ISI Admission Test 2016 Sample Paper JRF Physics, JRF QROR, MS(QMBA), M.Tech(QROR) MMA

FOR ISI ADMISSION TEST EXAM PREPARATION

ISI Admission Test
2016
Sample Paper · JRF
Physics, JRF QROR,
MS(QMBA),
Notes · Sample Papers · Previous Year Papers · Mock Tests

M.Tech(QROR) MMA

co SAMPLE QUESTIONS (MCQ -TYPE) m .
m . se
se l a

la ag
ag

m
. co
s em
g l a
a

m
m .co
c. o se
m
m
se

gla
For more Question Papers, Sample Papers, Notes & Syllabus visit Page 1 of 20

3. Let x be a positive real number. Then
(A) x2 + π 2 + x2π > xπ + (π + x)xπ
(B) xπ + π x > x2π + π 2x
(C) πx + (π + x)xπ > x2 + π 2 + x2π
(D) none of the above.

4. Suppose in a competition 11 matches are to be played, each having one of 3
distinct outcomes as possibilities. The number of ways one can predict the
outcomes of all 11 matches such that exactly 6 of the predictions turn out to
be correct is
� � � �
11 5 11
(A) ×2 (B) (C) 36 (D) none of the above.
6 6

5. A set contains 2n+1 elements. The number of subsets of the set which contain
at most n elements is

(A) 2n (B) 2n+1 (C) 2n−1 (D) 22n .

6. A club with x members is organized into four committees such that

(a) each member is in exactly two committees,
(b) any two committees have exactly one member in common.

Then x has
(A) exactly two values both between 4 and 8
(B) exactly one value and this lies between 4 and 8
(C) exactly two values both between 8 and 16
(D) exactly one value and this lies between 8 and 16.

7. Let X be the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Define the set R by

R = {(x, y) ∈ X × X : x and y have the same remainder when divided by 3}.

Then the number of elements in R is

(A) 40 (B) 36 (C) 34 (D) 33.

8. Let A be a set of n elements. The number of ways, we can choose an ordered
pair (B, C), where B, C are disjoint subsets of A, equals

(A) n2 (B) n3 (C) 2n (D) 3n .

2

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 2 of 20

m
m .co
c. o

em
em l a s
s ag

9. Let (1 + x)n = C0 + C1 x + C2 x2 + . . . + Cn xn , n being a positive integer. The
value of � �� � � �
C0 C1 Cn−1
1+ 1+ ... 1 +
C1 C2 Cn

m
co
is

m .
co
� �n � �n

. e m
n+1 nn n (n + 1)n

s
(A) (B) (C) (D) .

m a
n+2 n! n+1 n!

se g l
l a a
10. The value of the infinite product

ag
7 26 63 n3 − 1
P = × × × ··· × 3 × ···
9 28 65 n +1
is

(A) 1 (B) 2/3 (C) 7/3 (D) none of the above.

11. The number of positive integers which are less than or equal to 1000 and are
divisible by none of 17, 19 and 23 equals

c om
(A) 854 (B) 153 (C) 160 (D) none of the above.

.
em
12. Consider the polynomial x5 + ax4 + bx3 + cx2 + dx + 4 where a, b, c, d are real

l a s
numbers. If (1 + 2i) and (3 − 2i) are two roots of this polynomial then the

g
a (B) 524/65
value of a is

(A) −524/65 (C) −1/65 (D) 1/65.

13. The number of real roots of the equation
� 2 �
x +x
2 cos = 2x + 2−x
6
is

m
m .co
(A) 0 (B) 1 (C) 2 (D) infinitely many.

c. o 14. Consider the following system of equivalences of integers.

s em
s em g la
la a
x ≡ 2 mod 15

ag
x ≡ 4 mod 21.

The number of solutions in x, where 1 ≤ x ≤ 315, to the above system of
equivalences is

(A) 0 (B) 1 (C) 2 (D) 3.

3

m .
.co s e m
e m la
l a s a g
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 3 of 20

15. The number of real solutions of the equation (9/10)x = −3 + x − x2 is

(A) 2 (B) 0 (C) 1 (D) none of the above.

16. If two real polynomials f (x) and g(x) of degrees m (≥ 2) and n (≥ 1) respec-
tively, satisfy
f (x2 + 1) = f (x)g(x),

for every x ∈ R, then

(A) f has exactly one real root x0 such that f � (x0 ) �= 0
(B) f has exactly one real root x0 such that f � (x0 ) = 0
(C) f has m distinct real roots
(D) f has no real root.

