 1 is
A. Initial value problem
B
B. Boundary value problem
C. Eigen value problem
D. none of these
254) If p and q are the order and degree of DE 𝑦 𝑑𝑦 + 𝑥 2 (𝑑2 𝑦)3 + 𝑥𝑦 = 𝑐𝑜𝑠𝑥 , then
𝑑𝑥
𝑑𝑥 2
A. 𝑝 < 𝑞
B. 𝑝 = 𝑞
A
C. 𝑝 > 𝑞
D. none of these
255) A solution to the partial differential equation 𝜕2 𝑢 = 9 𝜕2 𝑢 is
𝜕𝑥 2
𝜕𝑦 2
A. cos (3𝑥 − 𝑦)
B. 𝑥 2 + 𝑦 2
C. sin (3𝑥 − 𝑦)
D. none of these
256) The partial differential equation 5 𝜕2𝑧 + 6
𝜕𝑥 2
𝜕2𝑧
𝜕𝑦 2
A
= 𝑥𝑦 is classified as
A. Elliptic
B. Parabola
C. Hyperbola
D. none of these
Discipline: ________________________
D
Page 33 of 110
257) The partial differential equation 𝑥𝑦 𝜕𝑧 = 5
𝜕𝑥
𝜕2𝑧
𝜕𝑦 2
B
is
A. Elliptic
B. Parabolic
C. Hyperbolic
D. none of these
258) The following is true for the following partial differential equation under non linear mechanics known
as the Kortewege-de-vfies equation
𝜕𝑤
𝜕𝑡
+
𝜕3𝑤
𝜕𝑥 3
− 6𝑤 𝜕𝑥 = 0
A. Linear, 3rd order
B. Non-linear 3rd order
C. Linear first order
D. none of these
259) Solve 𝜕𝑢 = 6 𝜕𝑢 + 𝑢 using separation method of variable if 𝑢(𝑥, 0) = 10𝑒 −𝑥 , 𝑢 =
𝜕𝑥
𝜕𝑡
A.
𝑡
10𝑒 −𝑥 𝑒 −3
C. 10𝑒 𝑥3 𝑒 −𝑡
B.
A
𝑡
10𝑒 𝑥 𝑒 −3
D. none of these
260) While solving the partial differential equation by separable method we equate the ratio to constant
which?
A. Can be positive or negative integer or zero
B. Can be positive or negative rational number or
zero
C. Must be positive integer
D. none of these
261) When solving a 1-dimensional heat equation using a variable separable method we get the solution
A. k is positive
B. k is 0
C. k is negative
D. none of these
262) 𝑓(𝑥, 𝑦) = sin(𝑥𝑦) + 𝑥 2 ln(𝑦). Then 𝑓𝑥𝑦 at (0,𝜋) is
2
𝑑𝑥 2
+ 𝛼𝑦 = 0
Discipline: ________________________
C
A
A. 164
B. -164
C. 0
D. none of these
264) DE for 𝑦 = 𝐴𝑐𝑜𝑠𝛼𝑥 + 𝐵𝑠𝑖𝑛𝛼𝑥, where A and B are arbitrary constants is
𝑑2 𝑦
B
D
A. 33
B. 0
C. 3
D. none of these
263) 𝑓(𝑥, 𝑦) = 𝑥 2 + 𝑦 3 ; 𝑥 = 𝑡 2 + 𝑡 3 ; 𝑦 = 𝑡 3 + 𝑡 9 . Then 𝑑𝑓 at t=1 is
𝑑𝑡
A.
B
𝜕𝑤
B
B. 𝑑 2 𝑦
− 𝛼𝑦 = 0
𝑑𝑥 2
Page 34 of 110
C. 𝑑2 𝑦
− 𝛼2𝑦 = 0
𝑑𝑥 2
D. none of these
265) The order of DE is defined as
B
A. The highest degree of the variable
B. The order of the highest derivative
C. The power of variable in the solution
D. none of these
266) A primitive of an ODE is
C
A. Its general solution
B. Its particular solution
C. Its complementary solution
D. none of these
267) The solution of a DE subject to a condition satisfied at one particular point is called
C
A. A boundary value problem
B. A two-point boundary value problem
C. An initial value problem
D. none of these
268) A general solution of an nth order DE then
A
A. n can be zero
B. n is any non-negative integer
C.
D. none of these
n is any integer
269) The DE 𝑑𝑦 = 𝑎𝑥+𝑏𝑦+𝑐 is
𝑑𝑥
𝑎′ 𝑥+𝑏 ′ 𝑦+𝑐′
B
A. Homogeneous
B. Non-Homogeneous
C. Separable
D. none of these
270) The order of D.E where general solution is 𝑦 = 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 2𝑥 + 𝐶3 𝑒 3𝑥 + 𝐶4 𝑒 4𝑥 + 𝐶5 , where
𝐶1 , 𝐶2 , 𝐶3 , 𝐶4 , 𝐶5 , are arbitrary constant is
A. 5
B. 4
C. 7
D. none of these
2
2
271) The particular integral of DE (𝐷 − 𝑎 )𝑦 − 𝑐𝑜𝑠𝑎𝑥 is 𝑦 =
A.
𝑥
− 2𝑎 𝑐𝑜𝑠𝑎𝑥
C. − 𝑥 𝑠𝑖𝑛𝑎𝑥
2𝑎
B.
𝑥
2𝑎
C
𝑠𝑖𝑛𝑎𝑥
D. none of these
272) The equation 𝜕2𝑢 + 𝜕2𝑢 = 𝜕𝑢 where x, y, z are variable is a partial DE of order and degree
𝜕𝑥 2
𝜕𝑦 2
𝜕𝑧
A. 2, 1
Discipline: ________________________
A
B. 2, 2
Page 35 of 110
A
C. 1, 2
D. none of these
273) A PDE 𝐴
𝜕2𝑢
𝜕𝑥 2
+𝐵
𝜕2𝑢
𝜕𝑦 2
+𝐶
𝜕2𝑢
𝜕𝑥𝜕𝑦
= 𝑠𝑖𝑛𝑥 + 2𝑦
A. Linear
C. Non-Homogeneous
274) The linear partial DE
A
where A, B, C are real constants
B. Homogeneous
D. none of these
𝜕2𝑢
𝜕𝑥 2
𝜕2𝑢
B
+ 𝜕𝑦 2 = 0 is
A. 2- dimensional Poisson equation
B. 2-dimensional Laplace equation
C. 3-dimensional Laplace equation
D. none of these
275) The linear PDE 𝐴𝑈𝑥𝑥 + 2𝐵𝑈𝑥𝑦 + 𝐶𝑈𝑦𝑦 = 𝐹(𝑥, 𝑦, 𝑈, 𝑈𝑥 , 𝑈𝑦 ) is elliptic if
B
A. 𝐴𝐵 − 𝐶 2 > 0
C. 𝐴𝐶 − 𝐵 2 > 0
276) The wave equation 𝑈𝑡𝑡 = 𝑐 2 𝑈𝑥𝑥 is
C
277)
B. 𝐴𝐶 − 𝐵 2 > 0
D. none of these
A. Elliptic
B. Parabolic
C. Hyperbolic
D. none of these
𝜕2𝑈
𝜕𝑡 2
𝜕2𝑈
B
= 𝑐 2 𝜕𝑥 2 is
A. Heat equation
B. Wave equation
C. Equation of vibrating string
D. none of these
278) The D.E (1 − 𝑥 2 )𝑦 ′ − 2𝑥𝑦 ′ + 𝑛(𝑛 + 1)𝑦 = 0 is
A. Barrel equation
B
B. Legendre equation
C. Poisson equation
D. none of these
279) The partial differential 𝜕2𝑧 + 𝑧 𝜕2𝑧 + 𝜕2𝑧 = 0 is
𝜕𝑥 2
𝜕𝑥𝜕𝑦
𝜕𝑦 2
A. Hyperbolic
B. Parabolic
C. Elliptic
D. none of these
B
280) The PDE (𝑥 2 − 𝑦𝑧)𝑝 + (𝑦 2 − 𝑧𝑥)𝑞 = 𝑧 2 − 𝑥𝑦 where 𝑝 = 𝜕𝑧 , 𝑞 = 𝜕𝑧 is
𝜕𝑥
𝜕𝑦
A. Linear
Discipline: ________________________
A
B. Non-linear
Page 36 of 110
C. Algebraic
D. none of these
281) Let
b X be an arbitrary topological space and Y a Harsdorf space. Let f: X→ 𝑌be a Continues function.
Then the graph 𝐺 = {(𝑥, 𝑦): 𝑦 = 𝑓(𝑥) 𝑖𝑠 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝑋 × 𝑌} is ………. In 𝑋 × 𝑌
A. Open
B. closed
C. Continuous
D. None of these
282) A First countable space X is a ………... if and only if every convergent sequence has a unique limit
A. 𝑇0 𝑠𝑝𝑎𝑐𝑒
C. 𝑇2 𝑠𝑝𝑎𝑐𝑒
D
B. 𝑇1 𝑠𝑝𝑎𝑐𝑒
D. Harsdorf space
283) Let the product X of topological spaces be normal then each X is…………….
A. 𝑁𝑜𝑟𝑚𝑎𝑙
B. 𝑅𝑒𝑔𝑢𝑙𝑎𝑟
C. 𝐶𝑜𝑚𝑝𝑙𝑒𝑡𝑙𝑦 𝑅𝑒𝑔𝑢𝑙𝑎𝑟
D. 𝑇0 𝑠𝑝𝑎𝑐𝑒
284)If R is a ring and J be its Maximal Ideal, then what about R/J?
A R/J is vector
B R/J is field
C R/J is scaler
D R/J is ring
285) Gaussian integers are form a
A Vector
B Scalar
C Field
D None of these
286)Let I and J be the ideals of a commutative ring R Ideal of R is 𝐼 + 𝐽 = {𝑎 + 𝑏 | 𝑎 ∈ 𝐼 𝑎𝑛𝑑 𝑏 ∈ 𝐽}
then I+J is
A Principal Ideal
B Ideal of R
C Maximal Ideal
C None of these
287)Let A and B be the subspace of a vector space V(F). Then Dim(A+B) is
A. Dim A+ Dim B
B. Dim A+ Dim B-Dim (A∪ 𝐵)
C. Dim A + Dim B-Dim (A∩ 𝐵)
D None of these
2
2
288)𝑇: 𝑅 → 𝑅 𝑇(𝑥, 𝑦) = (𝑥 + 2, 𝑦 + 1) 𝑡ℎ𝑒𝑛 𝑇 𝑖𝑠
A T is not Linear Transformation
B. T is Linear Transformation
C. T is Kernel of Linear Transformation
D. None of these
289)Let I and J be the ideals of a commutative ring R Ideal of R is I . 𝐽 = {𝑎𝑏 | 𝑎 ∈ 𝐼 𝑎𝑛𝑑 𝑏 ∈ 𝐽} then IJ
is
A Ideal
B. Not Ideal
C. Principal Ideal
D. None of these
290)A ring Homomorphism is both (1-1) and onto, then it is called
A. Endomorphism
B. Epimorphism
C. 𝐼𝑠𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑠𝑚
D. None of these
291)Every ring R has surely
A. 𝑇𝑤𝑜 𝑅𝑖𝑛𝑔
B. 𝑂𝑛𝑒 𝑟𝑖𝑛𝑔
C. 𝑇ℎ𝑟𝑒𝑒 𝑅𝑖𝑛𝑔
D. None of these
292)If there will be trivial solution of homogenous equations, then there exist
A. Linearly Independent
B. Linearly Dependent
C. Basis
D. None of these
293)The vectors (3,0, -3), (-1,1,2),(4,2,-2) and (2,1,1) are
A. Linearly Independent
B. Bases
C. Linearly Dependent
D None of these
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B
Page 37 of 110
A
B
C
B
C
A
A
D
A
B
C
294)Let {𝑣1 , 𝑣2 , … 𝑣𝑛 } be the bases of a vector space V(F) and let {𝑤1 , 𝑤, … 𝑤𝑟 } be linearly independent
subset of V(F) then
A. (𝑖) 𝑛 < 𝑟 (𝑖𝑖) 𝑛 ≥ 𝑟 (𝑖𝑖𝑖)𝑛 ≤ 𝑟
B. (𝑖) 𝑛 > 𝑟 (𝑖𝑖) 𝑛 ≥ 𝑟 (𝑖𝑖𝑖)𝑛 ≤ 𝑟
C. (𝑖) 𝑛 = 𝑟 (𝑖𝑖) 𝑛 ≥ 𝑟 (𝑖𝑖𝑖)𝑛 ≤ 𝑟
D None of these
295)Let 𝑇: 𝑉1 (𝐹) → 𝑉2 (𝐹) be a linear transformation. Then R(T) is …………Where R(T) = Rank T
A. subspace of 𝑉1 (𝐹)
B. subspace of V2 (F)
C. Subspace of 𝑉1 (𝐹) ∩ V2 (F)
D. None of these
296)Converse of the Lagrange ‘s Theorem holds true in P groups due to
A. Lagrange ‘s Theorem
B. 3rd Sy low Theorem
C. Sy low Theorem
D. None of these
297)If H is the only One P-sylow subgroup of a finite group, then ………. H is in G
A. Cyclic
B. Abelian
C. Normal
D. None of these
298)Every Characteristic subgroup is a
A. Double Cosets
B. Normal subgroup
C. Invariant Subgroup
D. None of these
299)In group theory Z(G) is fully
A. Covariant
B. Cosets
C. Invariant
D. None of these
300)If a group is …………... there is no inner automorphism
A. Abelian
B. Cyclic
C. Non-Abelian
D. None of these
301)Every outer is inner automorphism ∅𝑎 : 𝑔 → 𝑎𝑔𝑎−1 = 𝑔 𝑖𝑛 … … … … ….
A. Abelian
B. Cyclic
C. Non-Abelian
D. None of these
𝐺
302)
𝐺
Let H and K be normal subgroup of G where H is subset of K then 𝐾 = 𝐻𝐾
A
B
A
C
B
C
A
A
C
𝐻
A. 2nd isomorphism Theorem
B. 3rd isomorphism Theorem
C. 1st isomorphism Theorem
D. None of these
303)Under the condition 𝐻𝐾 = 𝐻 holds in group G for its subgroup H & K
𝐾
𝐻∩𝐾
A. (i)K is normal in HK (ii) 𝐻 ∩ 𝐾 is normal in H B. (i)K is normal in HK (ii) 𝐻𝑈𝐾 is normal in H
C. (i)K is normal in K (ii) 𝐻 ∩ 𝐾 is normal in H D. None of these
304)The identity of the Quotient group of G by its normal subgroup H is always
A. G
B. K
C. H
D. None of these
305)Every automorphism is endomorphism its converse is
A. Not true
B. True
C. False
D. None of these
306)A mapping 𝜇: 𝐺 → 𝐺 , 𝜇(𝑔) = 𝑔𝑘 , ∀𝑔 ∈ 𝐺 then 𝜇 𝑖𝑠 𝑎𝑛
𝐾
A. Endomorphism of G to G/K
B. Epimorphism of G to G/K
C. Homomorphism of G to G/K
D. None of these
307)If H is a subgroup of index ……… then H is subgroup of G
A. 2
B. 3
C. 4
D. None of these
308)If H and K are two normal subgroups, then 𝐻 ∩ 𝐾 is
Discipline: ________________________
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A
C
A
B
A
C
A. Abelian
B. subgroup
C. Normal
D. None of these
309)A finite p group has a ………center
A. Trivial
B. Non-trivial
C. Normal
D. None of these
310) If G is ……...then all the elements of finite order in a group G form its subgroup.
A. Non-abelian
B. Normal
C. Abelian
D. None of these
311)Every subgroup of an abelian group is ……
A. Non-abelian
B. Normal
C. Abelian
D. None of these
312)The center of simple group is ……
A. Either identity or group itself
B. group itself
C. Identity
D. None of these
313)Let G be group of order 2p, where p is prime then g has a normal subgroup of order
A. p
B. 2p
C. p/2
D. None of these
314)If HK= KH, then HK is the ……...subgroup containing both H & K.
A. Normal
B. Largest
C. Smallest
D. None of these
315)Let H & K be subgroups of group G then 𝐻 ∩ 𝐾 is the …… subgroup of g contained in both H &K.
A. Normal
B. Largest
C. Smallest
D. None of these
316)The …… subgroup containing H & K is the subgroup generated by H∪K
A. Normal
B. Largest
C. Smallest
D. None of these
317)Every ……space is Lindelöf
A. 1st Countable
B. Countable
C. Second Countable
D. None of these
318)A discrete space X is if and only if it is countable.
A. Symmetric
B. Separable
C. Countable
D. None of these
319)All open intervals are
A. Separable
B. Isomorphism
C. Homeomorphism
D. None of these
320)The set of Z integers as a subspace of R has ……Topology
A. Indiscrete
B. Coffinite
C. Countable
D. None of these
321)Let X,Y,Z be topological spaces 𝑓: 𝑋 → 𝑌 , 𝑔: 𝑌 → 𝑍 be continues function & then …. Is Continues
A. 𝑔𝑜𝑓: 𝑋 → 𝑍
B. 𝑔𝑜𝑓: 𝑋 → 𝑌
C. 𝑔𝑜𝑓: 𝑌 → 𝑋
D. None of these
322)A function f is continuing on a topological space X if it is continues at ……. Point X
A. One
B. Every
C. Two
D. None of these
st
323)Any uncountable set X with ……. topology is not 1 Countable and so not 2nd Countable.
A. Usual
B. Cofinite
C. Discrete
D. None of these
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A
C
B
A
A
C
B
C
C
B
C
D
A
B
B
324)A space X is said to completely Normal if and only if every subspace of x is Normal is called
A. Completely Normal
B. Normal
C. completely Regular
D. None of these
325)Let X be normal space then every point of disjoint closed sets has neighborhoods whose closures are
also
A. Joint
B. Closures
C. Disjoint
D. None of these
326)A normal T1 space is
A. Normal
B. Not regular
C. Closed
D. None of these
327)Every Closed subspace of a T4 space is a
A. Normal
B. Not regular
C. Closed
D. None of these
328)Every subspace of completely Regular space is
A. Regular
B. Not Regular
C. Completely Regular
D. None of these
329)T3 space is a
A. Regular T1 space.
B. Regular T2 space.
C. Regular T4 space.
D. None of these
330)T2 space is called
A. Regular Space
B. Normal Space
C. Harsdorf Space
D. None of these
331) Let
b V be a vector space over field F. A subset W of V is called ……. Of V if under the operation of V,
W itself forms a vector space over F.
A. Vector
B. Scalar
C. subspace
D. None of these
332) Let W1 and W2 be the two subspaces of vector space V(F). Then V(F) is said to be ……
