ISI Admission Test 2023 Question Paper B.Stat B.Math UGA

FOR ISI EXAM PREPARATION

ISI 2023
Question Paper · B.Stat
B.Math UGA
EXAM YEAR TYPE SUBJECT

ISI 2023 Question Paper B.Stat B.Math UGA

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The following notations are used in the question paper:

R is the set of real numbers,
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C is the set of complex numbers,
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Z is .the set of integers, s e
s enm� l a
a ag
�
l
n!
g
= for all n = 1, 2, 3, . . . and r = 0, 1, . . . , n.
a r r!(n − r)!

1. For a real number x,

x3 − 7x + 6 > 0
m
if and only if
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s e
(A) x > 2.
l a
(C) x < −3 or 1 < xg< 2.
(B) −3 < x < 1.

a (D) −3 < x < 1 or x > 2.

2. Define a polynomial f (x) by
� �
� 1 x x�
� �
f (x) = �x 1 x�
� �
� �
�x x 1 �
m
m for all x ∈ R, where the right hand side above is a determinant.
.co
m .co Then the roots of f (x) are of the form
s em
s e g la root of
(A) α, β ± iγ where α, β, γ ∈ R, γ �= 0 and i is a square

g la −1. a
a (B) α, α, β where α, β ∈ R are distinct.
(C) α, β, γ where α, β, γ ∈ R are all distinct.
(D) α, α, α for some α ∈ R.

1
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s em l a
g la ag
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3. Let S be the set of those real numbers x for which the identity
∞
�
cosn x = (1 + cos x) cot2 x
n=2

is valid, and the quantities on both sides are finite. Then
(A) S is the empty set.
(B) S = {x ∈ R : x �= nπ for all n ∈ Z}.
(C) S = {x ∈ R : x �= 2nπ for all n ∈ Z}.
(D) S = {x ∈ R : x �= (2n + 1)π for all n ∈ Z}.

4. The number of consecutive zeroes adjacent to the digit in the
unit’s place of 40150 is

(A) 3. (B) 4. (C) 49. (D) 50.

5. Consider a right angled triangle �ABC whose hypotenuse AC
is of length 1. The bisector of ∠ACB intersects AB at D. If
BC is of length x, then what is the length of CD?
�
2x2 1
(A) 1+x
(B) √2+2x
� x x
(C) 1+x
(D) √1−x 2

6. Consider a triangle with vertices (0, 0), (1, 2) and (−4, 2). Let A
be the area of the triangle and B be the area of the circumcircle
of the triangle. Then B
A
equals

5π √3 π.
(A) π
2
. (B) 4
. (C) 2
(D) 2π.

2

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7. Let f, g be continuous functions from [0, ∞) to itself,
� 3x
m
.co
h(x) = f (t) dt , x > 0 ,
m
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2x

e m
and
em
� h(x)
l as
as F (x) = g(t) dt , x > 0 .
l If F is the derivative of F , then for x > 0, ag
g
0

a �

(A) F � (x) = g(h(x)).
(B) F � (x) = g(h(x)) [f (3x ) − f (2x )].
(C) F � (x) = g(h(x)) [x3x−1 f (3x ) − x2x−1 f (2x )].
(D) F � (x) = g(h(x)) [3x f (3x ) ln 3 − 2x f (2x ) ln 2].

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8. How many numbers formed by rearranging the digits of 234578
m
are divisible by 55?
s e
12l
(B) g
a
a
(A) 0 (C) 36 (D) 72

9. Let
�� � �
πθ 1 πθ
S= θ sin , cos : θ ∈ R, θ > 0
1+θ θ 1+θ

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and
m
� �
1

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T = (x, y) : x ∈ R, y ∈ R, xy = .
2
e m
e m How many elements does S ∩ T have?
las
las g
a (D) 3
ag (A) 0 (B) 1 (C) 2

3
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s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 3 of 10

10. The limit
3 � �1
lim n− 2 (n + 1)(n+1) (n + 2)(n+2) . . . (2n)(2n) n2
n→∞

equals
1 3
(A) 0. (B) 1. (C) e− 4 . (D) 4e− 4 .

11. Suppose x and y are positive integers. If 4x + 3y and 2x + 4y
are divided by 7, then the respective remainders are 2 and 5. If
11x + 5y is divided by 7, then the remainder equals

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

12. The value of
202 � � � �
�
k 202 kπ
(−1) cos
k=0
k 3
equals
� 202 � � 202 �
(A) sin 3
π . (B) − sin 3
π .
� 202
(D) cos202 3 .
� � �
(C) cos 3 π . π

13. For real numbers a, b, c, d, a� , b� , c� , d� , consider the system of
equations

ax2 + ay 2 + bx + cy + d = 0 ,
a� x2 + a� y 2 + b� x + c� y + d� = 0 .