1 1 1 1
17. Let X = + + + ··· + . Then,
1001 1002 1003 3001

(A) X < 1 (B) X > 3/2
(C) 1 < X < 3/2 (D) none of the above holds.

18. The set of complex numbers z satisfying the equation

(3 + 7i)z + (10 − 2i)z + 100 = 0

represents, in the complex plane,

(A) a straight line
(B) a pair of intersecting straight lines
(C) a point
(D) a pair of distinct parallel straight lines.
n �
� �
� 2πik 2πi(k−1) �
19. The limit lim �e n − e n � is
n→∞
k=1

(A) 2 (B) 2e (C) 2π (D) 2i.
� �n
1
20. The limit lim 1− 2 equals
n→∞ n

(A) e−1 (B) e−1/2 (C) e−2 (D) 1.

4

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 4 of 20

m
m .co
c. o

e m
em l a s
s a g

21. Let ω denote a complex fifth root of unity. Define
4
�
bk = jω −kj ,

m
co
j=0

m .
co m
4
�

. e
for 0 ≤ k ≤ 4. Then bk ω k is equal to

m
e (A) 5 l a s
k=0

l a s (B) 5ω (C) 5(1 + ω) (D) 0.
ag
g
a 22. Let a = 1 − √12 · · · 1 − √n1+ 1 , n ≥ 1. Then lim a
n
� � � �
n
n→∞

1
(A) equals 1 (B) does not exist (C) equals √ (D) equals 0.
π

23. Let X be a nonempty set and let P(X) denote the collection of all subsets of
X. Define f : X × P(X) → R by

m
o x ∈/ A.
�

c
1 if x ∈ A

.
f (x, A) =

em
0 if

Then f (x, A ∪ B) equals

l a s
(A) f (x, A) + f (x, B)
(B) f (x, A) + f (x, B) − 1a g
(C) f (x, A) + f (x, B) − f (x, A) · f (x, B)
(D) f (x, A) + |f (x, A) − f (x, B)|
∞
� 1
24. The series converges to
k(k − 1)
k=2

m
(A) −1 (B) 1 (C) 0 (D) does not converge.

m .co
c. o
� �4x

m
se
3x − 1
25. The limit lim equals

em la
x→∞ 3x + 1

las (A) 1 (B) 0 (C) e−8/3
a g (D) e4/9

ag 26. lim
1
�
n
+
n
+ ··· +
n
�
is equal to
n→∞ n n+1 n+2 2n

(A) ∞ (B) 0 (C) loge 2 (D) 1

5

m .
.co s e m
e m la
l a s ag
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 5 of 20

27. Let cos6 θ = a6 cos 6θ+a5 cos 5θ+a4 cos 4θ+a3 cos 3θ+a2 cos 2θ+a1 cos θ+a0 .
Then a0 is

(A) 0 (B) 1/32. (C) 15/32. (D) 10/32.

28. In a triangle ABC, AD is the median. If length of AB is 7, length of AC is
15 and length of BC is 10 then length of AD equals
√ √ √
(A) 125 (B) 69/5 (C) 112 (D) 864/5.
� �
� 1 ��
�
29. The set {x : �x + � > 6} equals the set
x
√ √
(A) (0, 3 − 2 2) ∪ (3 + 2 2, ∞)
√ √
(B) (−∞, −3 − 2 2) ∪ (−3 + 2 2, ∞)
√ √
(C) (−∞, 3 − 2 2) ∪ (3 + 2 2, ∞)
√ √ √ √
(D) (−∞, −3 − 2 2) ∪ (−3 + 2 2, 3 − 2 2) ∪ (3 + 2 2, ∞)

30. Suppose that a function f defined on R2 satisfies the following conditions:

f (x + t, y) = f (x, y) + ty,
f (x, t + y) = f (x, y) + tx and
f (0, 0) = K, a constant.

Then for all x, y ∈ R, f (x, y) is equal to

(A) K(x + y). (B) K − xy. (C) K + xy. (D) none of the above.

31. Consider the sets defined by the real solutions of the inequalities

A = {(x, y) : x2 + y 4 ≤ 1} B = {(x, y) : x4 + y 6 ≤ 1}.

Then
(A) B ⊆ A
(B) A ⊆ B
(C) Each of the sets A − B, B − A and A ∩ B is non-empty
(D) none of the above.