W2
A. Sum
C. subspace
of W1 and
C
D
D
C
A
C
C
B
B. Direct Sum
D. None of these
333)The set of all vectors in vector space V(F) which can be written as Linear Combination of
{v1,v2,…vn}is denoted by
L {v1,v2,…vn} is called
A. Linear Span
B. Direct Sum
C. Subspace
D. None of these
334)Pn(F)={1,x,x2,…xn} is
A. Linearly dependent
B. Linearly Independent
C. Basis
D. None of these
335)The set of polynomials is
A. Linearly dependent
B. Basis
C. Linearly Independent
D. None of these
336)If a vector is multiple of other vector in a set. Then it is always
A. Linearly dependent
B. Linearly Independent
C. Basis
D. None of these
Discipline: ________________________
A
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A
B
C
A
337)Let {v1,v2,…vn} generate the vector space V(F) then for any vector v∈ 𝑉(𝐹) ,the set {v1,v2,…vn} is
B
……. and generate V(F).
A. Linearly independent
B. Linearly dependent
C. Basis
D. None of these
338)The number of elements in the basis of a vector space V(F) is called ………. Vector space of V(F)
C
A. Linearly independent
B. Basis
C. Dimension
D. None of these
339)V(R ) = C the set of all complex numbers , {1,i} is …… generate V( R) Dim V( R) = 2
C
A. Dimension
B. Linearly dependent
C. Linearly independent
D. None of these
340)If C(C ) is a vector space over the field of Complex Number ,then the Basis and dimension is
C
A. Dim(C) =1, Basis =2
B. Dim(C) =2, Basis =1
C. Dim(C) =1, Basis =1
D. None of these
341)A linearly independent set is always a part of …… of V(F)
A
A. Basis
B. Subspace
C. Vector Space
D. None of these
342)If 𝑉 = 𝑀𝑛×𝑛 (𝐹), 𝐿𝑒𝑡 𝑇: 𝑀𝑛×𝑛 → 𝑀𝑛×𝑛 𝑏𝑒 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝑇(𝐴) = 𝐴𝑡 . 𝑇 𝑖𝑠 𝑎
A
A. Linear Transformation
B. Range of Linear of Transformation
C. Not a Linear Transformation
D. None of these
343)
𝑇: 𝑅 2 → 𝑅 2 , 𝑇(𝑥, 𝑦) = (𝑥 + 2 , 𝑦 + 1) 𝑡ℎ𝑒𝑛 𝑇 𝑖𝑠
C
A. Linear Transformation
B. Range of Linear of Transformation
C. Not a Linear Transformation
D. None of these
344)Linear Transformation is called if it is …... then it is called Monomorphism
A
A. One -One
B. Onto
C. One-one and onto
D. None of these
345)Linear Transformation is called if it is …... then it is called Isomorphism
C
A. One -One
B. Onto
C. One-one and onto
D. None of these
346) 𝐿𝑒𝑡 𝑇: 𝑉1 → 𝑉2 𝑏𝑒 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑡ℎ𝑒𝑛 𝑖𝑚𝑎𝑔𝑒 𝑜𝑓 𝑇 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑎𝑠 𝑅(𝑇) = {𝑇(𝑉1 | ) 𝑣1
B
∈ 𝑉1 } 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑
A. Linear Transformation
B. Range of Linear of Transformation
C. Not a Linear Transformation
D. None of these
347)𝐿𝑒𝑡 𝑉(𝐹) 𝑏𝑒 𝑎 𝑣𝑒𝑐𝑡𝑜𝑟 𝑖𝑓 dim 𝑛 . 𝑡ℎ𝑒𝑛
C
𝑛−1
A. 𝑉𝑛 (𝐹) = 𝐹
B. 𝑉𝑛 (𝐹) = 𝐹
C. 𝑉𝑛 (𝐹) = 𝐹 𝑛
D. None of these
348) 𝐿𝑒𝑡 𝑇: 𝑉1
B
→ 𝑉2 𝑏𝑒 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑣𝑒𝑐𝑡𝑜𝑟 𝑠𝑝𝑎𝑐𝑒 𝑉1 (𝐹) 𝑎𝑛𝑑 𝑉2 (𝐹), 𝑤ℎ𝑒𝑟𝑒 𝑉1 (𝐹) 𝑎𝑛𝑑 𝑉2 (𝐹)
ℎ𝑎𝑣𝑒 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 𝑡ℎ𝑒𝑛 𝑇 𝑖𝑠 (1 − 1)𝑖𝑓𝑓 𝑇 𝑖𝑠
A. One -One
B. Onto
C. One-one and onto
D. None of these
349)𝐿𝑒𝑡 𝑇: 𝑉1 → 𝑉2 𝑏𝑒 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑇ℎ𝑒𝑛 ker 𝑇 𝑖𝑠 𝑠𝑢𝑏𝑠𝑝𝑎𝑐𝑒 𝑜𝑓 ….
B
A. 𝑉2 (𝐹)
B. 𝑉1 (𝐹)
C. R(T)
D. None of these
350)𝐿𝑒𝑡 𝑇: 𝑉1 → 𝑉2 𝑏𝑒 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑇ℎ𝑒𝑛 R ( 𝑇 ) 𝑖𝑠 𝑠𝑢𝑏𝑠𝑝𝑎𝑐𝑒 𝑜𝑓 ….. Where R(T) = Ker
A
of T
A. 𝑉2 (𝐹)
B. 𝑉1 (𝐹)
C. R(T)
D. None of these
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351)𝐿𝑒𝑡 𝑇: 𝑉1 → 𝑉2 𝑏𝑒 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑇ℎ𝑒𝑛 T … … . 𝑖𝑠 𝐼𝑓𝑓 𝑁(𝑇) = {01 } ….
A. One -One
B. Onto
C. One-one and onto
D. None of these
352)𝐿𝑒𝑡 𝑇: 𝑉1 → 𝑉2 𝑏𝑒 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑡ℎ𝑒𝑛
A.
B.
𝐷𝑖𝑚 𝑉1 (𝐹) = 𝑁𝑢𝑙𝑙𝑖𝑡𝑦 (𝑇)
𝐷𝑖𝑚 𝑉1 (𝐹) = 𝑁𝑢𝑙𝑙𝑖𝑡𝑦 (𝑇) + 𝑅𝑎𝑛𝑔𝑒 (𝑇)
(𝐹)
(𝑇)
C. 𝐷𝑖𝑚 𝑉1
D. None of these
= 𝑁𝑢𝑙𝑙𝑖𝑡𝑦
+ 𝑅𝑎𝑛𝑘 (𝑇)
353)𝐿𝑒𝑡 𝑇 ∶ 𝑉 → 𝑉 𝑖𝑠 … … . . 𝐼𝑓𝑓 𝑇 −1 𝑒𝑥𝑖𝑠𝑡 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑇𝑇 −1 = 𝐼
A. Singular
B. Non-singular
C. Bijective
D. None of these
354)𝐿𝑒𝑡 𝑇: 𝑉 → 𝑉 𝑏𝑒 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑎𝑛𝑑 𝐷𝑖𝑚 𝑉 = 𝑛 𝑡ℎ𝑒𝑛 𝑇 𝑐𝑎𝑛 𝑛𝑜𝑡 𝑚𝑜𝑟𝑒 𝑡ℎ𝑒𝑛 𝑒𝑖𝑔𝑒𝑛 𝑣𝑎𝑙𝑢𝑒𝑠
A. n+1
B. n
C. n-1
D. None of these
355)An 𝑛 × 𝑛 matrix A is …. iff A has n real and distinct eigen values
A. Similar
B. Orthogonal
C. Diagonalizable
D. None of these
̅
̅̅̅
̅
̅
356) 𝐼𝑠 𝐺 = {1 , 2 , 3 , 4} 𝑎 𝑔𝑟𝑜𝑢𝑝 𝑜𝑓 𝑚𝑜𝑑8
A. Yes
B. No
C. Basis
D. None of these
357)𝐴𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑎 ∈ 𝐺 𝑖𝑠 𝑠𝑎𝑖𝑑 𝑎𝑛𝑑 𝑐𝑜𝑛𝑔𝑢𝑔𝑎𝑡𝑒 𝑡𝑜 𝑏 ∈ 𝐺 , 𝑖𝑓 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑔 ∈ 𝐺 𝑠. 𝑡. 𝑎 =
𝑔−1 𝑎𝑔 is called
A. Conjugate of an element in a group
B. Self-conjugate of an element in a group
C. Equivalence Relation
D. None of these
358)How many conjugate classes are there I symmetric group 𝑆3
A. 1
B. 2
C. 3
D. None of these
359)Every Group of prime order is a
A. Cyclic
B. Generator
C. Center of the group
D. None of these
360) (Q +) is
A. A cyclic group
B. Not cyclic group
C. Abelian group
D. None of these
361)A cyclic of length 2 is called
A. Permutation
B. Combination
C. Transposition
D. None of these
362)Each permutation can be expressed as product of
A. Not Cyclic Permutation
B. Cyclic Permutation
C. Transposition
D. None of these
363)𝐼𝑓 𝑔𝑟𝑜𝑢𝑝 𝑖𝑠 𝑎𝑏𝑒𝑙𝑖𝑎𝑛 , 𝑡ℎ𝑒𝑛 𝑤ℎ𝑎𝑡 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑁𝐺 (𝑋)?
A. 𝑁𝐺 (𝑋) = 𝐺
B. 𝑁𝐺 (𝑋) = 𝑋
C. 𝑁𝐺 (𝑋) = 𝑁
D. None of these
364)𝐿𝑒𝑡 ∅: (𝑍, +) → (𝑍, +)𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 ∅(𝑛) = 2𝑛 ∀ 𝑛 ∈ 𝑍 = 𝑠𝑒𝑡 𝑜𝑓 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑖𝑠 𝑒𝑥𝑎𝑚𝑝𝑙𝑒 𝑜𝑓
A. Isomorphism
B. Epimorphism
C. Monomorphism
D. None of these
365)𝐿𝑒𝑡 ∅: (𝑍, +) → (𝐺, . ) = {±1, ±𝑖}𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 ∅(𝑛) = 𝑖 𝑛 ∀ 𝑛 ∈ 𝑍 =
𝑠𝑒𝑡 𝑜𝑓 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑖𝑠 𝑒𝑥𝑎𝑚𝑝𝑙𝑒 𝑜𝑓
A. Isomorphism
B. Epimorphism
C. Monomorphism
D. None of these
Discipline: ________________________
Page 42 of 110
A
C
B
B
C
A
A
C
A
B
C
B
A
C
B
366)𝐿𝑒𝑡 ∅: (𝑍, +) → (𝐸, +) = 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 ∅(𝑛) = 2𝑛 ∀ 𝑛 ∈ 𝑍 = 𝑠𝑒𝑡 𝑜𝑓 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑎𝑛𝑑 𝐸 =
𝑆𝑒𝑡 𝑜𝑓 𝑒𝑣𝑒𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑖𝑠 𝑒𝑥𝑎𝑚𝑝𝑙𝑒 𝑜𝑓
A. Isomorphism
B. Epimorphism
C. Monomorphism
D. None of these
367)Every Characteristic subgroup is a
A. Subgroup
B. Invariant
C. Normal Subgroup
D. None of these
368)When are the subgroups of group its normal subgroup?
A. Left coset ≠ Right Coset
B. Left coset = Right Coset
C. Left coset × Right Coset
D. None of these
369)If p is prim divisor of a finite group G having order n then G has an element with order p is called
A. Sylow 2nd Theorem
B. Lagrange’s Theorem
C. Cauchy 2nd order Theorem
D. None of these
370)Intersection of two subrings be an
A. Quotient Set
B. Empty Set
C. Integer Set
D. None of these
371)Every ideal is a
A. Maximal
B. Ring
C. Subring
D. None of these
𝑟 0
372)
𝑎 𝑏
) , 𝑟1 , 𝑟2 ∈ 𝑅 𝑖𝑠
𝑅 = 𝑀2×2 𝑏𝑒 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔 𝑜𝑓 2 × 2 𝑚𝑎𝑡𝑖𝑟𝑐𝑒𝑠 , 𝑅 = (
) , 𝐼1 = ( 1
𝑟2 0
𝑐 𝑑
A. Right Ideal
B. Left Ideal
C. Sided ideal
D. None of these
𝑟 𝑟
𝑎 𝑏
373)𝑅 = 𝑀
) , 𝐼1 = ( 1 2 ) , 𝑟1 , 𝑟2 ∈ 𝑅 𝑖𝑠
2×2 𝑏𝑒 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔 𝑜𝑓 2 × 2 𝑚𝑎𝑡𝑖𝑟𝑐𝑒𝑠 , 𝑅 = (
0 0
𝑐 𝑑
A. Right Ideal
B. Left Ideal
C. Sided ideal
D. None of these
374)Irrational number is a
A. Ring
B. Not Ring
C. Ideal ring
D. None of these
375)A ring I which the non-zero element form a multiplication group is called
A. Ring
B. Ideal Ring
C. Division Ring
D. None of these
376)𝑇ℎ𝑒 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑢𝑙𝑙 𝑠𝑝𝑎𝑐𝑒 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑
A. Ring
B. Ideal Ring
C. Division Ring
D. None of these
377)𝑆𝑢𝑏𝑠𝑝𝑎𝑐𝑒 𝑜𝑓 𝑎 𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑠𝑝𝑎𝑐𝑒 𝑖𝑠
A. Topological Space
B. Indiscrete
C. Discrete
D. None of these
378)In particular,
I
the open intervals on the real lines are base for the ……...topology.
A. Discrete
B. Indiscrete
C. Usual
D. None of these
𝑛
379)𝐸𝑣𝑒𝑟𝑦 𝐶𝑜𝑚𝑝𝑎𝑐𝑡 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 𝑅 𝑖𝑠 … ….
A. Closed and bounded
B. Closed and Continuous
C. Open and interior
D. None of these
380)𝑇ℎ𝑒 𝑐𝑜𝑛𝑡𝑖𝑛𝑜𝑢𝑠𝑒 𝑖𝑚𝑎𝑔𝑒 𝑜𝑓 𝑎 𝑐𝑜𝑛𝑒𝑐𝑡𝑒𝑑 𝑠𝑝𝑎𝑐𝑒 𝑖𝑠
A. Discrete
B. Disconnected
Discipline: ________________________
Page 43 of 110
A
C
B
C
B
C
B
A
B
C
D
C
C
A
C
C. Connected
D. None of these
d2x
+ 2 x3 = 0 ?
dt 2
Which of the following is the degree of the differential equation
381)
b
A. 0
C. 2
C
B.
D.
1
3
3
The order and degree of differential equation
382)
A.
C.
3 and 2
3 and 3
The differential equation 2
383)
A.
C.
d3x
 dy 
+ 4   + y 2 = 0 are respectively
dt 3
 dx 
A
B.
D.
2 and 3
3 and 1
B.
D.
Non-Linear
Undetermined to be linear or non-linear
dy
+ x 2 y = 2 x + 3 is
dx
Linear
Linear with fixed constants
dy
= 5 y; y ( 0 ) = 2
dt
B.
y = 2e−10
D.
y = 2e5t
A
Which of the following is the solution of the differential equation
384)
A.
y = 5e−5t
C.
y = 3e−5t
D
3
d2y
 dy 
+ 4   + y 4 = e − t are respectively
2
dt
 dx 
385)
A. 2 and 1
B.
1 and 2
C. 4 and 3
D.
2 and 3
A differential equation is said to be ordinary differential equation if it has
386) A. one dependent variable
B.
More than one dependent variables
C. one independent variable
D.
More than one dependent variables
Which of the following is the trivial solution of a differential equation.
B. y  0
387) A. y  0
C. y  0
D. y  0
The order and degree of differential equation 3
The differential equation x 2
388)
A.
C.
( 0,  )
( −, 0 )  (1,  )
dy
− 2 xy = sin x is defined for
dx
The differential equation y  +
389)
A.
C.
( 0,  )
( −,  )
391)
D.
B.
( 0,1)  (1,  )
D
D.
All real line except 0,2 and -2
1
1
is defined for
y= 2
x
x −4
A. y = Ax + B
B.
B
x
D.
y=
C
A
+ Bx
x
The solution of a differential equation which is not obtained from the general solution is known as
Discipline: ________________________
A
D
d2y
dy
+x −y=0
2
dx
dx
y = Ax 2 + Bx
C. y = Ax +
C
( 0,1)  (1,  )
( −,  )
B.
Which of the following is the solution of the differential equation x 2
390)
A
Page 44 of 110
B
A. Particular solution
C. Complete solution
The differential equation
392)
393)
394)
395)
396)
397)
398)
399)
400)
401)
B.
D.
dy
= y 2 is:
dx
A. Linear
C. Quasi Linear
The equation a0 x 2 y  + a1 xy  + a2 y =  ( x ) is called:
A. y = sin ( x ) + c
C.
dy
= 1 + y2 ?
dx
B.
y = tan ( x + c )
Which of the following is the solution of
403)
B
B. Non-Linear
D. None of these
A. Legendre’s linear equation
B. Cauchy’s linear equation
C. Simultaneous equation
D.
None of these
Which of the following is wronskian of cos and sin  ?
A. 1
B. sin 
C. cos
D. None of these
Which of the following is wronskian of x, x 2 and x 3 ?
A. 2x
B. 2x3
C. 2
D. None of these
2
3
−2
Which of the following is wronskian of x , x and x ?
A. 20x
B. 20x2
C. 20
D. None of these
−x
x
x
Which of the following is wronskian of e , 2e and e ?
A. 1
B. e x
C. 0
D. None of these
sin 2x and cos 2x are
A. Linearly Independent
B. Linearly dependent
C. Singular
D. None of these
Which of the following is the differential equation of all the circles passing through origin having centers on x-axis.
A.
B.
dy
dy
2 xy + x 2 − y 2 = 0
2 xy + x 2 = 0
dx
dx
C. dy
D.
None
of these
+ x2 − y 2 = 0
dx
dy
If a differential equation
= f ( x, y ) can be written in the form of g ( y ) dy = f ( x ) dx then it is called
dx
A. Exact
B. Linear
C. Non-exact
D. Separable
dy
Which of the following is the solution of
= 8 x3 y 2 ?
dx
A.
B. y = 2 x 4 + c
−1
y= 4
2x + c
C. y = 2 x + c
D. None of these
Which of the following is the solution of
402)
Singular solution
Auxiliary solution
D.
B
C
C
A
A
D
A
None of these
B.
y = ce2 x
D.
None of these
B
dy
= x 1 − y2 ?
dx
Discipline: ________________________
A
C
dy
= 2y ?
dx
A. y = ce x
C. y = e x + c
404) Which of the following is the solution of
y = sin −1 ( x ) + c
B
B
Page 45 of 110
 x2

