ISI Admission Test 2022 Question Paper B.Stat B.Math UGA

FOR ISI EXAM PREPARATION

ISI 2022
Question Paper · B.Stat
B.Math UGA
EXAM YEAR TYPE SUBJECT

ISI 2022 Question Paper B.Stat B.Math UGA

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1. Suppose, for some θ ∈ [0, π2 ], cos 3θ 1
cos θ = 3 . Then (cot 3θ) tan θ equals

m
1 1
(A) 2 (B) 3

(C) 8
1

c o m (D) 7
1
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2. Any .positive real number x can be expanded as s e
s ex m l a
g la = a ·2 +a ·2 +· · ·+a ·2 +a ·2 +a ·2 +a ·2 +· · · ,
n
n
n−1
n−1
1
1
0
0
−1
−1
−2
−2
ag
a for some n ≥ 0, where each a ∈ {0, 1}. In the above-described
i
expansion of 21.1875, the smallest positive integer k such that
a−k �= 0 is:
(A) 3 (B) 2 (C) 1 (D) 4

3. Amongst all polynomials p(x) = c0 + c1 x + · · · + c10 x10 with real
coefficients satisfying |p(x)| ≤ |x| for all x ∈ [−1, 1], what is the
m
c. (C)o 2
maximum possible value of (2c0 + c1 )10 ?

em
(A) 410 (B) 310 10 (D) 1

l as
4. The locus of points z in the complex plane satisfying z 2 + |z|2 = 0 is
(A) a straight line a
g
(B) a pair of straight lines
(C) a circle
(D) a parabola

5. Let A and B be two 3 × 3 matrices such that (A + B)2 = A2 + B 2 .
Which of the following must be true?
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(A) A and B are zero matrices.

e m
e m (B) AB is the zero matrix.
las
las (C) (A − B)2 = A2 − B 2
ag
ag (D) (A − B)2 = A2 + B 2
6. Let Z denote the set of integers. Let f : Z → Z be such that
f (x)f (y) = f (x + y) + f (x − y) for all x, y ∈ Z. If f (1) = 3, then f (7)
equals

(A) 840 (B) 844 (C) 843 (D) 842

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7. The sides of a regular hexagon ABCDEF is extended by doubling
them to form a bigger hexagon A� B � C � D� E � F � as in the figure below.

Then the ratio of the areas of the bigger to the smaller hexagon is:
√ √
(A) 3 (B) 3 (C) 2 3 (D) 4

8. Let (n1 , n2 , · · · , n12 ) be a permutation of the numbers 1, 2, · · · , 12.
The number of arrangements with

n1 > n2 > n3 > n4 > n5 > n6

and
n6 < n7 < n8 < n9 < n10 < n11 < n12

equals:
�12� �12� �11� 11!
(A) 5 (B) 6 (C) 6 (D) 2
9. Suppose the numbers 71, 104 and 159 leave the same remainder r
when divided by a certain number N > 1. Then, the value of 3N + 4r
must equal:
(A) 53 (B) 48 (C) 37 (D) 23

10. In how many ways can we choose a1 < a2 < a3 < a4 from the set
{1, 2, . . . , 30} such that a1 , a2 , a3 , a4 are in arithmetic progression?

(A) 135 (B) 145 (C) 155 (D) 165

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11. What is the minimum value of the function |x−3|+|x+2|+|x+1|+|x|
for real x?

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(A) 3 (B) 5 (C) 6 (D) 8
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12. If x, y are positive real numbers such that 3x + 4y < 72, then the
s e m
s em possible value of 12xy(72 − 3x − 4y) is:
maximum
l a
g la (A) 12240 (B) 13824 (C) 10656 (D) 8640 ag
a 13. A straight road has walls on both sides of height 8 feet and 4 feet
respectively. Two ladders are placed from the top of one wall to the
foot of the other as in the figure below. What is the height (in feet)
of the maximum clearance x below the ladders?

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s e
l a
ag√ √
8
(A) 3 (B) 2 2 (C) 3 (D) 2 3

14. Consider a differentiable function u : [0, 1] → R. Assume the function
u satisfies
� a+r
1
u(a) = u(x) dx, for all a ∈ (0, 1) and all r < min(a, 1−a).
2r a−r

o m
m
Which of the following four statements must be true?
. c
m .co (A) u attains its maximum but not its minimum on the setm
s e 1}.
{0, 1}.

s e g la
(B) u attains its minimum but not maximum on the set {0,

g la a on the set
(C) If u attains either its maximum or its minimum
a {0, 1}, then it must be constant.
(D) u attains both its maximum and its minimum on the set {0, 1}.