If S denotes the set of all real solutions (x, y) of the above
system of equations, then the number of elements in S can
never be

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

4

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14. The limit
� � � � � �
1 1 1
lim cos(x) + cos − cos(x) cos −1
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x→0 x x x
m
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s e(A)m equals 0. (B) equals 12 . l a
g la (C) equals 1. (D) does not exist. ag
a
15. Let n be a positive integer having 27 divisors including 1 and
n, which are denoted by d1 , . . . , d27 . Then the product of
d1 , d2 , . . . , d27 equals
27
(A) n13 . (B) n14 . (C) n 2 . (D) 27n.
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16. Suppose F : R → R is aacontinuous function which has exactly
one local maximum.ag
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Then which of the following is true?
(A) F cannot have a local minimum.
(B) F must have exactly one local minimum.
(C) F must have at least two local minima.
(D) F must have either a global maximum or a local minimum.

m
m 17. Suppose z ∈ C is such that the imaginary part of z is non-zero
.co
m .co and z 25 = 1. Then
s e m
la
2023

s e
�

la
zk
ag
g
k=0

a equals

(A) 0 . (B) 1 . (C) −1 − z 24 . (D) −z 24 .

5
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s em l a
g la ag
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18. Let f : R → R be a twice differentiable one-to-one function. If
f (2) = 2, f (3) = −8 and
� 3
f (x) dx = −3 ,
2

then � 2
f −1 (x) dx
−8

equals

(A) −25. (B) 25. (C) −31. (D) 31.

19. If f : [0, ∞) → R is a continuous function such that
� x
f (x) + ln 2 f (t) dt = 1 , x ≥ 0 ,
0

then for all x ≥ 0,

(A) f (x) = ex ln 2. (B) f (x) = e−x ln 2.
� �x
(C) f (x) = 2x . (D) f (x) = 12 .

20. If [x] denotes the largest integer less than or equal to x, then
� √ �
(9 + 80)20

equals
√ √
(A) (9 + 80)20 − (9 − 80)20 .
√ √
(B) (9 + 80)20 + (9 − 80)20 − 20.
√ √
(C) (9 + 80)20 + (9 − 80)20 − 1.
√
(D) (9 − 80)20 .

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21. The limit � �2−n
−2n+1 −2n−1
lim 2 +2
m
n→∞

equals
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(A) .c e
em
1. (B) √1 .
2
(C) 0. (D) 1
4
.
l as
las ag
ag
22. In the following figure, OAB is a quarter-circle. The unshaded
region is a circle to which OA and CD are tangents.

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s e
l a
ag
If CD is of length 10 and is parallel to OA, then the area of the
shaded region in the above figure equals

(A) 25π. (B) 50π. (C) 75π. (D) 100π.

m
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in such a way that if the brackets are serially counted from the
e left, then the number of right brackets counted is aalways less
s g l In how
g la a
than or equal to the number of left brackets counted.
a many ways can this be done?

(A) 3 (B) 4 (C) 5 (D) 6

7
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24. The polynomial x10 + x5 + 1 is divisible by

(A) x2 + x + 1. (B) x2 − x + 1.
(C) x2 + 1. (D) x5 − 1.

25. Suppose a, b, c ∈ R and

f (x) = ax2 + bx + c , x ∈ R .

If 0 ≤ f (x) ≤ (x − 1)2 for all x, and f (3) = 2, then

(A) a = 12 , b = −1, c = 21 . (B) a = 13 , b = − 31 , c = 0.
(C) a = 23 , b = − 35 , c = 1. (D) a = 34 , b = −2, c = 45 .

26. As in the following figure, the straight line OA lies in the
second quadrant of the (x, y)-plane and makes an angle θ with
the negative half of the x-axis, where 0 < θ < π2 .

The line segment CD of length 1 slides on the (x, y)-plane in
such a way that C is always on OA and D on the positive side
of the x-axis. The locus of the mid-point of CD is
(A) x2 + 4xy cot θ + y 2 (1 + 4 cot2 θ) = 14 .
(B) x2 + y 2 = 14 + cot2 θ.
(C) x2 + 4xy cot θ + y 2 = 41 .
(D) x2 + y 2 (1 + 4 cot2 θ) = 14 .

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27. Suppose that f (x) = ax3 + bx2 + cx + d where a, b, c, d are real
numbers with a �= 0. The equation f (x) = 0 has exactly two
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distinct real solutions. If f � (x) is the derivative of f (x), then
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which of the following is a possible graph of f (x)? �

s e
s e(A)m l a
g la (B)
ag
a

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s e
l a
(C)
ag (D)

m
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m .co 28. Consider the function f : C → C defined by
s e m
e f (a + ib) = e (cos b + i sin b) , a, b ∈ Rl, a
las a

ag
ag where i is a square root of −1. Then
(A) f is one-to-one and onto.
(B) f is one-to-one but not onto.
(C) f is onto but not one-to-one.
(D) f is neither one-to-one nor onto.

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29. Suppose f : Z → Z is a non-decreasing function. Consider the
following two cases:

Case 1. f (0) = 2 , f (10) = 8 ,
Case 2. f (0) = −2 , f (10) = 12 .

In which of the above cases it is necessarily true that there exists
an n with f (n) = n?
(A) In both cases.
(B) In neither case.
(C) In Case 1. but not necessarily in Case 2.
(D) In Case 2. but not necessarily in Case 1.

30. How many functions f : {1, 2, . . . , 10} → {1, . . . , 2000}, which
satisfy
f (i + 1) − f (i) ≥ 20 , for all 1 ≤ i ≤ 9 ,

are there?
� � � �
1829 1830
(A) 10! (B) 11!
10 11
� � � �
1829 1830
(C) (D)
10 11

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