32. If a square of side a and an equilateral triangle of side b are inscribed in a
circle then a/b equals
� � √ √
(A) 2/3 (B) 3/2 (C) 3/ 2 (D) 2/3.

6

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 6 of 20

m
m .co
c. o

e m
em l a s
s a g

33. If f (x) is a real valued function such that

2f (x) + 3f (−x) = 15 − 4x,

m
co
for every x ∈ R, then f (2) is

m .
c. o √
(A) −15 (B) 22 (C) 11 (D) 0.

s e m
e m l a
l a s 3 sin x
ag
ag
34. If f (x) = , then the range of f (x) is
2 + cos x

√ √
(A) the interval [−1, 3/2] (B) the interval [− 3/2, 1]
(C) the interval [−1, 1] (D) none of the above.

35. If f (x) = x2 and g(x) = x sin x + cos x then
(A) f and g agree at no points

m
(B) f and g agree at exactly one point

c o
. points.
(C) f and g agree at exactly two points

e m
(D) f and g agree at more than two

36. For non-negative integers m,sn define a function as follows
l a
agn + 1 if m = 0
f (m, n) = f (m − 1, 1) if m �= 0, n = 0


f (m − 1, f (m, n − 1)) if m �= 0, n �= 0

Then the value of f (1, 1) is

(A) 4 (B) 3 (C) 2 (D) 1.

m
m .co
37. Let a be a nonzero real number. Define

c. o m
� �

se
� x a a a ��

em
�

la
� a x a ��

s
� a

la g
f (x) = � �

a
� a a x a ��

g
�
� a a x �

a
a

for x ∈ R. Then, the number of distinct real roots of f (x) = 0 is

(A) 1 (B) 2 (C) 3 (D) 4.

7

m .
.co s e m
e m la
l a s a g
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 7 of 20

38. A real 2 × 2 matrix M such that
� �
2 −1 0
M =
0 −1 − ε

(A) exists for all ε > 0
(B) does not exist for any ε > 0
(C) exists for some ε > 0
(D) none of the above is true

 
2 1 1
 
39. The eigenvalues of the matrix X =  1 2 1  are
1 1 2

(A) 1, 1, 4 (B) 1, 4, 4 (C) 0, 1, 4 (D) 0, 4, 4.

40. Let x1 , x2 , x3 , x4 , y1 , y2 , y3 and y4 be fixed real numbers, not all of them equal
to zero. Define a 4 × 4 matrix A by

 
x21 + y12 x 1 x 2 + y1 y2 x 1 x 3 + y1 y3 x 1 x 4 + y1 y4
 
 x 2 x 1 + y2 y1 x22 + y22 x 2 x 3 + y2 y3 x 2 x 4 + y2 y4 
 
A= .
 x 3 x 1 + y 3 y1 x 3 x 2 + y3 y2 x23 + y32 x 3 x 4 + y3 y4 
 
x 4 x 1 + y4 y1 x 4 x 2 + y4 y2 x 4 x 3 + y4 y3 x24 + y42

Then rank(A) equals

(A) 1 or 2. (B) 0. (C) 4. (D) 2 or 3.

41. Let k and n be integers greater than 1. Then (kn)! is not necessarily divisible
by

(A) (n!)k . (B) (k!)n . (C) n!.k!. (D) 2kn .

8

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 8 of 20

m
m .co
c. o

e m
em l a s
s a g

42. Let λ1 , λ2 , λ3 denote the eigenvalues of the matrix
 
1 0 0
 

m
A= 0 cos t sin t  .

co
0 − sin t cos t

m .
.co e m
√

s
If λ1 + λ2 + λ3 = 2 + 1, then the set of possible values of t, −π ≤ t < π, is

m
e (A) Empty set (B) 4 l a
a s g
�π� � π π� � π π�

g l a
(C) − , (D) − , .

a 43. The values of η for which the following system of equations
4 4 3 3

x + y + z = 1
x + 2y + 4z = η
x + 4y + 10z = η2

has a solution are

o m(C) η = 3, −3 (D) η = 1, 2.
. c
(A) η = 1, −2 (B) η = −1, −2

s em the planes defined by
laax + b y + c z = α
44. Let P , P and P denote, respectively,
1 2 3

g
a a x+b y+c z = α 1 1 1 1

2 2 2 2

a3 x + b3 y + c3 z = α3 .