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15. In the figure below, ABCD is a square and ΔCEF is a triangle with
given sides inscribed as in the figure. Find the length BE.

(A) √13 (B) √14
17 17

(C) √15 (D) √16
17 17

16. Let y = x + c1 , y = x + c2 be the two tangents to the ellipse
x2 + 4y 2 = 1. What is the value of |c1 − c2 |?
√ √ √
(A) 2 (B) 5 (C) 25 (D) 1

17. For n ∈ N, let an be defined as
� n
1
an = dx.
0 1 + nx2

Then limn→∞ an
(A) equals 0 (B) equals π4
(C) equals π2 (D) does not exist

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18. Let p and q be two non-zero polynomials such that the degree of
p is less than or equal to the degree of q, and p(a)q(a) = 0 for

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a = 0, 1, 2, . . . , 10. Which of the following must be true?

o m m
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(A) degree of q �= 10
(B) .degree of p �= 10 s e
se(C)m degree of q �= 5 l a
g la ag
a (D) degree of p �= 5
19. The number of positive integers n less than or equal to 22 such that
7 divides n5 + 4n4 + 3n3 + 2022 is

(A) 7 (B) 8 (C) 9 (D) 10

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20. A 3 × 3 magic square is a 3 × 3 rectangular array of positive integers
o
. c
such that the sum of the three numbers in any row, any column or any

e m
of the two major diagonals, is the same. For the following incomplete
magic square
l a s
ag
27 36

31

the column sum is

(A) 90 (B) 96 (C) 94 (D) 99
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21. Let 1, ω, ω 2 be the cube roots of unity. Then the product
s
e (1 − ω + ω )(1 − ω + ω )(1 − ω + ω ) · · · (1 − ωla+ ω )
las 2 2 22 22

a g
23 29 210

ag is equal to:

(A) 210 (B) 310 (C) 210 ω (D) 310 ω 2

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22. In a class of 45 students, three students can write well using either
hand. The number of students who can write well only with the right
hand is 24 more than the number of those who write well only with
the left hand. Then, the number of students who can write well with
the right hand is:

(A) 33 (B) 36 (C) 39 (D) 41

23. The number of triples (a, b, c) of positive integers satisfying the
equation
1 1 1 2
+ + =1+
a b c abc
and such that a < b < c, equals:

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

24. The function x2 loge x in the interval (0, 2) has:
(A) exactly one point of local maximum and no points of local minimum.
(B) exactly one point of local minimum and no points of local maximum.
(C) points of local maximum as well as local minimum.
(D) neither a point of local maximum nor a point of local minimum.

√ √ √
25. A triangle has sides of lengths 5, 2 2, 3 units. Then, the radius
of its inscribed circle is :
√ √ √ √ √ √
5+ 3+2 2 5+ 3+2 2
(A) 2 (B) 3
√ √ √ √ √ √
5+ 3−2 2
(C) 5+ 3+2 2 (D) 2

26. An urn contains 30 balls out of which one is special. If 6 of these balls
are taken out at random, what is the probability that the special ball
is chosen?
1 1 1 1
(A) 30
(B) 6
(C) 5
(D) 15

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27. If x1 > x2 > · · · > x10 are real numbers, what is the least possible
value of

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� x − x �� x − x � �x − x �
1 10 1 10 1 10

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··· ?
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x1 − x2 x2 − x3 x9 − x10

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(A) 1010 (B) 109 (C) 99 (D) 910
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28. Two ships are approaching a port along straight routes at constant
ag
ag velocities. Initially, the two ships and the port formed an equilateral
triangle. After the second ship travelled 80 km, the triangle became
right-angled.

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When the first ship reaches the port, the second ship was still 120 km
from the port. Find the initial distance of the ships from the port.

(A) 240 km (B) 300 km (C) 360 km (D) 180 km

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29. In the following diagram, four triangles and their sides are given.
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Areas of three of them are also given. Find the area x of the remaining

m s e
la
triangle.

s e g
g la a
a

(A) 12 (B) 13 (C) 14 (D) 15

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30. The range of values that the function

x2 + 2x + 4
f (x) =
2x2 + 4x + 9
takes as x varies over all real numbers in the domain of f is:
3 1 3 1
(A) < f (x) ≤ (B) ≤ f (x) <
7 2 7 2
3 4 3 1
(C) < f (x) ≤ (D) ≤ f (x) ≤
7 9 7 2

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