A. y = cos  + c 
2


B.
y = cos ( x + c )
C.
D. None of these
A differential equation that can be put in the form of
405)
A. Separable
C. Non Exact
A differential equation
406)
408)
dy
 y
= f   is known as
dx
x
B. Exact
D. Homogenous
D
dy y y 3
= − is known as
dx x x 3
A. Separable
C. Non Exact
B.
D.
Exact
Homogenous
A. Separable
B.
C. Non Exact
D.
A differential equation M ( x, y ) dx + N ( x, y ) dy = 0 is exact if
Exact
Homogenous
A differential equation
407)
 x2

y = sin  + c 
2


D
dy y
= + e y / x is known as
dx x
A.
M N
=
y
x
B.
C.
M N
=
x
y
D. None of these
D
M N

y
x
A
u
u
dx + dy is known as
x
y
409)
A. Exact differential
B. Laplace differential
C. Homogenous differential
D. None of these
The differential equation − ydx + xdy = 0 is
410) A. Separable
B. Exact
C. Non-Exact
D. A and C both
Which of the following integrating factor can be multiplied by − ydx + xdy = 0 to make it exact?
2
A. x
B. e x
411)
C. 1
D. None of these
du =
A
D
C
x2
If a differential equation M ( x, y ) dx + N ( x, y ) dy = 0 is not exact and
412)
A. e f ( x ) dx
C.  f ( x )dx
M y − Nx
N
= f ( x ) then integrating factor is
B.
ex
D.
None of these
A
The solution of an exact differential equation ( 3x y + 2 xy ) dx + ( 2 x3 y + x2 ) dy = 0 is
2
413)
A. x3 y 2 + x 2 y = C
C. xy 2 + x 2 y = C
2
A
D.
x3 y 2 + y = C
None of these
B.
D.
Exact
None of these
C
B.
D.
Non-linear
None of these
B.
The differential equation ( x4 + y 4 ) dx − xy3 dy = 0 is
414)
A. Linear
C. Homogenous
A differential equation
415)
dy
+ 2 y = 6e x is
dx
A. Linear
C. Homogenous
Discipline: ________________________
A
Page 46 of 110
A differential equation
416)
A. Linear
C. Homogenous
A differential equation
417)
419)
dy
+ y = 5 x is
dx
A. Linear
C. Homogenous
y = ce − x
2
A differential equation
421)
A differential equation x
423)
Non-linear
None of these
B
B.
y = ( x 2 + c ) e− x
D.
y = ( 2 x + c ) e− x
B.
D.
Non-linear
None of these
B.
D.
Bernouli
None of these
B.
D.
Bernouli
Riccati
B.
y = C (1 + x2 )
D
2
2
A
B
dy 1 2 1
2
= y + y − is
dx x
x
x
A. Linear
C. Homogenous
Which of the following is the solution of
424)
B.
D.
A
dy
+ y = xy3 is
dx
A. Linear
C. Homogenous
A differential equation
Non-linear
None of these
dx 2
+ x = 10 y 2 is
dy y
A. Linear
C. Homogenous
422)
B.
D.
A
2
2
dy
+ 2 xy = 2e− x with integrating factor e x ?
dx
x
A. y = ( 2 x + c ) e
C.
Non-linear
None of these
dy
+ 3xy 2 = sin x is
dx
What is the solution of a linear differential equation
420)
B.
D.
dy
+ 3xy = sin x is
dx
A. Linear
C. Homogenous
A differential equation
A
B. Non-linear
D. None of these
A. Linear
C. Homogenous
A differential equation
418)
dy 2
+ y = 9 is
dx x
y
2
A. 1 + e = C (1 + x )
ey
2x
dy =
dx
1+ ey
1 + x2
y
2
C. e = C (1 + x )
D
A
D. None of these
The solution of an exact differential equation ( 3x2 + y cos x ) dx + ( sin x − 4 y 3 ) dy = 0 is
425)
A. x3 + y sin x − y 4 = C
C. x3 + y sin x − y 4 = y
B.
D.
y sin x − y 4 = C
None of these
A
What should be I.F of a non-exact differential equation ( 6 x2 + 4 y3 + 12 y ) dx + ( 3x + 3xy 2 ) dy = 0 to be an exact?
426)
427)
A. 1 / x
C. x
B.
D.
A. 1 / y
B.
ex
x3
What should be I.F of a non-exact differential equation ( 2 x2 y 2 + e x y ) dx − ( e x + y3 ) dy = 0 to be an exact?
C. 1 / x
Discipline: ________________________
D.
y2
1/ y
D
D
2
Page 47 of 110
What should be I.F of a linear differential equation
428)
A. 1 / y
C. 1 / x
dx
1
1
+
x= ?
dy y ln y
y
B. ln y
D. 1/ y 2
B
4
1
+ Ce − x using y ( 0 ) = 1 ?
4
4
1
y = + 5e − x
4
None of these
What will be particular solution if general solution of an ODE is y =
429)
A.
y=
C.
y = 4 − 5e− x
B.
1 5 − x4
− e
4 4
D.
4
A
3
Determine the order and degree of the differential equation 2 x
430)
A. Fourth order first degree
C. First order first degree
Which of the following is the exact differential equation?
A. ( x2 + 1) dx − xydy = 0
431)
C. 2 xydx + ( 2 + x2 ) dy = 0
432)
d4y
 dy 
+ 5 x 2   − xy = 0 .
dx 4
 dx 
B. Fourth order third degree
D. Third order fourth degree
B.
xdy + ( 3 x − 2 y ) dx = 0
D.
x2 ydy − ydx = 0
Which of the following is the variable separable equation?
A. ( x + x2 y ) dy = ( 2 x + xy 2 ) dx
B.
C.
D.
2 ydx = ( x2 + 1) dy
The equation y 2 = cx is a general solution of
A.
dy 2 y
=
dx
x
433)
C.
dy
x
=
dx 2 y
A
C
C
( x + y ) dx − 2 ydy = 0
y 2 dx + ( 2 x − 3 y ) dy = 0
D
B.
dy 2 x
=
dx
y
D.
dy
y
=
dx 2 x
If dy = x 2 dx then what is the equation of y in terms of x if the curve passes through (1,1)?
434) A.
x2 − 3 y + 3 = 0
B.
B
x3 − 3 y + 2 = 0
x3 + 3 y 2 + 2 = 0
2 y + x3 + 2 = 0
C.
D.
What is the differential equation of the family of lines passing through origin?
A.
B.
ydx − xdy = 0
xdy − ydx = 0
435)
C.
D.
xdx + ydy = 0
ydx + xdy = 0
B
What is the differential equation of the family of parabolas having their vertices at the origin and their foci on the x-axis?
A. 2 xdy − ydx = 0
B.
xdy + ydx = 0
436)
C.
2 ydx − xdy = 0
D.
dy
−x=0
dx
What will be the particular integral of the differential equation ( D2 + 4 ) y = sin 3x ?
437)
A.
C.
sin 3x
− cos 3x
5
B.
D.
D
− cos3x
− sin 3x
5
What will be the particular integral of the differential equation ( D2 + 1) y = sin 2 x ?
438)
439)
A.
C.
sin 2x
− cos 2 x
5
B.
D.
D
− cos 2x
sin 2 x
9
What will be the particular integral of the differential equation ( D2 + 2D + 3) y = cos 2 x ?
A.
sin 2 x − cos 2 x
Discipline: ________________________
B.
A
C
− cos 2 x + 17 sin 2 x
Page 48 of 110
C.
D.
1
( 4sin 2 x − cos 2 x )
17
sin 2 x cos 2 x
−
9
17
What will be the particular integral of the differential equation ( D3 + D2 + 2D − 1) y = cos 2 x ?
440)
sin 2 x − cos 2 x
1
− ( 4sin 2 x + 5cos 2 x )
41
The general solution of y  + y  − 2 y = 0 is
B.
D.
− cos 2 x + 17 sin 2 x
sin 2 x cos 2 x
−
9
17
−2 x
B.
−x
D.
y = c1e + c2 e
None of these
B.
y = ( c1 + c2 x ) e
D.
None of these
B.
−2 x
A.
C.
441) A.
C.
y = c1e + c2 e
x
y = c1e + c2 e
x
A
The general solution of y  − 4 y  + 4 y = 0 is
y = c1e x + c2 e−2 x
442) A.
C.
y = c1e x + c2 e− x
443)
C.
1
y = c1e + c2 e
x
2
2x
−2 x
B
The general solution of y  + y  − 2 y = 0 is
A.
y = e− x c cos 2 x + c sin 2 x
(
C
2x
A
)
−x
D.
y=e
(c cos
1
3x + c2 sin 3x
)
None of these
444) What will be form of y p while solving y  − 5 y  + 4 y = 8e by UC method?
x
A
A.
C.
B.
Axe x
Ae x
2 x
D.
None of these
Ax e
445) What will be form of y p while solving a2 y  + a1 y  + a0 y = 1 by UC method?
A.
C.
A
2 x
B.
D.
A
x
Ae
None of these
Ax e
446) What will be form of y p while solving a2 y  + a1 y  + a0 y = 5 x + 7 by UC method?
B
Ax + B
A
B.
2 x
D.
None
of these
Ax e
2
447) What will be form of y p while solving a2 y + a1 y + a0 y = 3x − 2 by UC method?
C
Ax + B
A
B.
2
D.
None
of these
Ax + Bx + C
3
448) What will be form of y p while solving a2 y + a1 y + a0 y = 3x − 2 x by UC method?
A
Ax + B
B.
Ax3 + Bx2 + Cx + E
D.
None of these
Ax2 + Bx + C
449) What will be form of y p while solving a2 y  + a1 y  + a0 y = sin 5 x by UC method?
C
A cos x
A.
B.
A sin x + B
C.
D.
None of these
A cos 5 x + B sin 5 x
450) What will be form of y p while solving a2 y  + a1 y  + a0 y = cos 5 x by UC method?
C
A cos x
A.
B.
A sin x + B
C.
D.
None of these
A cos 5 x + B sin 5 x
451) What will be form of y p while solving a2 y + a1 y  + a0 y = e3 x by UC method?
C
A.
C.
A.
C.
A.
C.
A.
C.
A cos x
B.
D.
Ae3x + B
None of these
Ae3x
452) What will be form of y p while solving a2 y  + a1 y  + a0 y = xe3 x by UC method?
A.
A cos x
B.
B
( Ax + B ) e3x
C.
D.
None of these
Ae3x
453) What will be form of y p while solving a2 y + a1 y  + a0 y = x 2 e3 x by UC method?
Discipline: ________________________
C
Page 49 of 110
A.
A cos x
B.
C.
( Ax
D.
2
+ Bx + C ) e3x
( Ax + B ) e3x
None of these
454) What will be form of y p while solving a2 y + a1 y + a0 y = e3 x sin 4 x by UC method?
A.
C.
A cos 4 xe3 x
( A cos 4 x + B sin 4 x ) e3 x
B.
( Ax + B ) e3x
D.
None of these
C
455) What will be form of y p while solving a2 y + a1 y + a0 y = 5x sin 4 x by UC method?
C
2
A.
A cos 4 xe3 x
B.
( Ax + B ) e3x
C.
( Ax
D.
None of these
2
+ Bx + C ) cos 4x + (Cx2 + Ex + F ) sin 4x
456) What is y 2 if y1 = x 2 is a solution of x2 y − 3xy + 4 y = 0 ?
A.
B.
A cos 4 xe3 x
3
C.
D.
4x
2x
457) What is y 2 if y1 = e is a solution of y  − 4 y  + 4 y = 0 ?
2x
A.
C.
xe
4x3
458) What is y 2 if y1 = cos 4 x is a solution of y  + 16 y = 0 ?
A. sin 4x
C.
4x3
459) What is y 2 if y1 = cosh x is a solution of y  − y = 0 ?
A. sin 4x
C.
sinh x
460) What is y 2 if y1 = ln x is a solution of xy  + y  = 0 ?
A. sin 4x
C.
sinh x
461) The partial differential equation ∂u + 𝑢 ∂u = ∂2u is a
∂t
A.
C.
462)
∂x
B.
D.
B
x2 ln x
None of these
A
2
x ln x
None of these
A
B.
D.
2
x ln x
None of these
C
B.
D.
x2 ln x
None of these
D
B.
D.
2
x ln x
1
D
∂x2
Linear equation of order 2
Linear equation of order 1
B.
D.
Non-linear equation of order 1
Non-linear equation of order 2
𝑑2𝑦
𝑑𝑦 3
3
The order of the differential equation ( 2 )+ ( ) + 𝑦 4 = 𝑒 −𝑡 is
𝑑𝑡
A.
C.
B
𝑑𝑡
1
3
B.
D.
2
4
463) The differential equation 𝑑2 𝑦 + 16y=0 for y(x) with two boundary conditions 𝑑𝑦 (𝑥 = 0) = 1 and 𝑑𝑦 (x=𝜋)=-1 has
𝑑𝑥 2
𝑑𝑥
𝑑𝑥
2
A. No solution
B.
Exactly one solution
C. Exactly two solutions
D.
Infinitely many solutions
464) A differential equation is considered to be ordinary if it has
465) A. one dependent variable
B.
more than one dependent variable
466) C. one independent variable
D.
more than one independent variable
467) PDE has independent variable
A. 0
B.
1
C. Less than 1
D.
More than 1
468) In homogeneous first order linear constant coefficient ordinary DE is
A. 𝜕𝑢=0
B.
𝑐𝑢 + 𝑥 2 =0
A
C
D
C
𝜕𝑥
C.
𝜕𝑢
𝜕𝑥
D.
=cu+𝑥 2
469) The P.D.E
𝜕2 𝑈
𝜕𝑥 2
+
𝜕𝑢
𝜕𝑥
𝜕2 𝑈
𝜕𝑦 2
𝑐
= + 𝑥2
𝑢
A
= 𝑓(𝑥, 𝑦); is known as
Discipline: ________________________
Page 50 of 110
A. laplace equation
B.
heat equation
C. poission equation
D.
wave equation
470) The solution of a differential equation which is not obtained from the general solution is known as:
A. Particular solution
B.
Singular solution
C. Complete solution
D.
Auxiliary solution
471) The differential equation 𝑑𝑦 = y2 is
𝑑𝑥
A. Linear
B.
Non-linear
C. Quasilinear
D.
None of these
472) The DE formed by y= acosx+bcosx+4 where a and b are arbitrary constants is:
𝑑2 𝑦
A. (𝑑2 𝑦)+y=0
B.
( )-y=0
C.
(
𝑑𝑥 2
𝑑2 𝑦
𝑑𝑥 2
D.
)+y=4
473) The equation a0 x y’’+a1 xy’+a2y=𝜑(𝑥) is called:
A. Legendre’s linear equation
C. Simultaneous equation
474) Solution of the DE 𝑑𝑦=sin(x+y)+ cos(x+y), is
𝑑𝑥
A. Log|1+tan(𝑥+𝑦)|=y+c
(
𝑑𝑥 2
𝑑2 𝑦
𝑑𝑥 2
B
B
C
)-y=4
2
B
B.
D.
D
B.
2
C. Log |1+tan(x+y)|=y+c
475) If y=a cos (log x)+ b sin (log x), then
A. x2 𝑑2𝑦 + x𝑑𝑦 + y=0
C.
x
𝑑𝑥 2
2
2𝑑 𝑦
𝑑𝑥 2
+x
𝑑𝑥
𝑑𝑦
𝑑𝑥
−1
- y=0
476) If y=sin(a sin 𝑥), then
A. (1-x2 ) 𝑑2 𝑦 - x𝑑𝑦 +a2 y=0
𝑑𝑥 2
𝑑2 𝑦
C.
𝑑𝑥
𝑑𝑦
(1-x2 ) 2 - x -a2 y=0
𝑑𝑥
𝑑𝑥
477) The DE of the family of curves y2= 4a(x+a) is
A. y2=4𝑑𝑦 (𝑥 + 𝑑𝑦 )
𝑑𝑥
𝑑𝑥
C. y2𝑑𝑦 + 4y=0
D.
√1−𝑥 2
C.
−1
479) Inverse derivative of cos x is
1
A.
C.
480) If
√1−𝑥 2
−1
√1−𝑥 2
3
y=tan−1 𝑥 ⁄2 ,
A.
𝑑𝑦
𝑑𝑥
B.
𝑑2 𝑦
A
D.
𝑑𝑦
(1-x2 ) 2 + x -a2 y=0
𝑑𝑥
𝑑𝑥
None of these
B
B.
D.
2 𝑑𝑦 2
𝑑𝑦
𝑑𝑥
𝑑𝑦
𝑑𝑥
y ( ) +2xy
2y
𝑑𝑥
2
- y =0
+ 4a=0
A
B.
1
√1+𝑥 2
None of these
C
B.
1
√1+𝑥 2
None of these
A
=
3√𝑥
B.
2(1+𝑥 3 )
C.
𝑑𝑦
D.
D.
then
2
2𝑑 𝑦
x 2 - x + y=0
𝑑𝑥
𝑑𝑥
none of these
D.
√1−𝑥 2
(𝑥+𝑦)
Log|2+sec
|=x+c
2
None of these
B
B.
𝑑𝑥
478) Inverse derivative of sin x is
1
A.
Cauchy’s linear equation
Method of undetermined coefficients
3√𝑥
2√𝑥
(1+𝑥 3 )
D.
None of these
2(1−𝑥 3 )
481) The equation of the curves, satisfying the DE 𝑑2𝑦 (𝑥 2 + 1) = 2𝑥 𝑑𝑦 passing through the point (0,1) and having the slope of
𝑑𝑥 2
𝑑𝑥
tangent at x=0 as 6 is
A.
B.
𝑦 2 = 2𝑥 3 + 6𝑥 + 1
𝑦 = 2𝑥 3 + 6𝑥 + 1
2
3
C.
D.
None of these
𝑦 = 𝑥 + 6𝑥 + 1
482) A particle, initially at origin moves along x-axis according to the rule 𝑑𝑥 = x+4. The time taken by the particle to traverse a
𝑑𝑡
distance of 96 units is:
Discipline: ________________________
Page 51 of 110
B
C
A.
ln 5
C. 2 ln 5
483) If y=cos −1 (ln 𝑥), then the value of 𝑑𝑦 is
𝑑𝑥
1
A.
B.
D.
B
B.
−1
D.
𝑥√1 − (ln 𝑥)2
None of these
𝑥√1 − (ln 𝑥)2
−1
C.
𝑥√1+(ln 𝑥)2
484) If x=2
) ln cot( 𝑡) and y= tan(t) + cot(t), the value of 𝑑𝑦 is
𝑑𝑥
A. cot(2t)
C. cos(2t)
𝑑𝑦
485) Solution
)
of the DE ln( )= ax + by is
A.
487)
488)
489)
490)
1
𝑏
𝑎
1
- 𝑒 −𝑏𝑦 =
A.
C.
1
A
B.
D.
tan(2t)
sec(2t)
B.
1
A
𝑑𝑥
𝑒 𝑎𝑥 + 𝑐
𝑏
𝑒 −𝑏𝑦 =
1
𝑎
𝑒 𝑎𝑥 + 𝑐
1
1
D.
𝑒 −𝑏𝑦 = - 𝑒 𝑎𝑥 + 𝑐
- 𝑒 −𝑏𝑦 = - 𝑒 𝑎𝑥 + 𝑐
𝑎
𝑏
𝑎
If the general solution of a differential equation is (y+c) 2=cx, where c is an arbitrary constant, then the order and degree of
differential equation is
A. 1, 2
B.
2, 1
C. 1, 3
D.
None of these
Solution of (𝑥 2 sin3y – y2cos x) dx + (x3 cos y sin2y – 2y sin x) dy = 0 is
A. (X3sin3y/3 ) = c
B.
𝑥 3 sin3y = y2 sin x + c
3
3
2
C. (X sin y/3 )= y sin x +c
D.
None of these
𝑥 𝑑𝑦
𝑦
Solution of 2 2 = ( 2 2 − 1)dx is
𝑥 +𝑦
𝑥 +𝑦
𝑦
𝑦
A.
B.
−1
𝑥 − tan
tan−1 = 𝑐
𝑥
𝑥
𝑦
C.
D.
None
of
these
−1
𝑥 tan
=𝑐
𝑥
Solution of (𝑦 + 𝑥 √𝑥𝑦 (𝑥 + 𝑦))𝑑𝑥 + (𝑦 √𝑥𝑦 (𝑥 + 𝑦) − 𝑥)𝑑𝑦 = 0 is
𝑦
𝑦
A.
B.
𝑥 2 + 𝑦 2 = 2 tan−1 √ + 𝑐
𝑥 2 + 𝑦 2 = 4 tan−1 √ + 𝑐
𝑥
𝑥
𝑦
C.
D.
None
of
these
𝑥 2 + 𝑦 2 = tan−1 √ + 𝑐
𝑥
𝑑𝑦
Solution of the DE + 2𝑥𝑦 = 𝑦 is
C.
486)
1
log 5 𝑒
None of these
𝑏
𝑦 = 𝑐𝑒 𝑥−𝑥
𝑦 = 𝑐𝑒 𝑥
2
𝑑𝑥
B.
D.
C
C
D
2
𝑦 = 𝑐𝑒 𝑥 − 𝑥
None of these
491) Solution of the differential equation 𝑑𝑦 = sin(𝑥 + 𝑦) + cos(𝑥 + 𝑦), is
𝑑𝑥
(𝑥+𝑦)
A. log |1 + tan (𝑥+𝑦)| = y + c
B.
log |2 + sec
|=x+c
2
C.
A
D
2
log |1 + tan (𝑥 + 𝑦)| = y + c
D.
None of these
492) The1Blasius equation 𝑑3𝑓 + 𝑓 𝑑2𝑓 = 0 is a
3
2
𝑑η
B
2 𝑑η
A. Second order non-linear differential equation
B.
Third order non-linear ordinary differential equation
C. Third order linear ordinary differential equation
D.
Mixed order non-linear ODE
493) The1general solution of DE 𝑑𝑦 = cos(𝑥 + 𝑦)with c as a constant is
D
𝑑𝑥
A.
C.
y + sin (x + y) = x + c
cos(
𝑥+𝑦
2
B.
D.
)=𝑥+𝑐
494) The solution of the initial value problem 𝑑𝑦 = −2𝑥𝑦; 𝑦(0) = 2 is
A.
C.
1 + 𝑒 −𝑥
2
1 + 𝑒𝑥
tan(
tan(
𝑥+𝑦
2
𝑥+𝑦
2
B.
D.
)=𝑥+𝑐
B
𝑑𝑥
2
)=𝑦+𝑐
2𝑒 −𝑥
2
2𝑒 𝑥
2
495) The solution of ODE 𝑑2𝑦 + 𝑑𝑦 − 6𝑦 = 0 is
2
𝑑𝑥
A
𝑑𝑥
Discipline: ________________________
Page 52 of 110
B.
𝑦 = 𝑐1 𝑒 −3𝑥 + 𝑐2 𝑒 2𝑥
𝑦 = 𝑐1 𝑒 3𝑥 + 𝑐2 𝑒 2𝑥
3𝑥
−2𝑥
D.
𝑦 = 𝑐1 𝑒 + 𝑐2 𝑒
𝑦 = 𝑒 −3𝑥 + 𝑒 2𝑥
𝑑𝑦
496) The solution for the differential equation
= 𝑥 2 𝑦 with two condition that y=1 at x=0
𝑑𝑥
𝑥
2
𝑥
A.
B.
3𝑒 2
2𝑒 2
𝑥2
𝑥2
C.
D.
3𝑒 2
𝑒2
497) The solution of 𝑑𝑦 = − 𝑥 with initial condition y(1)=√3 is
A.
C.
𝑑𝑥
A.
C.
498)
C
C
𝑦
𝑥 3 + 𝑦3 = 4
𝑥 2 + 𝑦2 = 4
B.
D.
𝑦 = 4𝑎𝑥
None of these

A
If L  f ( t ) =  e − st f ( t ) dt , then what is L 1 ?
0
499)
A.
1
s
C.
s
B.
D.
1
s2
None of these

B
If L  f ( t ) =  e − st f ( t ) dt , then what is L t ?
0
500)
A.
1
s
C.
s
B.
D.
1
s2
None of these

C
If L  f ( t ) =  e − st f ( t ) dt , then what is L t n  ?
0
A.
C.
501)
D.