It is given that P1 , P2 and P3 intersect exactly at one point when α1 = α2 =
α3 = 1. If now α1 = 2, α2 = 3 and α3 = 4 then the planes

(A) do not have any common point of intersection

m
co
(B) intersect at a unique point

m .
c. o m
(C) intersect along a straight line

em se
(D) intersect along a plane.

s g la
la a
45. Angles between any pair of 4 main diagonals of a cube are

ag √ √
(A) cos−1 1/ 3, π − cos−1 1/ 3 (B) cos−1 1/3, π − cos−1 1/3
(C) π/2 (D) none of the above.

9

m .
.co s e m
e m la
l a s a g
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 9 of 20

46. If the tangent at the point P with co-ordinates (h, k) on the curve y 2 = 2x3
is perpendicular to the straight line 4x = 3y, then

(A) (h, k) = (0, 0)
(B) (h, k) = (1/8, −1/16)
(C) (h, k) = (0, 0) or (h, k) = (1/8, −1/16)
(D) no such point (h, k) exists.

47. Consider the family F of curves in the plane given by x = cy 2 , where c is a
real parameter. Let G be the family of curves having the following property:
every member of G intersects each member of F orthogonally. Then G is given
by

(A) xy = k (B) x2 + y 2 = k 2
(C) y 2 + 2x2 = k 2 (D) x2 − y 2 + 2yk = k 2

48. Suppose the circle with equation x2 + y 2 + 2f x + 2gy + c = 0 cuts the parabola
y 2 = 4ax, (a > 0) at four distinct points. If d denotes the sum of ordinates of
these four points, then the set of possible values of d is

(A) {0} (B) (−4a, 4a) (C) (−a, a) (D) (−∞, ∞).

49. The polar equation r = a cos θ represents

(A) a spiral (B) a parabola (C) a circle (D) none of the above.

50. Let
� �2
72 + 82 + 152 + 232 7 + 8 + 15 + 23
V1 = − ,
4 4
� �2
62 + 82 + 152 + 242 6 + 8 + 15 + 24
V2 = − ,
4 4
� �2
52 + 82 + 152 + 252 5 + 8 + 15 + 25
V3 = − .
4 4

Then

(A) V3 < V2 < V1 (B) V3 < V1 < V2
(C) V1 < V2 < V3 (D) V2 < V3 < V1 .

10

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 10 of 20

m
m .co
c. o

e m
em l a s
s a g

51. A permutation of 1, 2, ..., n is chosen at random. Then the probability that
the numbers 1 and 2 appear as neighbour equals

m
m .co
co m
1 2 1 1

. e
(A) (B) (C) (D) .

s
n n n−1 n−2

e m l a
a s
l 52. Two coins are tossed independently where P(head occurs when coin i is tossed) a g
ag = p , i = 1, 2. Given that at least one head has occurred, the probability that
i
coins produced different outcomes is

2p1 p2 p1 + p2 − 2p1 p2 2
(A) (B) (C) (D) none of the above.
p1 + p2 − 2p1 p2 p 1 + p 2 − p 1 p2 3

o m
c
53. The number of cars (X) arriving at .a service station per day follows a Poisson
m
distribution with mean 4. Theeservice station can provide service to a max-
s
a the expected number of cars that do not get
imum of 4 cars per day. lThen
a
service per day equals g

∞
� ∞
�
(A) 4 (B) 0 (C) iP (X = i + 4) (D) iP (X = i − 4).
i=0 i=4

m
m .co
c. o m
1 2 2 3 3 4

se
54. If 0 < x < 1, then the sum of the infinite series x + x + x + · · · is
2 3 4

s em g la
gla a x
a
1+x
(A) log (B) + log(1 + x)
1−x 1−x
1 x
(C) + log(1 − x) (D) + log(1 − x).
1−x 1−x

11

m .
.co s e m
e m la
l a s a g
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 11 of 20

55. Let {an } be a sequence of real numbers. Then lim an exists if and only if
n→∞

(A) lim a2n and lim a2n+2 exists
n→∞ n→∞
(B) lim a2n and lim a2n+1 exist
n→∞ n→∞
(C) lim a2n , lim a2n+1 and lim a3n exist
n→∞ n→∞ n→∞
(D) none of the above.

