ISI Admission Test 2024 Question Paper B.Stat B.Math UGA

FOR ISI EXAM PREPARATION

ISI 2024
Question Paper · B.Stat
B.Math UGA
EXAM YEAR TYPE SUBJECT

ISI 2024 Question Paper B.Stat B.Math UGA

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√ √ √ √ �30
1. If x = 1 + 5 2 + 5 4 + 5 8 + 5 16, then the value of 1 + x1
�
is

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(A) 2 (B) 5 (C) 32 (D) 64

o m m
c
.j be a number selected at random from {1, 2, . . . , 2024}. s e
m
eWhat is the probability that j is divisible by 9 and 15?
2. Let
l a
l as ag
ag
1 1
(A) (B)
23 46
1 1
(C) (D)
44 253

3. Let Sn be the set of all n-digit numbers whose digits are

o m
all 1 or 2 and there are no consecutive 2’s. (Example: 112 is
c
in S but 221 is not in S ). Then.the number of elements in S is
em
3 3 10

la s
(A) 512
a g
(B) 256 (C) 144 (D) 89

4. There are 30 True or False questions in an examination. A
student knows the answer to 20 questions and guesses the
answers to the remaining 10 questions at random. What is
the probability that the student gets exactly 24 answers correct?

m
105 105 105 4

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(A) (B) (C) (D)
m
29 28 210 210

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e lanumber of
5. Let T be a right-angled triangle in the plane whose side lengths

las are in a geometric progression. Let n(T ) denote g
a the

ag sides of T that have integer lengths. Then the maximum value
of n(T ) over all such T is

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

1
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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 1 of 7

6. Let x1 , x2 , ..., xn be non-negative real numbers such that
�n n √
�
xi = 1. What is the maximum possible value of xi ?
i=1 i=1

√
(A) 1 (B) n (C) n3/4 (D) n

7. The precise interval on which the function
f (x) = log1/2 (x2 − 2x − 3) is monotonically decreasing, is

(A) (−∞, −1) (B) (−∞, 1)

(C) (1, ∞) (D) (3, ∞)

8. The angle subtended at the origin by the common chord of the
circles x2 + y 2 − 6x − 6y = 0 and x2 + y 2 = 36 is

(A) π/2 (B) π/4 (C) π/3 (D) 2π/3

A
9. In �ABC, CD is the median and
BE is the altitude. Given that D E
CD = BE, what is the value of
∠ACD? B C

(A) π/3 (B) π/4 (C) π/5 (D) π/6

10. If the points z1 and z2 are on the circles |z| = 2 and |z| = 3,
respectively, and the angle included between these vectors is
|z1 +z2 |
60◦ , then the value of |z 1 −z2 |
is
� �
19 √ √ 7
(A) (B) 19 (C) 7 (D)
7 19

2

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11. Let n � 1. The maximum possible number of primes in the set
{n + 6, n + 7, ..., n + 34, n + 35} is
m
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(A) 7 (B) 8 (C) 12 (D) 13
e m
em l as
s
la different boxes such that each box gets exactly 10 balls. Out
12. Suppose 40 distinguishable balls are to be distributed into 4
ag
g
a of these 40 balls, 10 are defective and 30 are non-defective. In
how many ways can the balls be distributed such that all the
defective balls go to the first two boxes?

40! 30! · 20! 20! · 20! 30! · 10!
(A) (B) (C) (D)
(10!)4 (10!)5 (10!)5 (10!)4

m
13. The number of elements in the set
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s e
{x : 0 � x � 2, �x − x5 � = �x5 − x6 �}
� � � �

l a
ag
is

(A) 2 (B) 3 (C) 4 (D) 5

14. In a room with n � 2 people, each pair shakes hands between
themselves with probability n22 and independently of other pairs.
If pn is the probability that the total number of handshakes is
m
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at most 1, then lim pn is equal to
m
n→∞

m .co s e m
la
(A) 0 (B) 1 (C) e−1 (D) 2e−1
s e g
g la a
a 15. The number of positive solutions to the equation
√
ex sin x = log x + e x + 2
is