 1 
If L  f ( t ) =  e − st f ( t ) dt , then what is L 
?
0
 t 
A.
1
C.
502)
B.
1
s
n!
s n +1
s
n!
s n +1
1
s2
None of these
A
B.
D.
1
s2
None of these

C
If L  f ( t ) =  e − st f ( t ) dt , then what is L teat  ?
0
A.
B.
1
s
C.
(s − a)
503)
D.
1
1
s2
None of these
2

B
If L  f ( t ) =  e − st f ( t ) dt , then what is L cos at ?
0
A.
B.
1
s
C.
(s − a)
504)
D.
1
s
s2 + a2
None of these
2

B
If L  f ( t ) =  e − st f ( t ) dt , then what is L cosh at ?
0
Discipline: ________________________
Page 53 of 110
A.
B.
1
s
C.
D.
1
(s − a)
505)
s
s − a2
None of these
2
2

A
If L  f ( t ) =  e − st f ( t ) dt , then what is L eat sinb t ?
0
A.
(s − a)
C.
2
+b
2
+b
2
2
s
s − a2
2
D.
a
(s − a)
506)
B.
b
None of these

C
If L  f ( t ) =  e − st f ( t ) dt , then what is L t sina t ?
0
A.
(s − a)
C.
2
+b
2
+a
s
s − a2
2
2
D.
2as
(s
507)
B.
b
None of these
)
2 2

A
If L  f ( t ) =  e − st f ( t ) dt , then what is L 1 − cos at ?
0
A.
a
2
s (s + a
2
C.
2
+a
s
s2 − a2
D.
None of these
)
2as
(s
508)
2
B.
)
2 2

A
If L  f ( t ) =  e − st f ( t ) dt , then what is L t   ,   −1 ?
0
A.
 ( + 1)
s
2as
C.
(s
509)
2
+a

B.
n +1
D.
s2 −  2
None of these
)
2 2

C
If L  f ( t ) =  e − st f ( t ) dt , then what is L eat  ?
0
A.
a
s (s + a
2
C.
510)
B.
2
2
)
D.
1
(s − a)
s
s − a2
2
None of these

B
If L  f ( t ) =  e − st f ( t ) dt , then what is L sin at ?
0
A.
a
s (s + a
2
C.
511)
B.
2
2
)
D.
1
(s − a)
a
s + a2
2
None of these

B
If L  f ( t ) =  e − st f ( t ) dt , then what is L sinh at ?
0
Discipline: ________________________
Page 54 of 110
A.
B.
a2
s (s + a
2
C.
512)
2
)
D.
1
(s − a)
a
s − a2
2
None of these

C
If L  f ( t ) =  e − st f ( t ) dt , then what is L t n eat  ?
0
A.
a
B.
2
s (s + a
2
C.
)
a
s − a2
2
D.
n!
(s − a)
513)
2
None of these
n +1

B
If L  f ( t ) =  e − st f ( t ) dt , then what is L eat cos bt ?
0
A.
a
s (s + a
2
C.
514)
2
)
(s − a)
D.
n!
(s − a)
s−a
B.
2
2
+ b2
None of these
n +1

A
If L  f ( t ) =  e − st f ( t ) dt , then what is L  t cosa t ?
0
A.
s −a
2
(s
2
C.
+a
515)
C.
D.
+a
C.
+ b2
None of these
+a
(s − a)
D.
2
+ b2
None of these
B
B.
)
2 2

 F ( u ) du
s
D.
1
F (s)
s
s−a
B.
)

 f ( t ) 
If L  f ( t ) =  e − st f ( t ) dt , then what is L 
?
0
 t 
A.
s2 − a2
2
2
C
2 2
1
F (s)
s
(s
517)
2
(s − a)
n +1

t

If L  f ( t ) =  e − st f ( t ) dt , then what is L   f ( u ) du  ?
0

0
2
2
A.
s −a
(s
516)
)
2 2
n!
(s − a)
s−a
B.
2
None of these

A
If L  f ( t ) =  e − st f ( t ) dt , then what is L  f  ( t ) ?
0
A.
sF ( s ) − f ( 0 )
B.
1
F (s)
s
D.

 F ( u ) du
s
C.
518)
None of these

B
If L  f ( t ) =  e − st f ( t ) dt , then what is L at − sin t ?
0
Discipline: ________________________
Page 55 of 110
A.
C.
519)
sF ( s ) − f ( 0 )
B.
1
F (s)
s
D.
a3
s2 ( s2 + a2 )
None of these

C
If L  f ( t ) =  e − st f ( t ) dt , then what is L cosh at − cos at ?
0
A.
C.
520)
sF ( s ) − f ( 0 )
B.
2a 2 s
s4 − a4
D.
a3
s2 ( s2 + a2 )
None of these

A
If L  f ( t ) =  e − st f ( t ) dt and L−1 F ( s ) = f ( t ) then what is L−1 F ( s − a ) ?
0
A.
C.
521)
e f (t )
B.
2a 2 s
s4 − a4
D.
at
a3
s2 ( s2 + a2 )
None of these

B
If L  f ( t ) =  e − st f ( t ) dt and L−1 F ( s ) = f ( t ) then what is L−1 F ( cs ) ?
0
522)
A.
e at f ( t )
B.
C.
2a 2 s
s4 − a4
D.



C
If L  f ( t ) =  e − st f ( t ) dt and L−1 F ( s ) = f ( t ) then what is L−1 F ( n) ( s ) ?
0
523)
1 t
f 
c c
None of these
A.
e f (t )
B.
C.
( −1)
D.
at
n
t n f (t )
1 t
f 
c c
None of these

 F ( s ) 
If L  f ( t ) =  e − st f ( t ) dt and L−1 F ( s ) = f ( t ) then what is L−1 
?
 s 
0
A. e at f ( t )
B.
1 t
f 
c c
t
C.
D.
None of these
 f ( u ) du
C
0
524)

B
If L  f ( t ) =  e − st f ( t ) dt and L−1 F ( s ) = f ( t ) then what is L−1 e− as F ( s ) ?
0
A.
e f (t )
B.
ua ( t ) f ( t − a )
C.
t
D.
None of these
at
 f ( u ) du
0
525) The solution of ( D2 + 4D + 3) y = 0 is
A
A.
y = c1e− x + c2 e−3 x
B.
C.
y = c1e− x + c2 e−2 x
D.
y = c1e−4 x + c2 e−3 x
None of these
526) The solution of ( D3 − 5D2 + 7 D − 3) y = 0 is
A.
y = c1e− x + c2 e−3 x
Discipline: ________________________
B
B.
y = ( c1 + c2 x ) e x + c3 e3 x
Page 56 of 110
C.
y = c1e− x + c2 e−2 x + c3e−3 x
D.
527) The solution of ( D3 − D2 + D − 1) y = 0 is
None of these
C
A.
y = c1e− x + c2 e−3 x
B.
y = ( c1 + c2 x ) e x + c3e3 x
C.
y = c1e x + c2 sin x + c3 cos x
D.
None of these
528) The solution of ( D + D − 12) y = 0 is
A
2
A.
y = c1e3 x + c2 e−4 x
B.
C.
y = c1e− x + c2 e−2 x
D.
y = c1e−4 x + c2 e−3 x
None of these
529) The solution of ( D + 4D + 5) y = 0 is
C
2
A.
y = c1e3 x + c2 e−4 x
B.
y = c1e−4 x + c2 e−3 x
C.
y = e −2 x ( c1 sin x + c2 cos x )
D.
None of these
530) The solution of ( D3 − 3D2 + 4) y = 0 is
B
A.
y = c1e− x + c2 e−3 x
B.
y = ( c1 + c2 x ) e 2 x + c3 e − x
C.
y = c1e− x + c2 e−2 x + c3e−3 x
D.
None of these
531) The solution of ( 9D2 − 12D + 4 ) y = 0 is
A.
y = ( c1 + c2 x ) e 3
C.
y = c1e− x + c2 e−2 x + c3e−3 x
2
A
x
B.
y = ( c1 + c2 x ) e 2 x + c3 e − x
D.
None of these
532) The solution of ( D3 − 4D2 + D + 6) y = 0 is
A.
y = ( c1 + c2 x ) e 3
C.
y = c1e− x + c2 e2 x + c3e3 x
2
C
x
533) Which of these is the solution of differential equation
𝑑𝑥
𝑑𝑡
B.
y = ( c1 + c2 x ) e 2 x + c3 e − x
D.
None of these
B.
2𝑒 −3𝑡
D.
2𝑒 2𝑡
534) The general solution of DE 𝑑𝑦 = 𝑦 is
𝑑𝑥
𝑥
A. log y= kx
B.
C. y=𝑘
D.
𝑥
535) Integrating factor of DE cos𝑑𝑦 + 𝑦𝑠𝑖𝑛𝑥 = 1 is
𝑑𝑥
A. sin x
B.
C. tan x
D.
536) If 2xy dx + P(x, y)dy = 0 is exact then P(x, y) is
A. x – y
B.
C. x – y2
D.
537) A differential equation of first degree
A. Is of first order
B.
C. Always linear
D.
538) A general solution of an nth order differential equation contains
A. n – 1 arbitrary constants
B.
C. n+1 arbitrary constants
D.
A.
C.
A
+ 3𝑥 = 0
𝑒 −3𝑡
𝑒 −2𝑡
B
y=kx
y=k log x
B
sec x
cos x
D
x+y
x2 + y
B
May or may not be linear
All are false
B
n arbitrary constants
no constant
𝜕2 𝑦
A
The order of the differential equation 2 + 𝑦 2 = 𝑥 + 𝑒 𝑥 is
𝜕𝑥
A. 2
B.
3
C. 0
D.
1
539) ‘’Infinitely many differential equation have the same integrating factor’’. This statement is
A. Never true
B.
May be true
C. Semi true
D.
Always true
Discipline: ________________________
D
Page 57 of 110
540) If
𝑑𝑦
𝑑𝑥
=
𝑓(𝑥,𝑦)
𝜑(𝑥,𝑦)
is a homogeneous DE then it can be made in the form ‘’separable in variables’’ by putting
A. y2 =vx
B.
x2=vy
C. y=vx
D.
x=vy
541) The differential equation 𝑑𝑦 + 𝑃𝑦 = 𝑄𝑦 𝑛 , 𝑛 ≥ 2 can be reduced to linear form by substituting
C
D
𝑑𝑥
n-1
542)
543)
544)
545)
546)
547)
548)
549)
550)
551)
552)
553)
554)
555)
n
A. z= y
B.
z= y
C. x=yn+1
D.
x=y1-n
3
The differential equation (𝑦 − 2𝑥 )𝑑𝑥 − 𝑥(1 − 𝑥𝑦)𝑑𝑦 = 0 becomes exact on multiplication by
1
1
A.
B.
𝑥
𝑥2
1
1
C.
D.
𝑥3
𝑥4
If the DE 𝑓(𝑥, 𝑦)𝑑𝑥 + 𝑥𝑠𝑖𝑛𝑦𝑑𝑦 = 0 is exact then 𝑓(𝑥, 𝑦) equals
A. cos y
B.
–cos(y) +x2
C. –sin y
D.
Sin(y) +x
𝑑𝑦
The differential equation 𝑥 =𝑥 − 𝑦 is
𝑑𝑥
A. Exact
B.
Linear
C. Homogeneous
D.
All of above
The general solution of the equation 𝑥 ′ + 5𝑥 = 3 is
3
A.
B.
𝑥(𝑡) = 3 + 𝐶 sin 5𝑡
𝑥(𝑡) = + 𝑒 −5𝑡
5
3
C.
D.
𝑥(𝑡) = 𝐶 cos 3𝑡
𝑥(𝑡) = + 𝐶𝑒 −5𝑡
5
A solution of the initial value problem 𝑦 ′ + 8𝑦 = 1 + 𝑒 −6𝑡 is
1 1
5
1 1
5
A.
B.
𝑥(𝑡) = + 𝑒 6𝑡 − 𝑒 8𝑡
𝑥(𝑡) = + 𝑒 −6𝑡 − 𝑒 −8𝑡
8 2
8
8 2
8
C.
D.
𝑥(𝑡) = 4 − 𝑒 2𝑡 + 3𝑒 8𝑡
𝑥(𝑡) = 4 − 𝑒 −2𝑡 + 3𝑒 −8𝑡
Two linearly independent solutions of the equation 𝑦 ′′ + 𝑦 ′ − 6𝑦 = 0 are
A. 𝑒 −3𝑥 And 𝑒 2𝑥
B.
𝑒 −2𝑥 And 𝑒 3𝑥
−𝑥
6𝑥
C.
D.
𝑒 And 𝑒
𝑒 −6𝑥 And 𝑒 𝑥
′′
′
A particular solution for the differential equation 𝑦 + 2𝑦 + 𝑦 = 3 − 2 sin 𝑥is
A.
B.
𝐴 + 𝐵 sin 𝑥
𝐴 + 𝐵𝑥 2 + 𝐶 cos 𝑥 + 𝐷 sin 𝑥
C.
D.
𝐴 + 𝐵𝑥 cos 𝑥 + 𝐶𝑥 sin 𝑥
𝐴 + 𝐵 cos 𝑥 + 𝐶 sin 𝑥
The solution of the initial value problem 𝑥 2 𝑦 ′′ − 𝑥𝑦 ′ − 3𝑦 = 0, 𝑦(1) = 1, 𝑦 ′ (1) = −2 is
1
3
5 −1 1 3
A.
B.
𝑥 + 𝑥 −3
𝑥 − 𝑥
4
4
4
4
1 𝑥 3 −3𝑥
5 −𝑥 1 3𝑥
C.
D.
𝑒 + 𝑒
𝑒 − 𝑒
4
4
4
4
′′
′
The differential equation 𝑥 + 2𝑥 − 5𝑥 = sin 𝑡 is equivalent to the system
A.
B.
𝑥 ′ = 𝑦 , 𝑦 ′ = 5𝑥 − 2𝑦 + sin 𝑡
𝑥 ′ = 2𝑥 − 5𝑦 , 𝑦 ′ = sin 𝑡
′
′
C.
D.
𝑥 = 𝑦 , 𝑦 = 2𝑥 − 5𝑦 + sin 𝑡
𝑥 ′ = 5𝑥 − 2𝑦 , 𝑦 ′ = sin 𝑡
𝑦
𝑦
′′
′′
The system of DE 𝑥 = − 2 2 , 𝑦 = − 2 2 is equivalent to a 1st order system consisting of
𝑥 +𝑦
B
B
D
C
B
A
D
A
A
D
𝑥 +𝑦
A. One equation
B.
Two equations
C. Three equations
D.
Four equations
The series solution for the DE 𝑦 ′′ + 𝑥𝑦 ′ + 𝑦 = 0 is of the form ∑∞
=
0 𝐶𝑛 𝑋 𝑛 has recursion relation
𝑛
(𝑐
A.
B.
𝐶𝑛+2 + 𝐶𝑛+1 + 𝐶𝑛 = 0, 𝑛 ≥ 0
+ 2)𝐶𝑛+2 + 𝐶𝑛 = 0, 𝑛 ≥ 2
C.
D.
𝑛𝑐𝑛+1 − 𝑐𝑛 = 0, 𝑛 ≥ 1
𝐶𝑛 = 0, 𝑛 ≥ 3
The differential equation 𝑥𝑦 ′′ + (𝑥 − 2)𝑦 ′ + 𝑦 = 0 has a solution of the form
𝑛
𝑛
A. 𝑦 = 𝑥 2 ∑∞
B.
𝑦 = 𝑥 1/2 ∑∞
𝑛=0 𝑐𝑛 𝑥 , 𝑐0 ≠ 0
𝑛=0 𝑐𝑛 𝑥 , 𝑐0 ≠ 0
−1
∞
𝑛
3
∞
𝑛
C. 𝑦 = 𝑥 ∑𝑛=0 𝑐𝑛 𝑥 , 𝑐0 ≠ 0
D.
𝑦 = 𝑥 ∑𝑛=0 𝑐𝑛 𝑥 , 𝑐0 ≠ 0
𝑛
The solution of the DE(2𝑥 − 1)𝑦 ′ + 2𝑦 = 0, can be represented as a power series ∑∞
𝑛 = 0 𝐶𝑛 𝑋 with radius of convergence
equal to
A. 0
B.
1/2
C. 1
D.
∞
𝑠+4
The partial fraction decomposition of
is
2 2
(𝑠−1) (𝑠 +4)
Discipline: ________________________
Page 58 of 110
B
C
C
B
𝐴
𝐵𝑠 + 𝐶
𝐴
𝐵
𝐶𝑠 + 𝐷
B.
+
+
+
(𝑠 − 1)2 𝑠 2 + 4
𝑠 − 1 (𝑠 − 1)2 𝑠 2 + 4
𝐴
𝐵
𝐶
𝐴
𝐵
𝐶𝑠
C.
D.
+
+ 2
+
+ 2
2
2
𝑠 − 1 (𝑠 − 1)
𝑠 +4
𝑠 − 1 (𝑠 − 1)
𝑠 +4
Which of the following equations can be rearranged into separable equations?
(𝑥 + 𝑦)𝑦 ′ = 𝑥 − 𝑦
A.
B.
𝑦 ′ − 𝑒 𝑦 = 𝑒 𝑥=𝑦
′
C.
D.
None of these
𝑦 = 𝑙𝑛(𝑥𝑦)
The appropriate form of a particular solution 𝑦𝑝 of the equation 𝑦 ′′ + 2𝑦 ′ + 𝑦 = 𝑒 −𝑡 is
A.
B.
𝐴𝑒 −𝑡
𝐴𝑡𝑒 −𝑡
2 −𝑡
(𝐴 + 𝐵𝑡)𝑒 −𝑡
C.
D.
𝐴𝑡 𝑒
2
Particular integral of (𝐷 + 4)𝑦 = 𝑠𝑖𝑛3𝑥 is
1
1
A.
B.
− 𝑠𝑖𝑛3𝑥
− 𝑐𝑜𝑠3𝑥
5
5
1
C.
D.
None of these
− 𝑡𝑎𝑛3𝑥
5
𝑑𝑦
𝑥 2 +𝑥𝑦+2𝑦 2
The D.E =
is
2
2
A.
556)
557)
558)
559)
𝑑𝑥
A.
C.
560)
B
C
C
B
2𝑥 +𝑦
Exact
Cauchy
B.
D.
The order and degree of the D.E (
Homogenous
None of these
𝑑𝑦 2 3
)
𝑑𝑥
2
2
𝑑 𝑦
𝑑𝑥2
1+(
D
) =b are respectively
A. 1 and 3
B.
1 and 2
C. 3 and 2
D.
2 and 3
561) A general solution of 3rd order D.E contains
A. One constant
B.
Two constants
C. Three constants
D.
No constant
562) solving 𝑦 ′ + 𝑦 ′ + 2𝑦 = 0 with 𝑦(0), 𝑦(1) = 1 is
A. Initial value problem
B.
Boundary value problem
C. Eigen value problem
D.
None of these
563) If p and q are the order and degree of D.E 𝑦 𝑑𝑦 + 𝑥 2 (𝑑2𝑦)3 + 𝑥𝑦 = 𝑐𝑜𝑠𝑥 , then
B
A
𝑑𝑥 2
𝑑𝑥
A.
C.
C
𝑝<𝑞
𝑝>𝑞
B.
D.
𝑝=𝑞
None of these
564) A solution to the partial differential equation 𝜕2𝑢 = 9 𝜕2 𝑢 is
2
2
𝜕𝑥
A.
C.
cos (3𝑥 − 𝑦)
sin (3𝑥 − 𝑦)
B.
D.
565) The partial differential equation 5 𝜕2 𝑧 + 6
2
𝜕𝑥
A.
C.
D
𝜕𝑦
𝜕2 𝑧
𝜕𝑦 2
B.
D.
𝜕𝑥
𝜕2 𝑧
𝜕𝑦 2
A
= 𝑥𝑦 is classified as
Elliptic
Hyperbola
566) The partial differential equation 𝑥𝑦 𝜕𝑧 = 5
𝑥 2 + 𝑦2
𝑒 −3𝜋𝑥 sin (𝜋𝑥)
Parabola
None of above
B
is
A. Elliptic
B.
Parabola
C. Hyperbola
D.
None of above
567) The following is true for the following partial differential equation under non-linear mechanics known as the Kortewege-de𝜕𝑤
𝜕3 𝑤
B.
D.
Non-linear 3rd order
Non-linear first order
568) Solve 𝜕𝑢 = 6 𝜕𝑢 + 𝑢 using separation method of variable if 𝑢(𝑥, 0) = 10𝑒 −𝑥
𝜕𝑥
A.
B
𝜕𝑤
vfies equation
+ 3 − 6𝑤
=0
𝜕𝑡
𝜕𝑥
𝜕𝑥
rd
A. Linear, 3 order
C. Linear first order
𝜕𝑡
𝑡
−𝑥 −3
10𝑒 𝑒
B.
A
𝑡
10𝑒 𝑥 𝑒 −3
𝑥
𝑥
D.
10𝑒 3 𝑒 −𝑡
10𝑒 −3 𝑒 𝑡
569) While solving the partial differential equation by separable method we equate the ratio to constant which?
C.
Discipline: ________________________
Page 59 of 110
B
A. Can be positive or negative integer or zero
B.
Can be positive or negative rational number or zero
C. Must be positive integer
D.
Must be negative integer
570) When solving a 1-dimensional heat equation using a variable separable method we get the solution
A. k is positive
B.
k is 0
C. k is negative
D.
k can be anything
571) 𝑓(𝑥, 𝑦) = sin(𝑥𝑦) + 𝑥 2 ln(𝑦) . Find 𝑓𝑥𝑦 at (0,𝜋)
2
572)
573)
574)
575)
576)
577)
578)
A. 33
B.
0
C. 3
D.
1
𝑑𝑓
2
3
2
3
3
9
𝑓(𝑥, 𝑦) = 𝑥 + 𝑦 ; 𝑥 = 𝑡 + 𝑡 ; 𝑦 = 𝑡 + 𝑡 find at t=1
𝑑𝑡
A. 0
B.
1
C. -164
D.
164
D.E for 𝑦 = 𝐴𝑐𝑜𝑠𝛼𝑥 + 𝐵𝑠𝑖𝑛𝛼𝑥 , where A and B are arbitrary constants is
A.
B.
𝑑2 𝑦
𝑑2 𝑦
+ 𝛼𝑦 = 0
− 𝛼𝑦 = 0
2
𝑑𝑥
𝑑𝑥 2
C.
D.
𝑑2 𝑦
𝑑2 𝑦
− 𝛼 2𝑦 = 0
+ 𝛼 2𝑦 = 0
2
𝑑𝑥
𝑑𝑥 2
The order of D.E is defined as
A. The highest degree of the variable
B.
The order of the highest derivative
C. The power of variable in the solution
D.
None of these
A primitive of an ODE is
A. Its general solution
B.
Its particular solution
C. Its complementary solution
D.
None of these
The solution of a D.E subject to a condition satisfied at one particular point is called
A. A boundary value problem
B.
A two-point boundary value problem
C. An initial value problem
D.
A two point initial value problem
A general solution of an nth order D.E then
A. n can be zero
B.
n is any non-negative integer
C. n is any integer
D.
n is any natural number
𝑑𝑦
𝑎𝑥+𝑏𝑦+𝑐
The D.E = ′ ′
is
𝑑𝑥
𝜕𝑥
582)
𝜕𝑦
D
B
B
C
C
A
C
A
C
C
𝜕𝑧
2,1
1,2
B.
D.
2,2
None of these
If f ( x) = e2 x , f ''' ( x) =
C
6e2 x
B.
C. 8e2 x
583) d x
5 =
dx
D.
A.
D
𝑎 𝑥+𝑏 𝑦+𝑐′
A. Homogeneous
B.
Non-Homogeneous
C. Non-Linear
D.
None of these
579) The order of D.E where general solution is 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 2𝑥 + 𝐶3 𝑒 3𝑥 + 𝐶4 𝑒 4𝑥 + 𝐶5 , where 𝐶1 , 𝐶2 , 𝐶3 , 𝐶4 , 𝐶5 , are arbitrary
constant is
A. 5
B.
4
C. 3
D.
7
580) The particular integral of D.E (𝐷2 − 𝑎2 )𝑦 − 𝑐𝑜𝑠𝑎𝑥
𝑥
𝑥
A.
B.
− 𝑐𝑜𝑠𝑎𝑥
𝑠𝑖𝑛𝑎𝑥
2𝑎
2𝑎
𝑥
C.
D.
None of these
− 𝑠𝑖𝑛𝑎𝑥
2𝑎
581) The equation 𝜕2𝑢 + 𝜕2𝑢 = 𝜕𝑢 where x, y, z are variable is a partial D.E of order and degree
2
2
A.
C.
C
Discipline: ________________________
e2 x
6
none of these
B
Page 60 of 110
B.
5x ln 5
5x
ln 5
C. ln 5
D. none of these
x
5
584) If f ' (c) = 0 , then f has relative maximum value at x = c , if
A.
585)
A.
f '' (c)  0
B.
C.
f '' (c)  0
D.
A
f '' (c) = 0
none of these
The function f is neither increasing nor decreasing at a point, provided that f ' ( x) = 0 at that point, then
it is called
B
A. Critical point
B. Maximum point
C. Stationary point
D. none of these
586) The function f ( x) = ax 2 + bx + c has minimum value if
A
A. a  0
B.
a=0
C. a  0
D. none of these
587) 1 − x + x 2 − x3 + x 4 + ..... + (−1) n x n + .... is the expansion of
C
A.
B.
1
1− x
C.
D.
1
1+ x
588) If f ' ( x) = 0 , f '' (c)  0 at a point P, then P is called
1
1− x
none of these
A. Relative maximum
C. Point of inflection
589)
dy
If y = sinh −1 ( x3 ) ,
=
dx
Relative minimum
none of these
A.
x2
C.
1 + x3
3x 2
B.
D.
D
C
B.
D.
1
1 + x6
none of these
1 + x6
590) A function f ( x) is such that, f ' ( x)  0 at x = c , then f is said to be
A
A. Increasing
B. Decreasing
D.
C. Constant
none of these
'
591) A function f ( x) is such that, at a point x = 0 , f ( x) = 0 then f is said to be
C
B.
D.
A. Increasing
C. Constant
592)
If f ( x) = e
A.
x −1
Decreasing
none of these
B
, f (0) =
'
e−1x
Discipline: ________________________
B.