56. Let {an } be a sequence of non-negative real numbers such that the series
�∞ � √a n
an is convergent. If p is a real number such that the series
n=1
np
diverges, then

(A) p must be strictly less than 12
(B) p must be strictly less than or equal to 12
(C) p must be strictly less than or equal to 1 but can be greater than 12
(D) p must be strictly less than 1 but can be greater than or equal to 12 .

57. Suppose a > 0. Consider the sequence
√ √
an = n{ n ea − n a}, n ≥ 1.

Then

(A) lim an does not exist (B) lim an = e
n→∞ n→∞
(C) lim an = 0 (D) none of the above.
n→∞

58. Let {an }, n ≥ 1, be a sequence of real numbers satisfying |an | ≤ 1 for all n.
Define
1
An = (a1 + a2 + · · · + an ),
n
√
for n ≥ 1. Then lim n(An+1 − An ) is equal to
n→∞

(A) 0 (B) −1 (C) 1 (D) none of these.

59. In the Taylor expansion of the function f (x) = ex/2 about x = 3, the coeffi-
cient of (x − 3)5 is

1 1 1
(A) e3/2 (B) e3/2 5 (C) e−3/2 5 (D) none of the above.
5! 2 5! 2 5!

12

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 12 of 20

m
m .co
c. o

em
em l a s
s ag

60. Let σ be the permutation:

1 2 3 4 5 6 7 8 9
3 5 6 2 4 9 8 7 1,

m
m .co
co
I be the identity permutation and m be the order of σ i.e.

. e m
m a s
m = min{positive integers n : σ n = I}.

s e Then m is g l
g a
l (A) 8 a
a (B) 12 (C) 360 (D) 2520.

61. Let    
1 1 1 1 0 0
   
A= 1 2 2  and B= 1 1 0 .
1 2 3 1 1 1
Then

o m
c
(A) there exists a matrix C such that A = BC = CB

m .
(B) there is no matrix C such that A = BC

s ethat A = CB.
(C) there exists a matrix C such that A = BC, but A �= CB

l a
ag
(D) there is no matrix C such

62. If the matrix � �
a 1
A=
2 3

has 1 as an eigenvalue, then trace(A) is
(A) 4 (B) 5 (C) 6 (D) 7.

63. Let θ = 2π/67. Now consider the matrix

m
co
� �

m .
cos θ sin θ

c. o m
A= .

se
− sin θ cos θ

s em Then the matrix A2010 is

g la � �
gla a (B) 1 0
� �

a
cos θ sin θ
(A)
− sin θ cos θ 0 1
� � � �
cos30 θ sin30 θ 0 1
(C) (D) .
− sin30 θ cos30 θ −1 0

13

m .
.co s e m
e m la
l a s a g
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 13 of 20

64. Let the position of a particle in three dimensional space at time t be
(t, cos t, sin t). Then the length of the path traversed by the particle between
the times t = 0 and t = 2π is
√ √
(A) 2π. (B) 2 2π. (C) 2π (D) none of the above.

65. Let n be a positive real number and p be a positive integer. Which of the
following inequalities is true?

(n + 1)p+1 − np+1 (n + 1)p+1 − np+1
(A) np > (B) np <
p+1 p+1
p (n + 1)p+1 − np+1
(C) (n + 1) < (D) none of the above.
p+1

66. The smallest positive number K for which the inequality

| sin2 x − sin2 y| ≤ K|x − y|

holds for all x and y is
π
(A) 2 (B) 1 (C) (D) there is no smallest positive value of K;
2
any K > 0 will make the inequality hold.

67. Given two real numbers a < b, let

d(x, [a, b]) = min{|x − y| : a ≤ y ≤ b} for − ∞ < x < ∞.

Then the function
d(x, [0, 1])
f (x) =
d(x, [0, 1]) + d(x, [2, 3])
satisfies
(A) 0 ≤ f (x) < 12 for every x
(B) 0 < f (x) < 1 for every x
(C) f (x) = 0 if 2 ≤ x ≤ 3 and f (x) = 1 if 0 ≤ x ≤ 1
(D) f (x) = 0 if 0 ≤ x ≤ 1 and f (x) = 1 if 2 ≤ x ≤ 3.