(A) 0 (B) 1 (C) 2 (D) ∞

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 7

16. Let n > 1 be the smallest composite integer that is coprime to
10000!
9900!
. Then

(A) n � 100 (B) 100 < n � 9900

(C) 9900 < n � 10000 (D) n > 10000

17. Let P = {(x, y) : x + 1 � y, x � −1, y � 2x}. Then the
minimum value of (x + y) where (x, y) varies over the set P is

(A) −1 (B) −3 (C) 3 (D) 0

18. Let A = {1, ..., 5} and B = {1, ..., 10}. Then the number of
ordered pairs (f, g) of functions f : A → B and g : B → A
satisfying (g ◦ f ) (a) = a for all a ∈ A is

10! �10�
(A) × 55 (B) 510 × 5! (C) 10! × 5! (D) 5
× 105
5!

19. Let
1 1 1
S=√ +√ + ··· + √ .
10000 10001 160000
Then the largest positive integer not exceeding S is

(A) 200 (B) 400 (C) 600 (D) 800

20. The real number x satisfies
|x|2 − |x| − 2
>2
2 |x| − |x|2 − 2
if and only if x belongs to

(A) (−2, −1) ∪ (1, 2) (B) (−2/3, 0) ∪ (0, 2/3)

(C) (−1, −2/3) ∪ (2/3, 1) (D) (−1, 0) ∪ (0, 1)

4

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21. Consider points of the form (n, nk ), where n and k are integers
with n ≥ 0, k ≥ 1. How many such points are strictly inside the
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circle of radius 10 with centre at the origin?
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em
(A) 11 (B) 12 (C) 15 (D) 17

s l a
g a
l22. ag
a decreasing powers of x. Suppose the first three coefficients arein
� � 1/2 1 n
Let n > 1, and let us arrange the expansion of x + 2x1/4

in arithmetic progression. Then, the number of terms where x
appears with an integer power, is

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

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23. The limit
c
2 log 2 + 3 log. 3 + · · · + n log n
lim
s emn log n 2

la
n→∞

is equal to
a g
(A) 0 (B) 1/4 (C) 1/2 (D) 1

24. Let p < q be prime numbers such that p2 + q 2 + 7pq is a perfect
square. Then, the largest possible value of q is:

m
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(A) 7 (B) 11 (C) 23 (D) 29
m
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e a
2

l
21−x

s
25. The set of all real numbers x for which 3 is an integer has

la a g
ag (A) 3 elements (B) 15 elements

(C) 24 elements (D) infinitely many elements

5
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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 5 of 7

26. Let a, b, c be three complex numbers. The equation

az + bz̄ + c = 0

represents a straight line on the complex plane if and only if

(A) a = b (B) āc = bc̄

(C) |a| = |b| �= 0 (D) |a| = |b| �= 0 and āc = bc̄

27. In the adjoining figure, C is the A
centre of the circle drawn, A, F, E F
B
lie on the circle and BCDF is a
rectangle. If DE
AB
= 2, then FF E
A
equals C E
D
�
3 √
(A) (B) 2
2
�
5 √
(C) (D) 3
2

28. For every increasing function b : [1, ∞) → [1, ∞) such that
� ∞
dx
< ∞,
1 b(x)

we must have
√
∞
� log k ∞
� log k
(A) <∞ (B) <∞
k=1 b(k) k=3 b (log k)

�∞ ek ∞
� 1
(C) k
<∞ (D) � <∞
k=1 b(e ) k=3 b (log k)

6

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29. Consider the following two statements:

(I) There exists a differentiable function g : R → R such that
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g(x3 + x5 ) = ex − 100.
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em g(e ) = x + x .
(II) There exists a continuous function g : R → R such that

s l a
ag
x 3 5

g la Then
a
(A) Only (I) is correct.
(B) Only (II) is correct.
(C) Both (I) and (II) are correct.
(D) Neither (I) nor (II) is correct.
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em
�1
s function. Then
30. Let ψ : R → R be a continuous function with ψ(x)dx = 1.
a
−1

l
a1g�
Let f : R → R be a differentiable
1+ε
� �
1−y
lim f (y)ψ dy
ε→0 ε 1−ε ε

equals

(A) f (1) (B) f (1)ψ(0)

(C) f � (1)ψ(0) (D) f (1)ψ(1)
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