Page 61 of 110
C. e
593) d
(tan −1 x − cot −1 x) =
dx
A.
2
1 + x2
C.
2
1 + x2
594)
1
1
If f ( ) = tan x , f ' ( ) =
x

− 2
1
− 2
A.
C.
D.
none of these
C
B.
D.
2
1 + x2
none of these
−
A
B.
D.
1
none of these

595)
A.
C.
596)
-1
1
a
C.
598)

B.
D.
B.
C.
ln | f ' ( x) |
1
dx can be evaluated if
x+a + x
D.
x  0, a  0
x  0, a  0
B.
a
A.
C.
x2
f ' ( x)
none of these
C
D.
x  0, a  0
none of these
B
xdx =
2
ax
ln a
2
a x ln a
ax
'
600)
 e [af ( x) + f ( x)]dx =
A.
ax
 ln a
none of these
A
ln | f ( x) |
C.
none of these
B
A.

0
dx =

ax
ln a
'
f ( x)
dx =
f ( x)
A.
599)
x
B.
D.
ax
A.
597)
B
1
1
If f ( ) = Then a critical point of f is
x
x
eax f ' ( x)
Discipline: ________________________
B.
D.
2
ax
2 ln a
none of these
B
B.
eax f ( x)
Page 62 of 110
C.
601)
D.
none of these
 e [sin x + cos x]dx =
C
x
A.
C.
602)
aeax f ' ( x)
−e x sin x
e x sin x
B.
D.
e x cos x
none of these
B
2
x
 a dx =
1
A.
(a 2 − a) ln a
B.
(a 2 − a) log a
603)
1
 x ln x
D.
ln(ln x) + c
C.
ln x
604)
x+2
 x + 1 dx
B.
C.
A.
A.
C.
605)
ln( x + 1)
ln( x + 1) − x
(a 2 − a)
ln a
none of these
A
D.
ln x + c
none of these
B
B.
D.
x + ln( x + 1)
none of these
C
3
1
0 x3 + 9dx

4
C.

12
606)
x 1
 e [ x + ln x]dx =
A.
A.
e x ln x
C.
−e x
1
x
607)
1
1
x
 e [ x − x 2 ]dx =
A.
B.
D.

2
none of these
A
B.
D.
1
x
none of these
ex
A
B.
1
e x ln x
x
C.
D.
none of these
1
ex 2
x
608) If x  0, y  0 then the point P(− x, − y) lies in the quadrant
A.
ex
II
Discipline: ________________________
B.
C
II
Page 63 of 110
C.
IV
D. none of these
609) The centroid of a triangle divides each median in the ratio of
B
A.
C.
B.
1: 2
2 :1
D. none of these
1:1
610) If x and y have opposite signs then the point P( x, y) lies in the quadrants
A
A.
C.
II&IV
B. I&III
II&IV
D. none of these
611) The two intercepts form of the equation of a straight line is
A.
y = mx + c
B.
C.
D.
x y
+ =1
a b
612) The slope of the line perpendicular to ax + by + c = 0 is
A.
b
a
C.
a
b
613) The line 2 x + y + 2 = 0 and 6 x + 3 y − 8 = 0 are
B.
D.
C
y − y1 = m( x − x1 )
none of these
A
a
b
none of these
−
B
A.
C.
Perpendicular
B. Parallel
Non coplanar
D. none of these
614) If three lines pass through one common point then the lines are called
A.
C.
Parallel
Concurrent
615) 2 x + y + k = 0 (k being a parameter) represent
B.
D.
A. Two line
C. Intersecting lines
616) Equation of vertical line through (−5,3) is
B.
D.
C
Congruent
none of these
B
Family of lines
none of these
A
B.
A. x + 5 = 0
x −5 = 0
D.
C. x + 3 = 0
none of these
617) Equation of line through (−8,5) and having slope undefined is
C
B.
A. x + 8 = 0
x −5 = 0
D.
C. x − 8 = 0
none of these
618) Two lines l and l with the slope m and m , are perpendicular if
1
1
2
2
619)
A.
m1m2 = −1
B.
C.
m1m2 = 0
D.
A
m1m2 = 1
none of these
Two lines represented by ax 2 + 2hxy + by 2 = 0 are real and distinct if
A.
C.
h2 − ab  0
h2 − ab  0
Discipline: ________________________
B.
D.
C
h=0
none of these
Page 64 of 110
620)
Two lines represented by ax 2 + 2hxy + by 2 = 0 are coincident if
A
B.
h2 − ab = 0
h2 − ab  0
C.
D. none of these
h2 − ab  0
621) The lines 3 y = 2 x + 5 and 3x + 2 y − 8 = 0 intersect at an angle of
B
B.


3
2
C.
Intersect at an angle
D. none of these
622) The perpendicular distance of the line 3x + 4 y + 10 = 0 from the origin is
C
A.
A.
A.
C.
B.
1
D.
none
of these
2
2
2
623) The lines represented by ax + 2hxy + by = 0 are orthogonal if
A.
C.
624)
0
a −b = 0
a+b  0
The distance of the point (3, 7) from the y-axis is
B.
D.
B
a+b = 0
none of these
A
B.
3
−7
D. none of these
−3
625) The equation 9 x 2 + 24 xy + 16 y 2 = 0 represents a pair of lines which are
A.
C.
A.
C.
Real and distinct
B.
Real and coincident
D.
626) If a straight line is parallel to x-axis then its slope is
imaginary
none of these
A. −1
C. undefined
627) Intercept form of equation of line is
0
none of these
B.
D.
C
B
A
A.
B.
x y
+ =1
a b
C.
D.
x y
+ =0
a b
628) The perpendicular distance of a line 12 x + 5 y = 7 from
x y
− =0
a b
none of these
B
(0, 0) is
A.
B.
1
7
13
13
C.
D.
none of these
13
7
629) Line passes through the point of intersection of two lines l and l is
1
2
A.
k1l1 = k2l2
B.
C
l1 + kl2 = 2
none of these
D.
l1 + kl2 = 0
630) If 2 x + 5 y + k = 0 and kx + 10 y + 3 = 0 are parallel lines then k =
C.
Discipline: ________________________
C
Page 65 of 110
A.
C.
25
3
631) The solution of ax + by  c is
A.
C.
Closed half plane
parabola
632) The symbols used for inequality are
A.
C.
B.
D.
B
B.
D.
B.
D.
a = 0, b = 0
a = 0, b  0
634) x = 0 is the solution of the inequality
B.
D.
x0
x+40
635) The angle inscribed in a semi-circle is
B.
D.
C.
A.
C.
Open half plane
none of these
C
1
4
633) ax + by  c is not a linear inequality if
A.
2
none of these
2
none of these
A
a  0, b  0
none of these
B
2x + 3  0
none of these
C

B.

3
C.
D. none of these

2
636) The number of tangents that can be drawn from a point P ( x , y ) to a circle are
1
1
A.
A.
C.
One
B.
More than two
D.
637) Congruent chords of a circle are equi-distant form the
A.
C.
Center
B.
Tangent
D.
638) x = a cos t , y = a sin t are the parametric equations of
A.
C.
parabola
B.
circle
D.
639) x = a sec t , y = b tan t are the parametric equations of
A.
C.
parabola
ellipse
640) The parabola y 2 = −12 x opens
A.
C.
upwards
leftward
641) In the case of an ellipse it is always true that
A.
C.
a 2  b2
a 2 = b2
Discipline: ________________________
B.
D.
B
Two
none of these
A
Origin
none of these
C
ellipse
none of these
B
hyperbola
none of these
C
B.
D.
downwards
none of these
A
B.
D.
a 2  b2
none of these
Page 66 of 110
642)
B
If the associative law holds in a set, the set
A.
C.
Is a group
B. May be a group
Is not a group
D. none of these
643) An example of a group under multiplication is the set of
A.
C.
Integers
4th roots of unity
644) A group
A.
C.
B.
D.
B.
D.
Z Q
C.
RC
646) An example of a vector space is
B.
D.
A.
Q( R )
Q(Q)
647) A rational number
C.
A.
C.
May not be a real number
Is a real number
648) A real number
A.
C.
Is a rational number
May be an irrational number
649) The set of real numbers is a subset of
A.
Z
C.
C
Whole numbers
none of these
A
Is closed set
May be an empty set
645) Which one is not true
A.
C
May not be closed
none of these
D
QR
none of these
B
B.
D.
R(Q)
none of these
C
B.
D.
Is not a real number
none of these
C
B.
D.
Is not an irrational number
none of these
C
B.
D.
Q
none of these
650) [0,1] =
D
[0, [
[1, 2]
B.
C.
] − ,0]
D. none of these
651)
2
an =
is the nth term of a sequence. The sequence (an )n=1
2
n +3
A.
652)
A.
Converges
B.
Diverges
C.
May or may not converge
D.
None of these
an =
A
n +1
is the nth term of a sequence. The sequence (an )n=1
n
A. Diverges
B.
Converges
C. May or may not converge
D.
None of these
Discipline: ________________________
B
Page 67 of 110
653)
654)
A.
1 + (−1) n
is the nth term of a sequence. The sequence (an )n=1
n
converges
B.
diverges
C.
may not converge
an =
5n
an =
( n + 1)
2
D.
None of these
converges
B.
diverges
C.
may not diverge
D.
None of these

The series
B
is the nth term of a sequence. The sequence (an )n=1
A.
655)
A
5n + 2
B
 3n − 1
1
A.
converges
B.
diverges
C.
may not diverge
D.
None of these
656)
The series
657)
658)
659)


1
1
 an converges if
B
 f ( x)dx ----------
A.
Diverges
B.
Converges
C.
May not converges
D.
None of these


1
1
 an diverges if
A
 f ( x)dx --------
A.
Diverges
B.
Converges
C.
May not diverge
D.
None of these
1
n
p
B
is convergent for -----
A.
p 1
B.
C.
p =1
D.
1
n
p
p 1
None of these
A
is divergent for --------
Discipline: ________________________
Page 68 of 110
660)
A.
p 1
B.
C.
p =1
D.
2
p 1
None of these
3
C
2  2  2
+
Find the sum of th 1 +
 +
 +
3  3   3 
A.
B.
3
3+ 2
C.
D.
3
3
3− 2
None of these
3− 2
661)

The series

1
B
ln n
------n
A. Converges
C. May not converge

662)
ln n
The series 
--------1 1 + ln n
B.
D.
A. Diverges
C. May not converge
663) A sparse matrix has most of its entries are
B.
D.
A. zero
C. negative
664) A dense matrix has most of its entries are
B.
D.
Diverges
None of these
A
Converges
None of these
A
positive
None of these
A
B.
positive
A. non-zero
D. None of these
C. negative
665) A system of linear equation is said to be consistent if it has either one solution or
A. no solution
B. infinitely many solution
C. two solutions
D. None of these
666) A system of linear equation is inconsistent if it has
B
A
A. no solution
C. two solutions
667)
B. infinitely many solution
D. None of these
2 x − x2 = h, − 6x1 + x2 = k
For what values of h and k is the system 1
consistent?
A. h = −3, k = 1
C. h = 3, k = 1
668)
669)
A linear system x1 + x2 = 4, 3x1 + 3x2 = 6 has
A. no solution
C. two solutions
B.
D.
A
h = −3, k = 2
None of these
A
B.
D.
infinitely many solution
None of these
A linear system 2 x1 − 3x2 = a, 4x1 − 6 x2 = b has infinitely many solutions if
Discipline: ________________________
A
Page 69 of 110
A. 2a − b = 0
B.
C. a = b
D.
670) One of the solutions of the 10 x − 3 x − 2 x = 0 is
1
2
3
A.  1 ,1,1


3

C.  1 ,1,1 


2

671)
2a + b = 0
None of these
C
B.
1

 ,1,3 
2

D.
None of these
Which of the following is a diagonal matrix?
A. 1
0

 0
C. 0
0

0
C
0 0
2 6 
8 0 
B.
0 0
0 0 
0 0 
D.
1 0 0 
0 2 0 


 6 8 0 
None of these
1
then which of the following is true for A ?
0 
1
0 −1
A=
B.