68. Let � 2 2
e−1/(x +y ) if (x, y) �= (0, 0)
f (x, y) =
0 if (x, y) = (0, 0).
Then f (x, y) is
(A) not continuous at (0, 0)
(B) continuous at (0, 0) but does not have first order partial derivatives

14

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 14 of 20

m
m .co
c. o

e m
em l a s
s ag

(C) continuous at (0, 0) and has first order partial derivatives, but not differ-
entiable at (0, 0)
(D) differentiable at (0, 0)

m
co
69. Consider the function

m .
co
 � x

. e m

 {5 + |1 − y|}dy if x > 2

s


e m a
0

l
f (x) =

s g



a


l Then a
5x + 2 if x ≤ 2

ag
(A) f is not continuous at x = 2
(B) f is continuous and differentiable everywhere
(C) f is continuous everywhere but not differentiable at x = 1
(D) f is continuous everywhere but not differentiable at x = 2.
2 2
70. Let w = log(u2 + v 2 ) where u = e(x +y) and v = e(x+y ) . Then

m
�
∂w ��

. co
∂x �

em
x=0,y=0

s
is

g l a
a
(A) 0 (B) 1 (C) 2 (D) 4

71. Let �
1, if xy = 0,
f (x, y) =
xy, if xy �= 0.
Then
(A) f is continuous at (0, 0) and ∂f
∂x (0, 0) exists

(B) f is not continuous at (0, 0) and ∂f
∂x (0, 0) exists

m
c. o
(C) f is continuous at (0, 0) and ∂f
∂x (0, 0) does not exist

m
c. o
(D) f is not continuous at (0, 0) and ∂f

72. The map f (x) = a cos |x| + a sin |x| + a |x| is differentiable at m
∂x (0, 0) does not exist.

s e
em la
3
0 1 2 x = 0 if and

las (B) a = 0 and a
g
only if

ag (A) a = 0 and a = 0
1

(C) a1 = 0
2 a =0 0 1

(D) a0 , a1 , a2 can take any real value.

73. f (x) is a differentiable function on the real line such that lim f (x) = 1 and
x→∞
lim f � (x) = α. Then
x→∞

15

m .
.co s e m
e m la
l a s a g
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 15 of 20

(A) α must be 0 (B) α need not be 0, but |α| < 1
(C) α > 1 (D) α < −1.

74. Let f and g be two differentiable functions such that f � (x) ≤ g � (x) for all
x < 1 and f � (x) ≥ g � (x) for all x > 1. Then
(A) if f (1) ≥ g(1), then f (x) ≥ g(x) for all x
(B) if f (1) ≤ g(1), then f (x) ≤ g(x) for all x
(C) f (1) ≤ g(1)
(D) f (1) ≥ g(1).

75. The length of the curve x = t3 , y = 3t2 from t = 0 to t = 4 is
√ √
(A) 5 5 + 1 (B) 8(5 5 + 1)
√ √
(C) 5 5 − 1 (D) 8(5 5 − 1).
� ∞
2 √
76. Given that e−x dx = π, the value of
−∞
� ∞� ∞
2 2
e−(x +xy+y ) dxdy
−∞ −∞

is
� √ � √
(A) π/3 (B) π/ 3 (C) 2π/3 (D) 2π/ 3.

77. Let R be the triangle in the xy-plane bounded by the x-axis, the line y = x,
and the line x = 1. The value of the double integral
� �
sin x
dxdy
x
R

is
(A) 1 − cos 1 (B) cos 1 (C) π2 (D) π.

78. The value of � 1
� �
lim (n + 1) xn ln(1 + x) dx
n→∞ 0
is
(A) 0 (B) ln 2 (C) ln 3 (D) ∞.

79. Let g(x, y) = max{12 − x, 8 − y}. Then the minimum value of g(x, y) as (x, y)
varies over the line x + y = 10 is

(A) 5 (B) 7 (C) 1 (D) 3.

16

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 16 of 20

m
m .co
c. o

e m
em l a s
s ag

80. Let 0 < α < β < 1. Then

∞ � 1/(k+α)
� 1
dx

m
1+x

co
k=1 1/(k+β)

m .
c. o m
is equal to

m β s e
s e (A) log α l a
a ag
1+β 1+α

l
(B) loge (C) loge (D) ∞.

g
e
1+α 1+β

a 81. If f is continuous in [0, 1] then
� 1 �j �
[n/2]
lim f
n→∞
j=0
n n

(where [y] is the largest integer less than or equal to y)

m
co
(A) does not exist

.
�

em
1 1

s
(B) exists and is equal to f (x) dx
2 0

l a
� 1

a � g
(C) exists and is equal to f (x) dx
0
1/2
(D) exists and is equal to f (x) dx.
0

82. The volume of the solid, generated by revolving about the horizontal line
y = 2 the region bounded by y 2 ≤ 2x, x ≤ 8 and y ≥ 2, is

√

m
co
(A) 2 2π (B) 28π/3 (C) 84π (D) none of the above.

m αβ̄ + ᾱβ .
c. o s e mis
em a
83. If α, β are complex numbers then the maximum value of

l
|αβ|

las ag
g
(A) 2

a (B) 1
(C) the expression may not always be a real number and hence maximum
does not make sense
(D) none of the above.