0
0 0 
0 1 
C. A = 
D. None of these

1 0 
673) Let A =  a 
 ij  and B = bij  n p , then (i, j )th element of AB is
672)
0
If A2 = 
0
0
A. A = 
0
D
A
mn
A.
B.
n
 aik bkj
k =1
C.
k =1
674)
ik
D.
b jk
b
ki kj
None of these
If order of A is 8  7 , then the order of AAt is:
A. 7  7
C. 8  8
675)
a
k =1
n
a
n
4
0
The rank of the matrix A = 
0

0
A. 1
C. 3
C
B.
D.
78
None of these
1 8
7 7 
is
0 3

0 1
Discipline: ________________________
C
B.
D.
2
None of these
Page 70 of 110
676)
3
1
 0 −2 
 is
The rank of the matrix A = 
 5 −1


 −2 3 
B
A. 1
C. 3
677) Which of the following is an involutory matrix
A. 1 1 −1
 4 −3 4 


 3 1 1 
C.  4 1 −1
 0 −1 0 


 3 0 4 
678)
2
None of these
B
B.
D.
 0 1 −1
 4 −3 4 


 3 −3 4 
None of these
 1 −2 −6 
The matrix A =  −3 2 9  is periotic matrix having period
 2 0 −3
A. 1
C. 3
679) Let x =  x
1
x2
x3  , y =  y1
x  y =  x2 y3 − y2 x3
y2
x3 y1 − y3 x1
y3 
A
B.
2
D. None of these
be two 1  3 matrices. Its product is
A
x1 y2 − y1 x2  , then x  x =
A. 0
C. 3
680)
B.
D.
B.
D.
2
None of these
 6 3 −4 
The echelon form of matrix A =  −4 1 −6  is
 1 2 −5 
A.
C.
7

1 0 9 


0 1 2 

9


0 0 0 


7 

1 0 9 


1 1 − 6 

9


0 0 0 


Discipline: ________________________
B
B.
D.
7 

1 0
9 


0 1 − 26 

9


0 
0 0


None of these
Page 71 of 110
681)
1 0 3
The invers of the matrix A =  2 4 1  is
 1 3 0 


 − 1 3 −4 


1
5
−1

A =
−1
3
3
2
4


−1
3
3
C.
 −1 3 −4 
A−1 =  3 −1 2 
 2 −4 1 
682) Which of the following is diagonal matrix?
A.
1 0 0 
A = 0 2 6 
0 8 0 
A.
A
B.
D.
D
B.
A. Equal
C. Not equal
684)
1 0 0 
A = 0 2 0 
6 8 0 
None of these
D.
1 0 1 
A = 0 0 0 
1 0 1 
If a m  n matrix B is obtain from a m  n matrix A by a finite number of elementary row and column operations, B
then B is said to be … to A .
C.
683)


 − 1 3 −4 


5
−1

A = 3 −1

3
2


−4 9 
3

None of these
B.
D.
Equivalent
None of these
 I 0
Every nonzero m  n matrix is equivalent to a m  n matrix D =  r
 . Then D is called … form of A .
 0 0
A. Normal
B.
Canonical
C. Both A and B
D. None of these
685) A system of m linear equations Ax = B in n unknowns has a unique solution if and only if
rank( A) = rank( B) =
n
A. m
B.
C. 0
D. None of these
686) If A and B be m  n matrices over a field F and a, b  F . Then
A.
C.
687)
( a + b ) A = aA + bA
a ( bA)  ( ab ) A
B.
D.
( A + B)
2
= A2 + 2 AB + B 2
Discipline: ________________________
B.
A
A
a ( A + B )  aA + aB
None of these
If the matrices A and B are conformable for addition and multiplication, then
A.
C
C
A2 − B 2 = ( A − B )( A + B )
Page 72 of 110
C.
688)
2
 A2 + 2 AB + B 2
D.
C.
2
A =  −1
 1
2
A =  1
 0
−2 −4 
3 4 
−2 −3
2 4
0 0 
2 3 
D.
2 2 4 
A = 1 3 4 
1 2 −3
None of these
 1 − 3 −4 
The matrix A =  −1 3 4  is nilpotent. What is its nilpotency index?
 1 −3 −4 
If A is matrix over
( )
and A A
T
B.
D.
B
2
None of these
A
B.
D.
A0
None of these
B
= 0 then
A. A  0  A
C. A  0, A = 0
692)
A
B.
A. 1
C. 3
690) If A is matrix over R and AAT = 0 then
A. A = 0
C. A = I
691)
None of these
Write the matrix A that is idempotent
A.
689)
( A + B)
B.
D.
A=0= A
None of these
If rank ( A) = rank ( Ab ) , then the system Ax = b
A
A. Is consistent
B.
has unique solution
C. has infinite solutions
D. None of these
693) Let Ax = b be a system of 3 linear equations in 7 variables, then which of the following can be the
maximum value of rank ( Ab ) ?
A. 3
B.
4
C. 6
D. None of these
694) Let A be a matrix of order 4  5 and rank ( A ) = rank ( A ) = 3 , then the system Ax = b has
b
A. Unique solution
C. infinitely many solutions
695)
 −3 3   x1  0 
The system 
   =   has
 1 −1  x2  0 
A. Unique solution
C. infinitely many solutions
Discipline: ________________________
B.
D.
A
C
no solution
None of these
C
B.
D.
no solution
None of these
Page 73 of 110
696)
1 2 1 0 
If the augmented matrix of a system is 1 1 0 2  , then the system has
0 1 1 1 
A. Unique solution
C. infinitely many solutions
B.
D.
B
no solution
None of these
A system Ax = b of n equations and n unknowns has a unique solution if A is
A. Singular
B.
non-singular
C. non invertible
D. None of these
698) Let 𝐴 and 𝐵 are square matrices such that 𝐴𝐵 = 𝐼, then zero is an eigen value of
697)
A. 𝐴 but not of 𝐵
C. both 𝐴 and 𝐵
699) The eigen values of a skew-symmetric matrix are
A. negative
C. purely imaginary or zero
700) If lim a = l , a → l as
n
n
B.
D.
neither 𝐴 nor 𝐵
None of these
B.
D.
real
None of these
B
B
C
A
n →
A. n → 
C. n → 1
701)
B.
D.

B
1
The sequence   is
 n  n =1
A. divergent
C. oscillating
702) log
2 (27) =
3
A. 9
C. 6
703) A bounded sequence of real numbers
B.
D.
A. converges
C. may converge
704) A Cauchy sequence of real numbers is
B.
D.
A. not bounded
C. divergent
705) Which of the following is a linear equation?
B.
D.
A. xy = e
y = 3x
3
4
Let A = 
 −2

2
convergent
None of these
D
B.
D.
24
None of these
C
diverges
None of these
D
oscillating
None of these
B
B.
C.
706)
n →0
None of these
D.
x + y = e
None of these
2 1 −1
5 1 2 
, then A =
3 0 1

1 3 5
Discipline: ________________________
C
Page 74 of 110
A. 141
C. 149
707) Let A and B be matrices of order 6 such that
B.
D.
139
None of these
C
det ( AB 2 ) = 72 , det ( A2 B 2 ) = 144 . Then det ( A) =
A.
C.
708)
1
1
1
A.
C.
709)
0
2
a a2
b b2 = ?
c c2
B.
D.
( a − b )( a − c )( b − c )
( a − b )( c − b )( c − a )
B.
1
None of these
B
D.
( a − b )( b − c )( c − a )
None of these
1 2 3 
If A =  2 3 1  , then adj A =
 3 1 2 
 5 −1 −7 


A.  −1 −7 5 
 −7 5 −1
1 2 3 


C.  2 3 1 
 3 1 2 
710)
a−b b−c c−a
b−c
A
B.
 5 −1 0 
1 −7 5 


7 6 −8
D.
None of these
A
c−a a −b =
c−a a −b b−c
A.
C.
711)
1
2!
1
3!
1
4!
A.
C.
0
2
B.
D.
1
1
None of these
D
0
1
1 =
2!
1 1
3! 2!
0!
2!
Discipline: ________________________
B.
D.
1!
None of these
Page 75 of 110
1 2+ x
712)
A
3
1
3 + x = 0 , then the values of x
If 2
3 2+ x
1
x = −1, x = −6
x = 2, x = 3
A.
C.
B.
D.
 3 −1 0 
The eigenvalues of the matrix  −1 2 −1 are
 0 −1 3 
A. 1, 2, 3
C. 1, 4, 5
714) Let A be a square matrix of order 4  4 , then A =
x = 1, x = 6
None of these
713)
715)
B
B. 1, 3, 4
D. None of these
C
A.
−A
B. − At
C.
At
D. None of these
A
Row expansion of A ------------ column expansion of A
A. =
B. 
C. There is no comparison
D. None of these
716) Let A =  a  be a n  n triangular matrix, then A =
 ij 
B. a11 + a22 + + ann
D. None of these
A. a11 a22 ann
C. a11 − a22 − − ann
717)
For an invertible matrix A , A−1 =
A
C.
C
B. − A
A
A.
718)
A
−1
D. None of these
Let A be a square matrix of order n . A matrix obtained from A by deleting its ith row and jth column is A
again a matrix of order n − 1 which is called
A. ijth minor of A
B. ijth cofactor of A
C. Determinant of A
D. None of these
719) Let M be the ijth minor of a square matrix A of order n . Then ijth cofactor of A is
ij
B.  M ij
A. M ij
C.
720)
( −1)
i+ j
D. None of these
M ij
If A is any matrix of order n  n and k is a nonzero real number, then
A. kA = k A
B. kA = k A
kA = k n A
C.
721)
C
a11
a12
a21
a22
+
a11
C
D. None of these
b12
a21 b22
B
=
Discipline: ________________________
Page 76 of 110
A.
C.
a11
a12
a21
b22
B.
0
a12 + b12
a21
a22 + b22
D. None of these
2 1 −1
5 1 2 
, then 33th cofactor of A is
3 0 1

1 3 5
B. 34
D. None of these
3
4
Let A = 
 −2

2
A. 43
C. 56
723) 1 0 5 6
722)
0 5
a11
0
8
0 0 −1 8
B
B
=
0 0 0 3
A. 3
C. 28
724) 0
a −b
−a
0
c =
b
−c
0
B. −15
D. None of these
B
A. 1
C. −1
725)
B. 0
D. None of these
 0

If a, b, c are different numbers. For what value of x , the matrix  x − b
 x3 − c

A. 0
C. b
726)
 k 4k
Let A =  0 4
 0 0
A. 1
x+b
0
x+a
x2 + c 

x 2 − c  is singular?
0 
A
B. a
D. None of these
4
4k  . If A2 = 16 , then value of k is
4 
B
B. 1
4
C. 16
D. None of these
727) The discrete matrix on a non-empty set X is defined as
A.
B.
0 if x = y
0 if x  y
d ( x, y ) = 
d ( x, y ) = 
1 if x  y
1 if x = y
C
D. None of these
0 if x = y
d ( x, y ) = 
−1 if x  y
Discipline: ________________________
A
Page 77 of 110
728)
1
C.
729)
Minkowski’s inequality
B. Cauchy-Schwarz inequality
D. None of these
Which of the following is a system of nonhomogeneous linear equations?
A.
C.
730)
B
1
n
 n
2 2 
2 2
xk yk    xk    yk  , it is called

k =1
 k =1
  k =1

A. Cauchy inequality
n
x1 + 2 x2 = 1
2 x1 + x2 = 2
x1 + 2 x2 = 0
2 x1 + x2 = 0
B.
C
x1 − 6 x2 = 0
6 x1 + x2 = 20
D. None of these
If x1 − x2 + 2 x3 = 0, 4 x1 + x2 + 2 x3 = 1, x1 + x2 + x3 = −1 , then
A
A. x1 = 1, x2 = −1, x3 = −1
B. x1 = 0, x2 = 1, x3 = −1
C. x1 = 1, x2 = 1, x3 = 1
D. None of these
731) The system Ax = 0 of m equations and n unknowns has nontrivial solution if and only if
732)
733)
rank ( A) ------rank ( Ab ) .
B
A. =
B. 
C. 
D. None of these
734) If c  2a − 3b then this system of linear equations 2 x − x + 3x = a,3x + x − 5 x = b,
1
2
3
1
2
3
−5 x1 − 5 x2 + 21x3 = c is called
A. is consistent
B. is inconsistent
C. A and B both
D. None of these
735)
1 2 4 0 


1 1 1 0

This matrix
has
1 2 1 0 


1 3 3 0 
A. trivial solutions
B. nontrivial
C. no solution
D. None of these
736) For any matrix A the collection  x : Ax = 0 is called ----- of A
A. rank
B. solution space
C. both A and B
D. None of these
737) Which of the following is a linear equation in the variables x, y, z ?
A. x − 2 y = 0
B. x + cos y = z
C. sin x + cos y + tan z = 0
D. None of these
738) If a system of 2 equations and 2 unknown has no solution, then the graph looks like
A. intersecting lines
B. non intersecting lines
C. same lines
D. None of these
739) In Gauss-Jordan elimination method, we reduce the augmented matrix into
A. Echelon form
B. Reduced echelon form
Discipline: ________________________
B
B
B
A
B
B
Page 78 of 110
C. both A and B
D. None of these
740) Every homogenous system of linear equations
A. is consistent
B. is inconsistent
C. has only trivial solution
D. None of these
741) Let A be a 4  4 matrix and the system Ax = b has infinitely many solutions, then
A. rank ( A) = 4
B. rank ( A)  4
C. rank ( A)  4
A
C
D. None of these
If Ax = b does not have any solution, then the system is called
A. consistent
B. inconsistent
C. both A and B
D. None of these
743)
The equations of the type a11 x1 + a12 x2 = b1 , a21 x1 + a22 x2 = b2 , with b = 0 are called a system of....
A
A. homogeneous linear equations
B. non-homogeneous linear equations
C. non-linear equations
D. None of these
744) The solution of system of equations x + 5 x + 2 x = 9, x + x + 7 x = 6, −3x + 4 x = −2 is
1
2
3
1
2
3
2
3
A
742)
A. x1 = −3, x2 = 2, x3 = 1
C. x1 = 1, x2 = 1, x3 = 0
745)
746)
B. x1 = −3, x2 = 1, x3 = 0
D. None of these
The solution of system of equations 2 x1 − x2 + 3x3 = 3,3x1 + x2 − 5 x3 = 0, 4 x2 − x3 + x3 = 3 is
A. x1 = 1, x2 = 2, x3 = 1
C. x1 = 1, x2 = 1, x3 = 0
A
B. x1 = −3, x2 = 1, x3 = 0
D. None of these
 2 −1 3 4 


The solution of this matrix  0 4 1 8  is
 0 0 2 0 
A. x1 = 3, x2 = 2, x3 = 0
C. x1 = 1, x2 = 1, x3 = 0
B
A
B. x1 = −3, x2 = 1, x3 = 0
D. None of these
Let 𝑊 be the set of all points (𝑥, 𝑦) in 𝑅 2 for which 𝑥 ≥ 0 and 𝑦 ≥ 0 is
A. subspace of 𝑅 2
B. not subspace of 𝑅 2
C. not defined
D. None of these
748)
1 1
2 −4
1 −5
The dimension of the subspace of 𝑀2×2 spanned by (
),(
) and (
) is
−1 5
−5 7
−4 2
747)
A
B
A. 1
B. 2
C. 3
D. None of these
749) A is an upper triangular with all diagonal entries zero, then 𝐼 + 𝐴 is
A. invertible
C. nilpotent
750) A Newtonian fluid is defined as the fluid which
A. obeys Hook’s law
Discipline: ________________________
C
B. Idempotent
D. None of these
C
B.
is incompressible
Page 79 of 110
751)
752)
00)
753)
754)
C. obeys Newton’s law of viscosity
D. None of these
If the Reynolds number is less than 2000, the flow in a pipe is
A. Turbulent
B. Laminar
C. Transition
D. None of these
1
The continuity equation is the result of application of the following law to the flow field
A. Conservation of mass
B. Conservation of energy
C. Newton’s second law of motion
D. None of these
When a problem states “The velocity of the water flow in a pipe is 20 m/s”, which of the following
velocities is it talking about?
A. RMS velocity
B. Average velocity
C. Relative velocity
D. None of these
1
Power set topology is ----------- then any other.
B
A
B
A
A. finer
B. coarser
C. weaker topology
D. None of these
755) Let 𝜏1 & 𝜏2 are two topologies on X 𝜏1 ⊆ 𝜏2 then 𝜏1 is said to -------------.
C
A. stronger topology
B. finer topology
C. coarser topology
D. None of these
756) Let 𝜏1 & 𝜏2 are two topologies on X 𝜏1 ⊈ 𝜏2 then they are said to be -------------.
A
A. In compare able topology
B. Compare able topology
C. Finer topology
D. None of these
757) 𝜏 = {𝜑, 𝑋} be indiscrete topological space 𝐴 ⊆ 𝑋 then relative topology on A is ------.101
A. 𝜏𝐴 = {𝜑, 𝑋}
B.
𝜏𝐴 = { 𝑋}
C. 𝜏𝐴 = {𝜑, 𝐴}
D. None of these
758) 1) A={a,b} X={a,b,c} and 𝜏 = {𝜑, {𝑏, 𝑐}, 𝑋} then 𝐴̅ is equal to ------------.
A. {b}
B. X
C. {a}
D. None of these
759) 2) Let (X, 𝜏) be a topological space and 𝐴 ⊆ 𝑋 then A is closed iff -----------------.
B
A
A. 𝐴∘ = 𝐴
B.
𝐴̿ = 𝐴̅
C. 𝐴∘ = 𝐴̅
D. None of these
762) Let (X, 𝜏) be a topological space Ext A is the largest open set contain in ---------.
A. X
B.
𝐴̅
∘
C. 𝐴
D. None of these
763) Int(X-A)
1
is equal to ------------------.
Discipline: ________________________
B.
D.
B
C
A. 𝐴̿ = 𝐴̅
B.
𝐴̿ = 𝐴
̅
C. 𝐴 = 𝐴
D. None of these
760) Interior of A is union of all open set contain in ----------A. 𝐴̅
B. A
𝑑
C. 𝐴
D. None of these
761) 3) Let (X, 𝜏) be a topological space and 𝐴 ⊆ 𝑋 then A is open iff -----------------.
A. X
C. Ext(A)
C
B
A
Int(A)
None of these
Page 80 of 110
764)
Let (X, 𝜏) be a topological space and 𝐴 ⊆ 𝑋 then( ̅̅̅̅̅̅̅̅̅
𝑋 − 𝐴) =----------------.
A. X-A
C. X-Int(A)
765) Disjoint union of B and 𝐴𝑑 = -------------------.
B.
D.
A. 𝐴̅
C. A
766) Every
1
𝜏1 − 𝑠𝑝𝑎𝑐𝑒 is also a
B.
D.
A. 𝜏1 − 𝑠𝑝𝑎𝑐𝑒
C. 𝜏2 − 𝑠𝑝𝑎𝑐𝑒
767) If1 lim a = l , a → l as
n
n
B.
D.
C
Int(A)
None of these
A
Ext(A)
None of these
C
𝜏4 − 𝑠𝑝𝑎𝑐𝑒
None of these
A
n →
A. n → 
C. n → 1
768)
1
B.
D.
n →0
None of these

B
1
The sequence   is
 n  n =1
A. divergent
C. oscillating
769) log (27) =
3
A. 9
C. 6
770) A bounded sequence of real numbers
B.
D.
A. converges
C. may converge
771) A1 Cauchy sequence of real numbers is
B.
D.
A. not bounded
C. divergent
772) The set R + is equivalent to
B.
D.
A. Q
C. R
773) The interval  0,1 is equivalent to
B.
D.
A. Q
convergent
None of these
D
B.
D.
24
None of these
C
diverges
None of these
D
oscillating
None of these
C
Q+
None of these
B
B.
 2,5
C. Z
D. None of these
1 area bounded by the curves y = sin x , y = 0 , x = 0 and x =  is
The
A
A. 2
B. 0
C. undefined
D. None of these
775) The area bounded by the curves y = sin x , y = 0 , x = 0 and x = 2 is
B
774)
22)
A. 2
Discipline: ________________________
B.
4
Page 81 of 110
C. 0
D. None of these
776) If 1a  b and c  d ,
A. a + c  b + d
B.
a+c b+d
C. a + c  b + d
D. None of these
777) A1 part of many complex valued function which is single value and analytic is known as
778)
779)
780)
781)
782)
A Branch
B. Point
.
C Section
D. None of these
.
A 1function is analytic if it is a function of
A |𝑧| 𝑎𝑙𝑜𝑛𝑒
B. z alone
.
C 𝑧̅ alone
D. None of these
.
Piecewise smooth curve is also known as
A Smooth Curve
B. Regular Curve
.
C Contour
D. None of these
.
The product of complex number z and its conjugate is
A
|𝑧|3
B. |𝑧|1
.
C
|𝑧|2
D. None of these
.
1
Every
entire bounded function is
A Constant
B. Polynomial
.
C Monomial
D. None of these
.
𝑧+3
1 Singularity of 𝑓(𝑧) =
The
are
(𝑧−1)(𝑧−2)
A
B
C
C
A
C
A z = 1 ,3
B. z = 1 ,0
.
C z = 1 ,2
D. None of these
.
783) An integral curve along a simple closed curve is called a
C
A Multiple Curve
B. Jordan Curve
.
C Contour Curve
D. None of these
.
784) Bernoulli’s
1
equation cannot be applied when the flow is
A
rotational
B. turbulent
.
C
unsteady
D. all of the above
.
785) If1 a vector space 𝑉 has a basis 𝐵 = {𝑏1 , … , 𝑏𝑛 } then any set in V containing more than 𝑛 vectors must
Discipline: ________________________
C
Page 82 of 110
D
B
A Linearly independent
.
C Both A and B
.
786) A finite set that contains 0 is
B. Linearly dependent
A Linearly independent
.
C Both A and B
.
y
787)
1
If x = 10 , y =
B. Linearly dependent
D. None of these
B
D. None of these
D
A.
B.
1
ln( x)
D. None of these
1
ln(10)
C. e
788) A1 decreasing sequence
C
A. is convergent
B. is divergent
C. may diverge
D. None of these
789) For two non-empty sets A and B , the Cartesian product of A and B is denoted by
A
A. A B
B. B  A
C. AB
D. None of these
790) If x is an element of a metric space ( X , d ) and r  0 , {x  X : d ( x, x )  r} =
0
0
B
A. X − B( x0 ; r )
B. X − B( x ; r )
0
D. None of these
C. X − S ( x0 ; r )
791)
If a convergent sequence ( an ) n =1 consists of finitely many distinct elements and A = {a1 , a2 , a3 ,...} , the

A
limit of the sequence
A.  A
C. is undefined
792) Which
1
of the following is true
A.   e
793)
C.  is rational
A convergent sequence of real numbers is
A. unbounded
C. oscillating
794)
1 sin(2 x)
=
42) lim
x →0
3x
A. 1
C. 3
2
Discipline: ________________________
B.  A
D. None of these
D
22
7
D. None of these
B.  =
B
B. Cauchy
D. None of these
D
B. 0
D. None of these
Page 83 of 110
795)
+
1
The function f : R → R defined by f ( x) = ln x is
A. decreasing
C. increasing
796)
B. constant
D. None of these
x
+
The function f : R → R defined by f ( x) = e is
A. one to one
C. decreasing
797)
798)
799)
800)
801)
802)
C
A
B. not one to one
D. None of these
𝜋1⁄
∫0 4 𝜃𝑠𝑒𝑐 2 𝜃 𝑑𝜃=
A. 𝜋 1
+ 𝑙𝑛2
4 2
C. 𝜋 1
+ 𝑙𝑛2
4 2
1 partial differential equations in 𝑝 + 𝑞 = 𝑧 2 , is
The
A. of order 1 and is linear
C. of order 2 and is not linear
1 vertex of the equation 𝑦 2 = 4𝑎𝑥 is?
The
A. (1, 1)
C. (0, 0)
1 is the axis of the parabola 𝑦 2 = 4𝑎𝑥 ?
What
A. x=0
C. x=a
1 𝑎𝑘 diverges then
If ∑
A. ∑|𝑎𝑘 | diverges
C. ∑|𝑎𝑘 | absolutely converges
1
Which
of the following statement is not true?
A. Any sequence has a unique limit.
C
B. 𝜋 + 𝑙𝑜𝑔2
4
D. None of these
A
------B. of order 1 and is not linear
D. None of these
C
B. (2, 2)
D. None of these
B
B. y=0
D. None of these
A
B. ∑|𝑎𝑘 | converges
D. None of these
B
B. The set S = {0, 1} has exactly two accumulation
points.
C. There exist a sequence of rational numbers that D. None of these
has an irrational limit.
803) The continuity equation is the result of application of the following law to the flow field
A. Conservation of mass
B. Conservation of energy
C. Newton’s second law of motion
D. None of these
804) The series converges absolutely if
A.
B.
x 1
x 1
C.
both A and B
D.
805)The series diverges absolutely if
A.
x 1
C.
both A and B
A
None of these
B
B.
D.
x 1
None of these

806)
If the power series  cn x n converges for x = x1 , then it converges absolutely for all x such that
n =0
A.
x  x1
B.
C.
x = x1
D.
Discipline: ________________________
A
x  x1
None of these
Page 84 of 110
A

807)
If the power series  cn x n diverges for x = x1 , then it diverges for all x such that
B
n =0
A.
x  x1
B.
C.
x = x1
D.
None of these

808)
Let the power series
c x
n =0
n
n
C.

+ cn x n +
then
f is continuous
f is integrable
A.
x
D
radius convergence R and
f ( x ) = c0 + c1 x + c2 x 2 +
809)
x  x1
B.
D.
f is differentiable
all of these
A
n
 n! = …
n =0
A.
ex
B.
C.
0
D.
A
810) 1
=…
1− x
A. 1 + x + x 2 + x3 +
C.
811)
Both A and B
tan −1 ( x ) = x −
3
e2x

5
7
x
x
x
+
−
+
3
5
7
+ ( −1)
n
2 n +1
x
+
2n + 1
B.
1 − x − x 2 − x3 −
D.
None of these
B
, −1  x  1
This is known as …
A.
C.
812)
Cauchy series
both A and B


 a x + b x
n
n =0
A.
n
n =0
n
n
B.
D.
None of these
B
=

 ( an − bn ) x n
B.
both A and B
D.
n =0
C.
Gregory’s Series

(a
n
n =0
+ bn ) x n
None of these

813) 
n 
n 
a
x
  n   bn x  =
 n =0
 n =0


A

A.
 ( cn ) x n
B.
x
C.
both A and B
D.
None of these
n=0
814) 1
n
n=0
C
2 −x
 x e dx =
2
0
A.
C.
0.155
0.187
Discipline: ________________________
B.
0.167
D.
None of these
Page 85 of 110
A
815) 1
2
1
1+ x
4
dx =
0
B.
0.494
1.454
C. 2.434
D.
None of these
816)Let U and V be two vector spaces over the same field F . Then a map T : U → V is called a linear
A.
transformation if:
A.
a) T (u + v) = T (u) + T (v)
for
all B.
a) T (au + v) = aT (u) + T (v)
D
for
all
a  F and u, v  U
u, v  U
T (au) = aT (u) for all a  F , u U
C.
a) T (au + bv) = aT (u ) + bT (v)
for all D.
All of these
a  F and u, v  U
817)The linear transformation is called
A. Linear mapping
B.
linear function
C. vector space homomorphism
D.
all of these
818)A linear transformation T : V → V is called a zero transformation if:
A. T (v) = v for all v  V
B.
T (v) = 0 for all v  V
2
C.
D.
T (v) = 1 for all v  V
T (v) = v for all v  V
D
819)T : U → V be linear transformation then:
C
A.
T (u + v) = T (u) + T (v) for all u, v  U
B.
C.
both A and B
D.
820)A mapping T : R3 → R3 is not a linear transformation if:
A. T ( x, y, z ) = ( x − y, y − z , z − x )
B.
C.
T ( x, y, z ) = ( x + y, x − y, z + 1)
D.
A
T (u − v) = T (u) − T (v) for all u, v  U
None of these
C
T ( x, y, z ) = ( x + y,3z, 0 )
None of these
821)A one-one linear transformation is called …
A. Injective linear transformation
B.
bijective linear transformation
C. surjective linear transformation
D.
None of these
822)A onto linear transformation is called …
A. Injective linear transformation
B.
bijective linear transformation
C. surjective linear transformation
D.
None of these
823)A one-one and onto linear transformation is called …
A. Injective linear transformation
B.
bijective linear transformation
C. surjective linear transformation
D.
None of these
824)A one-one and onto linear transformation is called …
A. bijective linear transformation
B.
vector space homomorphism
C. both A and B
D.
None of these
825)T1 : U → V and T2 : U → V are said to be equal linear transformations if
Discipline: ________________________
Page 86 of 110
A
B
C
C
B
A.
T1 ( u )  T2 ( u ) for all u U
B.
T1 ( u ) = T2 ( u ) for all u U
C.
T1 ( u )  T2 ( u ) for all u U
D.
T1 ( u )  T2 ( u ) for all u U
826) For any field F , F n  F m if and only if
A.
n=m
C.
A
B.
D.
both A and B
nm
None of these
827) Two finite dimensional vector spaces U and V over F are isomorphic and dimU = 5 then
B.
dimV = 5
D.
dimV = 3
828)There is no one-one onto linear transformation from
A.
B.
R 2 to R 3
A.
C.
C.
both A and B
D.
829)The vectors (1, −2,3), (5,6, −1) and (3,2,1) are
A. Linearly independent
dimV = 4
None of these
C
3
4
R to R
None of these
B
C.
B.
D.
A.
C.
B.
D.
Both A and B
830)The vectors (1,2,2, −1), (4,9,9, −4) and (5,8,9, −5) are
Linearly independent
Both A and B
Linearly dependent
None of these
A
Linearly dependent
None of these
2
831) The polynomials 𝑝1 = 1 − 𝑥, 𝑝2 = 5 + 3𝑥 − 2𝑥 and 𝑝3 = 1 + 3𝑥 − 𝑥 2 are
A.
C.
Linearly independent
Both A and B
832)A finite set that contains 0 is
A. Linearly independent
Both A and B
833)A set of vectors {𝑥, 𝑠𝑖𝑛𝑥} is
C.
A.
C.
Linearly independent
Both A and B
834)A set of vectors {𝑠𝑖𝑛2𝑥, 𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥} is
A.
C.
Linearly independent
Both A and B
B.
D.
B
Linearly dependent
None of the
B
B.
D.
Linearly dependent
None of these
A
B.
D.
Linearly dependent
None of these
B
B.
D.
Linearly dependent
None of these
1
835)
For what value(s) of ℎ will 𝑦 be in the subspace of 𝑅 3 spanned by 𝑣1 , 𝑣2 , 𝑣3 if 𝑣1 = (−1) , 𝑣2 =
−2
−4
−3
5
(−4) , 𝑣3 = ( 1 ) and 𝑦 = ( 3 )
0
ℎ
−7
A.
C.
ℎ = −5,5
ℎ = −1,0, −1
B.
D.
Null set
Non trivial
B.
D.
None of these
A
trivial solution
None of these
837)𝑁𝑢𝑙𝑙(𝐴) = {0} if and only if the equation 𝐴𝑥 = 0has only the
A.
C.
Null set
Non trivial
B.
D.
B
trivial solution
None of these
838) 𝑁𝑢𝑙𝑙(𝐴) = {0} if and only if the linear transformation 𝑥 → 𝐴𝑥 is
Discipline: ________________________
B
ℎ=5
836)The set of all solutions of the homogenous equation 𝐴𝑥 = 0 is known as
A.
C.
A
A
Page 87 of 110
A.
C.
one-one
Subspace
B.
D.
onto
None of these
839)If a vector space 𝑉 has a basis 𝐵 = {𝑏1 , … , 𝑏𝑛 } then any set in V containing more than 𝑛 vectors must
A. Linearly independent
B.
Linearly dependent
C.
Both A and B
D.
None of these
840)If a vector space 𝑉 has a basis of 𝑛 vectors, then every basis of 𝑉 must consist of exactly
A. 𝑛 vectors
B.
𝑛 − 1 vectors
C. 𝑛 + 1 vectors
D.
None of these
841) The dimension of zero vector space is
A.
C.
Zero
Not defined
B.
D.
Equal
B.
3
8
B.
D.
A
A
A
Not equal
D
5
None of these
847)Could a 6 × 9 matrix have a two dimensional null space
A.
C.
Yes
may or may not be
B.
D.
B
No
None of these
848) Let 𝐴 be a an 𝑛 × 𝑛 invertible matrix then Null(A) is
{0}
A.
C.
Infinite
849)
𝑢=(
A.
C.
850)
Yes
may or may not be
𝑢=(
A.
C.
6
1
) is an eigen vector of 𝐴 = (
−5
5
3
1
) is an eigen vector of 𝐴 = (
−2
5
Yes
may or may not be
A
{}
B.
D.
None of these
6
) or not?
2
B.
D.
A
No
None of these
6
) or not?
2
B.
D.
B
No
None of these
851)The set {sin2 𝑥 , cos2 𝑥 , 5} is
A. Linearly independent
B.
Linearly dependent
C. Both A and B
D.
None of these
852) If 𝑢1 = (−1,2,4) and 𝑢2 = (5, −10, −20)then the set {𝑢1 , 𝑢2 } is
A.
C.
Linearly independent
Both A and B
Discipline: ________________________
A
B
D.
None of these
𝑚 and 𝑛
846)If 𝐴 is a 7 × 9 matrix with a two-dimensional null space, then rank of 𝐴 is
A.
C.
A
Infinite
None of these
842)The standard basis for the polynomial of degree 𝑛 has
A. 𝑛 vectors
B.
𝑛 + 1 vectors
C. infinite vectors
D.
None of these
843)Any subspace spanned by a single nonzero vector. Such subspaces are
A. lines through origin
B.
planes through origin
C. not defined
D.
None of these
844)The rank of A is the dimension of the
A. column space of A
B.
planes through origin
C. Polynomials
D.
None of these
845)The dimensions of the column space and the row space of an 𝑚 × 𝑛 matrix A are
A.
C.
B
B.
D.
B
B
Linearly dependent
None of these
Page 88 of 110
853) If 𝑢1 = (3, −1), 𝑢2 = (4,5) and 𝑢3 = (−4,7) then the set {𝑢1 , 𝑢2 , 𝑢3 } is
B
A.
C.
B.
Linearly independent
Linearly dependent
D.
Both A and B
None of these
2
2
854) If 𝑝1 = 3 − 2𝑥 + 𝑥 and 𝑝2 = 6 − 4𝑥 + 2𝑥 then the set {𝑝1 , 𝑝2 } is
B
A.
C.
B.
Linearly independent
Linearly dependent
D.
Both A and B
None of these
855) If 𝐴 = (−3 4) and 𝐵 = ( 3 −4) in 𝑀 , then the set {𝐴, 𝐵} is
22
2 0
−2 0
B
A.
C.
B.
Linearly independent
Linearly dependent
D.
Both A and B
None of these
856) The vectors (3,8,7, −3), (1,5,3, −1), (2, −1,2,6), (4,2,6,4)in 𝑅 4 are
A
A.
C.
B.
Linearly independent
Linearly dependent
D.
Both A and B
None of these
857)The Wronskian of 𝑓1 = sin 𝑥 , 𝑓2 = cos 𝑥 and 𝑓3 = 𝑥 cos 𝑥 is
2 sin 𝑥
A.
C.
zero
858)The Wronskian of 𝑓1 = 1, 𝑓2 = 𝑥 and 𝑓3 = 𝑒 𝑥 is
A.
C.
𝑥𝑒
𝑥
zero
859)The Wronskian of 𝑓1 = 1, 𝑓2 = 𝑥 and 𝑓3 = 𝑥 2 is
A.
C.
2
Zero
B.
D.
1
1
1
1
1
2
2
2
2
2
2
sin 2𝑥
None of these
B
B.
D.
𝑒
𝑥
None of these
A
B.
D.
1
A
𝑒
𝑥
None of these
A
860)If (𝜆, − , − ) , (− , 𝜆, − ) , (− , − , 𝜆) are linearly independent then
A.
C.
1
1
𝜆 = 1, −
2
D.
𝜆 = −2
861)The vectors (−3,7) and (5,5) in 𝑅 2 form
A. Basis for 𝑅 2
𝜆 = 1, −1
B.
None of these
A
Linearly dependent set
None of these
862) If 𝑊 is subspace of a finite dimensional vector space 𝑉 then 𝑊 is
C.
Infinite set
B.
D.
A
A.
C.
B.
Infinite dimensional
finite dimensional
D.
Basis for 𝑉
None of these
863) 𝑣3 can be added to linearly independent sets (1, −2,3), (0,5, −3) to form basis then
A.
C.
𝑣3 = (0,0,1)
B.
A
𝑣3 = (0,0,0)
D.
Both A and B
None of these
864) The area of the triangle formed by the tangent and the normal to the parabola y2 = 4ax both drawn at
the same end of the latus rectum and the axis of the parabola is
Discipline: ________________________
Page 89 of 110
C
A.
2
C.
B.
2a2
D.
None of these
2
a
4a2
865) The locus of the point from which tangents to a parabola are at right angles is a
A. Straight line
B.
Circle
C.
D.
None of these
Pair of straight line
A
866) Given the two ends of the latus rectum, the maximum number of parabolas that can be drawn is
A. 1
B.
2
C.
D.
Infinity
0
867) If the focus of the parabola is (–2, 1) and the directrix has the equation x + y = 3 then the vertex is
B
C
(–1, 1/2)
A. (0, 3)
B.
C.
D.
(–1, 2)
(2, –1)
868) The shortest distance between the parabola y2 = 4x and the circle x2 + y2 + 6x – 12y + 20 = 0 is
A.
B.
0
D.
1
A
4
C.
3
+5
869) If line y = 2x + 1/4 is tangent to y2 = 4ax, then a is equal to
A. 1/ 2
B.
1
C.
D.
None of these
2
A
870) The Cartesian equation of the curve whose parametric equations are x = t2 + 2t + 3 and y = t + 1 is
A.
y = ( x– 1) 2 + 2( y–1) + 3
B.
x = ( y – 1)2 + 2( y–1) +5
C.
x = y2 +2
D.
None of these
871) Normal at point to the parabola y2 = 4ax where abscissa is equal to ordinate, will meet the parabola
again at a point
Discipline: ________________________
Page 90 of 110
C
D
A. (6a, – 9a)
B.
(–6a, 9a)
D.
(9a, – 6a)
(–9a, 6a)
C.
872) The tangents from the origin to the parabola y2 + 4 = 4x inclined of
A. 𝜋/6
B.
𝜋/4
𝜋/3
D.
𝜋/2
C.
B
873) If the line y = x + k is a normal to the parabola y2 = 4x then k can have the value
A. 2√2
B.
4
C.
D.
3
-3
C
874) The number of tangents to the parabola y2 = 8x through (2, 1) is
A. 0
B.
1
C.
D.
None of these
2
A
875) The graph represented by the equations x = sin2t, y = 2 cost is
A.
Parabola
B.
Circle
C.
Hyperbola
D.
None of these
A
876) The point of intersection of the tangents of the parabola y2 = 4x at the points, where the parameter t
has the value 1 and 2 are
A.
(3, 8)
B.
(4, 5)
C.
(2, 3)
D.
(4, 6)
877) If (2, 0) is the vertex and y– axis the directrix of the parabola, then the focus is
A. (2, 0)
B.
(-2, 0)
C.
D.
(-4, 0)
(4, 0)
C
878) The equation of the parabola whose vertex and focus lie on the x– axis at distances a and a1 from the
origin respectively, is
A. y2 = 4(a1 – a)x
C.
y2 = 4(a1 – a) (x – a1)
Discipline: ________________________
B.
y2 = 4(a1 – a) (x – a)
D.
None of these
C
Page 91 of 110
B
879) If the line x – 1 = 0 is the directrix of the parabola y2 – kx + 8 = 0, then one of the values of k is
A. 1/8
B.
8
C.
D.
1/4
4
880) If the point P (4, – 2) is one end of the focal chord PQ of the parabola y2 = x, then the slope of the
tangent at Q is
A. -1/4
B.
1/4
C.
D.
-4
4
881) The line
intersects the circle
at the most of ------------points.
A. 1
B.
2
C.
D.
4
3
882) The eccentricity of an ellipse is
e=1
B.
e<1
C.
e>1
D.
0<e<1
883) The perpendicular distance from the point (3,−4) to the line
A. 7
B.
8
C.
D.
10
A. 8
B.
12
C.
D.
None of these
B
885) The line perpendicular to the tangent line is called
A
A. normal line
B.
secant line
C.
D.
derivative
limit
886) The point of a parabola which is closest to the focus is the __________ of the parabola.
A. vertex
B.
latus rectum
C.
D.
eccentricity
directrix
Discipline: ________________________
B
A
884) What is the length of latus rectum If the distance between vertex and focus is 3?
4
C
D
A.
9
C
Page 92 of 110
A
887) The center of the circle
A. (2,−3)
B.
(−2,3)
(−4,6)
D.
(4,−6)
C.
D
is ?
888) Which point of a parabola is closest to the focus is?
A
A.
directrix
B.
vertex
C.
eccentricity
D.
latus rectum
889) If the distance between vertex and focus is 3, then the length of latus rectum is?
A. 6
B.
8
C.
D.
12
10
890) The focus of the parabola
B
is ?
A. (0,0)
B.
(1,0)
C.
D.
(1,1)
(0,1)
891) If the discriminant of a conic is
, then it represents a
A. circle
B.
parabola
C.
D.
ellipse
hyperbola
892) The radius of the circle
B
C
is ?
A. √57
B.
√67
C.
D.
√87
√77
D
893) A line which is perpendicular to base of cone and passes through vertex of cone is called........... of
cone
A. rulings
B.
nap
C.
D.
axis
vertex
894) If the cutting plane is parallel to the generator of the cone and cut only one nap is called
A. Circle
B.
hyperbola
C.
D.
ellipse
parabola
Discipline: ________________________
Page 93 of 110
D
895) The perpendicular distance from the point (3, -4) to the line 3x – 4y + 10 = 0
A. 7
B.
8
C.
D.
10
9
A
896) The locus of the point from which the tangent to the circles x² + y² – 4 = 0 and x² + y² – 8x + 15 = 0
are equal is given by the equation
A. 8x + 19 = 0
B.
8x – 19 = 0
4x – 19 = 0
D.
4x + 19 = 0
C.
897) The number of tangents that can be drawn from (1, 2) to x² + y² = 5 is
A. 0
B.
1
C.
D.
More than 2
2
B
898) The equation of parabola whose focus is (3, 0) and directrix is 3x + 4y = 1 is
D
A. 16x² – 9y² – 24xy – 144x + 8y + 224 = 0
B.
16x² + 9y² – 24xy – 144x + 8y – 224 = 0
16x² + 9y² – 24xy – 144x – 8y + 224 = 0
D.
16x² + 9y² – 24xy – 144x + 8y + 224 = 0
C.
899) The center of the ellipse (x + y – 2)² /9 + (x – y)² /16 = 1 is
A.
(0, 0)
B.
(0, 1)
C.
(1, 0)
D.
(1, 1)
D
900) The equation of parabola with vertex at origin the axis is along x-axis and passing through the point
(2, 3) is
A.
y² = 9x
B.
y² = 9x/2
C.
y² = 2x
D.
y² = 2x/9
901) At what point of the parabola x² = 9y is the abscissa three times that of ordinate
A.
(1, 1)
B.
(3, 1)
C.
(-3, 1)
D.
(-3, -3)
Discipline: ________________________
B.
B
B
902) A man running a race course notes that the sum of the distances from the two flag posts from him is
always 10 meter and the distance between the flag posts is 8 meter. The equation of posts traced by
the man is
A. x²/9 + y²/5 = 1
B
x²/9 + y2 /25 = 1
Page 94 of 110
D
C.
x²/5 + y²/9 = 1
D.
x²/25 + y²/9 = 1
903) In an ellipse, the distance between its foci is 6 and its minor axis is 8 then its eccentricity is
A. 4/5
B.
1/√52
C.
D.
1/2
3/5
904) If the length of the tangent from the origin to the circle centered at (2, 3) is 2 then the equation of the
circle is
A.
(x + 2)² + (y – 3)² = 3²
B.
(x – 2)² + (y + 3)² = 3²
C.
(x – 2)² + (y – 3)² = 3²
D.
(x + 2)² + (y + 3)² = 3²
905) The parametric representation (2 + t², 2t + 1) represents
B.
a hyperbola
C.
D.
a circle
906) If a parabolic reflector is 20 cm in diameter and 5 cm deep then the focus of parabolic reflector is
A. (0 0)
B.
(0 5)
C.
D.
(5 5)
(5 0)
907) The parametric coordinate of any point of the parabola y² = 4ax is
A.
(-at², -2at)
B.
(-at², 2at)
C.
(a sin²t, -2a sin t)
D.
(a sin t, -2a sin t)
A. 10y = x² + 4x + 16
B.
12y = x² + 4x + 16
C.
D.
12y = x² + 4x + 8
B
909) The equation of a hyperbola with foci on the x-axis is
B
A. x²/a² + y²/b² = 1
B.
x²/a² – y²/b² = 1
C.
D.
x² – y² = (a² + b²)
x² + y² = (a² + b²)
910) The line lx + my + n = 0 will touches the parabola y² = 4ax if
A. ln = am²
B.
ln = am
C.
D.
ln = a² m
ln = a² m²