17

m .
.co s e m
e m la
l a s a g
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 17 of 20

84. For positive real numbers a1 , a2 , . . . , a100 , let
100
� �
p= ai and q= ai aj .
i=1 1≤i<j≤100

Then
p2 p2 p2
(A) q = (B) q 2 ≥ (C) q < (D) none of the above.
2 2 2
85. The differential equation of all the ellipses centred at the origin is

(A) y 2 + x(y � )2 − yy � = 0 (B) xyy �� + x(y � )2 − yy � = 0
(C) yy �� + x(y � )2 − xy � = 0 (D) none of these.

86. The coordinates of a moving point P satisfy the equations
dx dy
= tan x, = − sin2 x, t ≥ 0.
dt dt
If the curve passes through the point (π/2, 0) when t = 0, then the equation
of the curve in rectangular co-ordinates is

(A) y = 1/2 cos2 x (B) y = sin 2x
(C) y = cos 2x + 1 (D) y = sin2 x − 1.

87. If x(t) is a solution of
(1 − t2 ) dx − tx dt = dt
and x(0) = 1, then x( 21 ) is equal to
(A) √23 ( π6 + 1) (B) √23 ( π6 − 1) π
(C) 3√ 3
(D) √π3 .

88. Let f (x) be a given differentiable function. Consider the following differential
equation in y
dy
f (x) = yf � (x) − y 2 .
dx
The general solution of this equation is given by

x+c f (x)
(A) y = − (B) y 2 =
f (x) x+c
f (x) [f (x)]2
(C) y = (D) y = .
x+c x+c
89. Let y(x) be a non-trivial solution of the second order linear differential equa-
tion
d2 y dy
+ 2c + ky = 0,
dx2 dx
where c < 0, k > 0 and c2 > k. Then

18

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 18 of 20

m
m .co
c. o

e m
em l a s
s a g

(A) |y(x)| → ∞ as x → ∞
(B) |y(x)| → 0 as x → ∞
(C) lim |y(x)| exists and is finite

m
x→±∞

co
(D) none of the above is true.

m .
.co m
90. The differential equation of the system of circles touching the y-axis at the

m s e
a
origin is

se g l
a a
dy dy

l
2 2 2 2
(A) x + y − 2xy =0 (B) x + y + 2xy =0

ag
dx dx
dy dy
(C) x2 − y 2 − 2xy =0 (D) x2 − y 2 + 2xy = 0.
dx dx
91. Suppose a solution of the differential equation
dy
(xy 3 + x2 y 7 ) = 1,
dx
dy
satisfies the initial condition y(1/4) = 1. Then the value of when y = −1
dx
is

o m
4 4 16 16

c
(A) (B) − (C) (D) − .

.�
3 3 5 5

e m
92. Consider the group

G = as
�� �

g l 0 a : a, b ∈ R, a > 0
a b

a
−1

with usual matrix multiplication. Let
�� � �
1 b
N= :b∈R .
0 1
Then,
(A) N is not a subgroup of G

m
(B) N is a subgroup of G but not a normal subgroup

m c. o
(C) N is a normal subgroup and the quotient group G/N is of finite order

c. o m
(D) N is a normal subgroup and the quotient group is isomorphic to R+ (the

s e
em a
group of positive reals with multiplication).

s satisfying x y = x y = e, then which of the following g
l
la a conditions is true?
93. Let G be a group with identity element e. If x and y are elements in G

g
5 3 8 5

a (A) x = e, y = e
(B) x �= e, y = e
(C) x = e, y �= e
(D) x �= e, y �= e

19

m .
.co s e m
e m la
l a s a g
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 19 of 20

94. Let G be the group {±1, ±i} with multiplication of complex numbers as com-
position. Let H be the quotient group Z/4Z. Then the number of nontrivial
group homomorphisms from H to G is

(A) 4 (B) 1 (C) 2 (D) 3.

20

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 20 of 20