Discipline: ________________________
C
C
908) The equation of parabola with vertex (-2, 1) and focus (-2, 4) is
12y = x² + 4x
C
A
A. a parabola
an ellipse
C
A
Page 95 of 110
911) The center of the circle 4x² + 4y² – 8x + 12y – 25 = 0 is
A
A.
(2,-3)
B.
(-2,3)
C.
(-4,6)
D.
(4,-6)
912) The radius of the circle 4x² + 4y² – 8x + 12y – 25 = 0 is
C
A. √57/4
B.
√77/4
√77/2
D.
√87/4
C.
913) If (a, b) is the mid point of a chord passing through the vertex of the parabola y² = 4x, then
A. a = 2b
B.
2a = b
C.
D.
2a = b²
a² = 2b
914) A rod of length 12 CM moves with its and always touching the co-ordinate Axes. Then the equation
of the locus of a point P on the road which is 3 cm from the end in contact with the x-axis is
A.
x²/81 + y²/9 = 1
B.
x²/9 + y²/81 = 1
C.
x²/169 + y²/9 = 1
D.
x²/9 + y²/169 = 1
915) If the diameter of cylinder is 8cm and its height is 16cm then the volume of cylinder is
A. 804.352cm³
B.
1000cm³
C.
D.
850cm³
900cm³
916) If the volume of cylinder is 900cm² with the height of 20cm then the diameter of the cylinder is
A. 24cm²
B.
7.57cm²
C.
D.
12.23cm²
9.57cm²
917) If a rectangular tank is 21cm long, 13cm wide and 18cm high and contains water up to a height of
11cm then the total surface area is
A. 1450cm²
Discipline: ________________________
B.
1350cm²
Page 96 of 110
D
A
A
B
D
1200cm²
C.
1021cm²
D.
918) If circular metal sheet is 0.65cm thick and of 50cm in diameter is melted and recast into cylindrical
bar with 8cm diameter then the length of bar will be
A. 24.41cm
B.
35.41cm
C.
D.
30.41cm
40.41cm
919) If a cuboid is 3.2 cm high, 8.9cm long and 4.7 wide then total surface area is
A. 170.7cm²
B.
180cm²
C.
D.
325.8cm²
205.7cm²
A
920) By converting the 5.6m² into the cm², the answer will be
B
A. 0.0056cm²
B.
5600cm²
C.
D.
560cm²
56000cm²
921) A rectangular field is 40m long and 30m wide. The perimeter of rectangular field is
200m²
A.
A
D
180m²
B.
160m²
C.
140m²
D.
922) By converting the 0.96km² into m²(meter square), the answer will be
A. 9600m²
Discipline: ________________________
B.
B
960m²
Page 97 of 110
C.
0.96m²
D.
960000m²
923) By converting the 4.8mm² into the cm², the answer will be
A. 0.048cm²
B.
0.48cm²
C.
D.
480cm²
48cm²
A
924) By converting the 80.2km² into the hectare, the answer will be
A. 0.08020ha
B.
8020ha
C.
D.
0.802ha
802ha
B
925) The flat surface in which two points are joined by using straight line is classified as
line
A.
D
ray
B.
intersecting line
C.
plane
D.
926) If the line segment is extended in two directions indefinitely from each of the two points then it is
classified as
A. intersecting line
B.
plane
C.
D.
ray
line
927) The type of quadrilateral which has one pair of parallel sides is called
A. triangle
Discipline: ________________________
B.
C
D
semi-circle
Page 98 of 110
C.
parallelogram
D.
trapezium
928) If the base of parallelogram is 19cm and the height is 11cm then the area of parallelogram is
A. 105cm²
B.
209cm²
C.
D.
170cm²
110²
929) If the width of rectangle is 10cm lass than its length and its perimeter is 50cm then the width of
rectangle is
58cm²
A.
B
C
64cm²
B.
15cm²
C.
30cm²
D.
930) Converting the cm² into the m², the 6.5cm² is equal to
A
A. 0.00065m²
B.
0.0065m²
C.
D.
65m²
0.65m²
931) By converting the 78580m² into hectare(ha), the answer will be
A. 785.80ha
B.
0.0007858ha
C.
D.
78.580ha
0.07858ha
B
932) If the length of a square field is 12cm then the perimeter of square will be
A. 48cm²
Discipline: ________________________
B.
A
24cm²
Page 99 of 110
C.
36cm²
D.
50cm²
933) If the area of circle is 112m² then the circumference of the circle is
27.68m²
A.
B
37.68m²
B.
50.68m²
C.
55.68m²
D.
934) By converting the 62.7m² into km², the answer will be
A
A. 0.0000627km²
B.
0.0627km²
C.
D.
6270km²
0.00627km²
935) If the length of rectangle is 15cm and width of rectangle is 5cm then the area of rectangle is
A. 35cm²
B.
40cm²
C.
D.
70cm²
75cm²
936) By converting the 85600mm² into m²(meter square), the answer will be
A. 8560m²
B.
856m²
C.
D.
8560000m²
0.0856m²
C
937) By converting the 60.8cm² into the mm², the answer will be
A. 0.0608mm²
Discipline: ________________________
B.
C
C
0.608mm²
Page 100 of 110
6080mm²
C.
608mm²
D.
938) If the diameter of a truck wheel is 0.65m and truck is travelling at 40km⁄h then the number of
revolutions per minute made by truck wheel is
A. 3223
B.
5223
C.
D.
8500
6223
939) If the height of trapezium is 8cm and the sum of parallel sides is 16cm then the area of trapezium is
A. 85cm²
B.
54cm²
C.
D.
32cm²
64cm²
940) The kind of quadrilateral in which opposite pairs of the sides are parallel and equal is called
A. parallelogram
B.
trapezium
C.
D.
semi-circle
triangle
941) Volume of seashells, pebbles and keys can be measured by
A
C
A
B
displacement method
A. measuring cylinder
B.
Vernier caliper
C.
measuring flask
D.
942) The apparatus commonly used to measure volume of liquids is
A. measuring cylinder
Discipline: ________________________
B.
A
measuring tapes
Page 101 of 110
C.
jar
D.
cylinder
943) The amount of 1 liter contains
B
A. 100ml
B.
1000ml
C.
D.
10kg
10mm
944) The volume is measured by help of a curve made in measuring cylinder called
A. meniscus curve
B.
round curve
C.
D.
volume curve
slanting curve
A
945) The volume of liquids can be measured by using different instruments which includes
D
cylinders
A.
B.
burettes or pipettes
C.
volumetric flasks
all of them
D.
946) A cylindrical pencil sharpened at one edge is the combination of
A
A.
a cone and a cylinder
B.
frustum of a cone and a cylinder
C.
a hemisphere and a cylinder
D.
two cylinders
947) A cone is cut through a plane parallel to its base and then the cone that is for medon one side of that
plane is removed. The new part that is left over on the other side of the plane is called
A.
a frustum of a cone
B.
cone
C.
cylinder
D.
sphere
948) During conversion of a solid from one shape to another, the volume of the new shape will
Discipline: ________________________
Page 102 of 110
A
C
A. increase
B.
decrease
C.
D.
be doubled
remain unaltered
949) A right circular cylinder of radius r cm and height h cm (h>2r) just encloses a sphere of diameter
A.
r cm
B.
2r cm
C.
h cm
D.
2h cm
950) A hollow cube of internal edge 22cm is filled with spherical marbles of diameter 0.5 cm and it is
assumed that 1/8th space of the cube remains unfilled. Then the number of marbles that the cube can
accommodate is
A. 142244
B.
142396
C.
D.
142596
142496
951) . A metallic spherical shell of internal and external diameters 4 cm and 8 cm respectively, is melted
and recast into the form of a cone with base diameter 8cm. The height of the cone is
A. 12cm
B.
14cm
C.
D.
18cm
15cm
952) A solid piece of iron in the form of a cuboid of dimensions 49cm × 33cm × 24cm, is mound to form a
solid sphere. The radius of the sphere is
A. 21cm
B.
23cm
C.
D.
19cm
25cm
953) If two solid hemispheres of same base radii r, are joined together along their bases, then curved
surface area of this new solid is
A. 4πr2
B.
6πr2
3πr2
D.
8πr2
C.
954) A solid cylinder of radius r and height h is placed over other cylinder of same height and radius. The
total surface area of the shape so formed is
A. 4πrh + 4πr2
B.
4πrh − 4πr2
4πrh + 2πr2
D.
4πrh − 2πr2
C.
955) The radii of the top and bottom of a bucket of slant height 45cm are 28cm and 7 cm respectively. The
curved surface area of the bucket is
Discipline: ________________________
Page 103 of 110
B
A
B
A
A
C
A
A. 4950 cm2
B.
4951 cm2
4952 cm2
D.
4953 cm2
C.
956) A medicine-capsule is in the shape of a cylinder of diameter 0.5 cm with two hemispheres stuck to
each of its ends. The length of entire capsule is 2 cm. The capacity of the capsule is
A. 0.36 cm3
B.
0.35 cm3
0.34 cm3
D.
0.33 cm3
C.
A
957) Twelve solid spheres of the same size are made by melting a solid metallic cylinder of base diameter 2 C
cm and height 16 cm. The diameter of each sphere is
A. 4 cm
B.
3 cm
C.
D.
6 cm
2 cm
958) The diameters of the two circular ends of the bucket are 44 cm and 24 cm. The height of the bucket is
35 cm. The capacity of the bucket is
A.
32.7 litres
B.
33.7 litres
C.
34.7 litres
D.
31.7 litres
959) Volumes of two spheres are in the ratio 64:27. The ratio of their surface areas is
A.
3:4
B.
4:3
C.
9 : 16
D.
16 : 9
D
960) A mason constructs a wall of dimensions 270cm× 300cm × 350cm with the bricks each of size
22.5cm × 11.25cm × 8.75cm and it is assumed that 1/8 space is covered by the mortar. Then the
number of bricks used to construct the wall is:
A. 11100
B.
11200
C.
D.
11300
11000
961) O is the point of intersection of two equal chords ABand CD such that OB = OD, then triangles OAC
and ODB are
A. Equilateral but not similar
B.
Isosceles but not similar
C.
D.
Isosceles and similar
Equilateral and similar
962) D and E are respectively the midpoints on the sides AB and AC of a triangle ABC and BC = 6 cm. If
DE || BC, then the length of DE (in cm) is
Discipline: ________________________
A
Page 104 of 110
B
D
B
A.
2.5
B.
3
C.
5
D.
6
963) Areas of two similar triangles are 36 cm2 and 100 cm2. If the length of a side of the larger triangle is
20 cm, then the length of the corresponding side of the smaller triangle is
A. 12cm
B.
13cm
C.
D.
15cm
14cm
964) Inverse Laplace transformation 𝑓(𝑠) = 4/(𝑠 2 − 2𝑠 − 3) of
A.
C.
2𝑒 3𝑡− 𝑒 𝑡
a2
𝑒 −3𝑡 𝑒 −𝑡
B.
𝑒 −3𝑡− 𝑒 𝑡
D.
𝑒 3𝑡− 𝑒 −𝑡
C
965) Divergence operation result will always
B
A. vector
B.
Scalar
C.
D.
None of these
Vector or Scalar
966) The rectangular coordinate system is also called
A. Polar coordinate system
C.
Cylindrical coordinate system
B
B.
Cartesian coordinate system
D.
Spherical coordinate system
967) What is the minimum number of coplanar vectors with different magnitudes that can be added to get
a resultant of zero?
A. 𝑖̂ + 𝑗̂ + 5𝑘̂
B.
𝑖̂ + 𝑗̂
D.
C.
𝑖̂ + 𝑗̂ + 10𝑘̂
B
A. Distance
B.
Momentum
C.
D.
Acceleration
969) Cross product is also known as?
A. scalar product
Discipline: ________________________
C
2𝑖̂ + 4𝑗̂ + 6𝑘̂
968) ---------- is a scalar quantity
Torque
C
B
B.
vector product
Page 105 of 110
C.
dot product
D.
multiplication
970) What is the area of the parallelogram which represented by vectors P⃗ =2𝑖̂ + 3𝑗̂ and Q⃗ =𝑖̂ + 4𝑗̂
A.
5 units
B.
10 units
C.
15 units
D.
20 units
971) If it is not possible to draw any tangent from the point (1/4, 1) to the parabola y2 = 4x cosθ + sin2θ ,
then θ belongs to
A.
[-π/2 π/2]
B.
[-π/2 π/2] – {0}
C.
(-π/2 π/2) – {0}
D.
none of these
972)
A
C
B
The number of focal chord(s) of length 4/7 in the parabola 7y2 = 8x is
A.
1
B.
C.
infinite
D.
zero
none of these
973) The ends of line segment are P (1, 3) and Q (1, 1). R is a point on the line segment PQ such that PR :
RQ = 1 : λ . If R is an interior point of parabola y2 = 4x, then
A.
λ ∈ (0, 1)
B.
λ ∈ (-3/5 , 1)
C.
λ ∈ (1/2 , 3/5)
D.
none of these
974) A set of parallel chords of the parabola y2 = 4ax have their mid points on
A.
any straight line through the vertex
B.
C.
a straight line parallel to the axis
D.
C
any straight line through the focus
another parabola
975) The equation of the line of the shortest distance between the parabola y2 = 4x and the circle x2 + y2 –
4x – 2y + 4 = 0 is
A.
x+y=3
Discipline: ________________________
B.
A
x–y=3
Page 106 of 110
A
C.
2x + y = 5
D.
none of these
976) If normals are drawn from the extremities of the latus rectum of a parabola then normals are
A.
parallel to each other
B.
C.
intersect at the 450
D.
B
perpendicular to each other
none of these
977) The triangle formed by the tangent to the parabola y = x2 at the point whose abscissa is k where k ∈ [1, C
2] the y-axis and the straight line y = k2 has greatest area if k is equal to
A.
1
B.
C.
2
D.
3
none of these
978) A parabola y2 = 4ax and x2 = 4by intersect at two points. A circle is passed through one of the
intersection point of these parabola and touch the directrix of first parabola then the locus of the centre
of the circle is
A.
straight line
C.
circle
B.
D.
ellipse
parabola
979) A circle with centre lying on the focus of the parabola y2 = 2px such that it touches the directrix of the
parabola. Then a point of intersection of the circle and the parabola is
A.
(p/2 , p)
B.
C.
(-p/2 , p)
D.
2
B.
C.
6
D.
(-p/2 , -p)
B
4
none of these
981) If AFB is a focal chord of the parabola y2 = 4ax and AF = 4, FB = 5, then the latus-rectum of the
parabola is equal to
Discipline: ________________________
A
(p/2 , 2p)
980) The point (1, 2) is one extremity of focal chord of parabola y2 = 4x. The length of this focal chord is
A.
D
Page 107 of 110
A
A.
80/9
B.
9/80
C.
9
D.
80
982) If three normals can be drawn from (h, 2) to the parabola y2 = -4x, then
A.
h < -2
–2 < h < 2
C.
B.
D.
A
h>2
h is any real number
983) If the line y – √ x +3 = 0 cuts the parabola y2 = x + 2 at A and B, and if P≡ (√3 , 0), then PA. PB is
equal to
A.
2(√3+2)/3
B.
C.
4(2-√3)/3
D.
4√3/2
4(√3+2)/3
984) If the normal to the parabola y2 = 4ax at the point (at2, 2at) cuts the parabola again at (aT2, 2aT), then
A.
T2 ≥ 8
B.
C.
–2 ≤ T ≤ 2
D.
T2 < 8
A.
x=1
B.
x=2
C.
x=0
D.
none of these
986) The set {1, −1, i, −i}
987)
B.
C.
D.
Is a group under ‘.’
none of these
−1 + −3
, the set {1, w, w2 }
2
A
A. Is a group under ‘.’
B.
C.
D.
Is not a group
Discipline: ________________________
B
B
A. Is a group under ‘+’
If w =
A
T ∈ (- ∞, -8) ∪ (8, ∞)
985) The locus of point of intersection of any tangent to the parabola y2 = 4a (x – 2) with a line
perpendicular to it and passing through the focus, is
Is not a group
D
Is a group under ‘+’
none of these
Page 108 of 110
988) The set of complex numbers is
C
A. Not a group under ‘+’
B.
C.
D.
Is a filed
Not a group under ‘+’
none of these
989) Which one is not a filed
A
A.
Z
B.
C.
R
D.
Q
none of these
990) The set {1, −1, i, −i}
B
A. Not a group
B.
Is a cyclic group
C.
D.
none of these
In not abelian group
991) Which one is a semi group
B
A.
P under ‘+’
B.
C.
P under ‘.’
D.
N under ‘+’
none of these
992) Over the field of real numbers,
D
A.
Z is a vector space
B.
C.
E is a vector space
D.
N is a vector space
none of these
993) A group (G,*)
C
A. Is not closed under ‘ * ’
B.
Is closed under ‘ * ’
D.
C.
May not be closed under ‘ * ’
none of these
994) The set G is a group under ‘+’ for
C
A.
G=N
B.
C.
G=Z
D.
G =W
none of these
995) A set which is a group under ‘+’,
A. is a group under “.”
C.
May not be a group under “.”
B
B.
D.
is not a group under “.”
none of these
996) A cyclic group
Discipline: ________________________
C
Page 109 of 110
A. Is not abelian group
B.
C.
D.
Is an abelian group
May not be an abelian group
none of these
997) A group generated by one of its elements, is called
B
A. An abelian group
B.
A cyclic group
C.
D.
none of these
A filed
998) Which one is not true
A. A field is a ring
C.
A ring is a group under ‘+’
D
B.
D.
A ring may not be a field
none of these
999) The set {−3n : n  Z } is an abelian group under
A
A. Addition
B.
C.
D.
Multiplication
Subtraction
none of these
1000)A monoid is always
C
A. a group
B.
C.
D.
Groupoid
Discipline: ________________________
a commutative group
none of these
Page 110 of 110