ISI Admission Test 2018 Question Paper B.Stat B.Math UGA

FOR ISI EXAM PREPARATION

ISI 2018
Question Paper · B.Stat
B.Math UGA
EXAM YEAR TYPE SUBJECT

ISI 2018 Question Paper B.Stat B.Math UGA

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1. Let 0 < x < 16 be a real number. When a certain biased dice is rolled, a
particular face F occurs with probability 61 − x and and its opposite face
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occurs with probability 16 + x; the other four faces occur with probabil-
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l as ity 61 . Recall that opposite faces sum to 7 in any dice. Assume that the
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Then, the value of x is:
96 .

1 1 1 1
(A) 8 (B) 12 (C) 24 (D) 27 .

2. An office has 8 officers including two who are twins. Two teams, Red and
Blue, of 4 officers each are to be formed randomly. What is the probability
that the twins would be together in the Red team?

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(A) 6 (B) 7 (C) 4 (D) 14

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3. Suppose Roger has 4 identical green tennis balls and 5 identical red tennis
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balls. In how many ways can Roger arrange these 9 balls in a line so
that no two green balls are next to each other and no three red balls are
together?

(A) 8 (B) 9 (C) 11 (D) 12

4. The number of permutations σ of 1, 2, 3, 4 such that |σ(i) − i| < 2 for
every 1 ≤ i ≤ 4 is

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(A) 2 (B) 3 (C) 4 (D) 5.
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s e be the

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number of real zeroes of f , and e be the number of local extrema (i.e.,
local maxima or minima) of f . Which of the following is a lpossible (z, e)

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a (A) (4, 4) (B) (3, 3) (C) (2, 2) (D) (0, 0)
6. A number is called a palindrome if it reads the same backward or forward.
For example, 112211 is a palindrome. How many 6-digit palindromes are
divisible by 495?

(A) 10 (B) 11 (C) 30 (D) 45

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7. Let A be a square matrix of real numbers such that A4 = A. Which of
the following is true for every such A?
(A) det(A) �= −1
(B) A must be invertible.
(C) A can not be invertible.
(D) A2 + A + I = 0 where I denotes the identity matrix.

8. Consider the real-valued function h : {0, 1, 2, . . . , 100} → R such that
h(0) = 5, h(100) = 20 and satisfying h(i) = 12 (h(i + 1) + h(i − 1)), for
every i = 1, 2, . . . , 99. Then, the value of h(1) is:

(A) 5.15 (B) 5.5 (C) 6 (D) 6.15.

9. An up-right path is a sequence of points a0 = (x0 , y0 ), a1 = (x1 , y1 ), a2 =
(x2 , y2 ), . . . such that ai+1 − ai is either (1, 0) or (0, 1). The number of
up-right paths from (0, 0) to (100, 100) which pass through (1, 2) is:

(A) 3 · 197 (B) 3 · 100 (C) 2 · 197 (D) 3 · 197
� � � � � � � �
99 50 98 100 .

10. Let f (x) = 21 x sin x − (1 − cos x). The smallest positive integer k such
lim f (x) �= 0 is:
that x→0 xk

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

11. Nine students in a class gave a test for 50 marks. Let S1 ≤ S2 ≤ · · · ≤
S5 ≤ · · · ≤ S8 ≤ S9 denote their ordered scores. Given that S1 = 20 and
�9
Si = 250, let m be the smallest value that S5 can take and M be the
i=1
largest value that S5 can take. Then the pair (m, M ) is given by
(A) (20, 35) (B) (20, 34) (C) (25, 34) (D) (25, 50) .
12. Let 10 red balls and 10 white balls be arranged in a straight line such
that 10 each are on either side of a central mark. The number of such
symmetrical arrangements about the central mark is
10! 10!
(A) 5! 5! (B) 10! (C)(D) 2 · 10!
5!
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13. If z = x+iy is a complex number such that � z+i � < 1, then we must have

(A) x > 0 (B) x < 0 (C) y > 0 (D) y < 0.

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14. Let S = {x − y | x, y are real numbers with x2 + y 2 = 1}. Then the
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maximum number in the set S is

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√ √ √
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(A) 1 (B) 2 (C) 2 2 (D) 1 + 2.
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g a In a factory, 20 workers start working on a project of packing consign-
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a 4ments. They need exactly 5 hours to pack one consignment. Every hour
new workers join the existing workforce. It is mandatory to relieve a
worker after 10 hours. Then the number of consignments that would be
packed in the initial 113 hours is

(A) 40 (B) 50 (C) 45 (D) 52.

16. Let ABCD be a rectangle with its shorter side a > 0 units and perimeter

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2s units. Let P QRS be any rectangle such that vertices A, B, C and D

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respectively lie on the lines P Q, QR, RS and SP . Then the maximum
area of such a rectangle P QRS in square units is given by
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(A) s2
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(B) 2a (s − a) a (C) s2
(D) a (s − a). 5

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17. The number of pairs ofaintegers (x, y) satisfying the equation
2

xy(x + y + 1) = 52018 + 1 is:

(A) 0 (B) 2 (C) 1009 (D) 2018.

18. Let p(n) be the number of digits when 8n is written in base 6, and let
q(n) be the number of digits when 6n is written in base 4. For example,

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lim p(n)q(n)
82 in base 6 is 144, hence p(2) = 3. Then n→∞ equals:

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n2

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

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las 19. For a real number α, let S denote the set of those real g
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numbers
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a satisfy α sin(β) = β sin(α). Then which of the following
true ?
(A) For any α, Sα is an infinite set.
(B) Sα is a finite set if and only if α is not an integer multiple of π.
(C) There are infinitely many numbers α for which Sα is the set of all
real numbers.
(D) Sα is always finite.

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20. If A = and A2018 = , then a + d equals:
0 i c d
(A) 1 + i (B) 0 (C) 2 (D) 2018.

21. Let f : R → R and g : R → R be two functions. Consider the following
two statements:
lim f (x) exists and lim f (x)g(x) exists, then lim g(x) must exist.
P(1): If x→0 x→0 x→0
P(2): If f, g are differentiable with f (x) < g(x) for every real number x,
then f � (x) < g � (x) for all x.
Then, which one of the following is a correct statement?

(A) Both P(1) and P(2) are true.
(B) Both P(1) and P(2) are false.
(C) P(1) is true and P(2) is false.
(D) P(1) is false and P(2) is true.

22. The number of solutions of the equation sin(7x) + sin(3x) = 0 with
0 ≤ x ≤ 2π is

(A) 9 (B) 12 (C) 15 (D) 18.

23. A bag contains some candies, 25 of them are made of white chocolate and
the remaining 53 are made of dark chocolate. Out of the white chocolate
candies, 13 are wrapped in red paper, the rest are wrapped in blue paper.
Out of the dark chocolate candies, 23 are wrapped in red paper, the rest
are wrapped in blue paper. If a randomly selected candy from the bag is
found to be wrapped in red paper, then what is the probability that it is
made up of dark chocolate?
2 3 3 1
(A) 3 (B) 4 (C) 5 (D) 4

24. A party is attended by twenty people. In any subset of four people, there is
at least one person who knows the other three (we assume that if X knows
Y , then Y knows X). Suppose there are three people in the party who
do not know each other. How many people in the party know everyone?
(A) 16 (B) 17 (C) 18
(D) Cannot be determined from the given data.

25. The sum of all natural numbers a such that a2 − 16a + 67 is a perfect
square is:

(A) 10 (B) 12 (C) 16 (D) 22.

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26. The sides of a regular hexagon ABCDEF are extended by doubling them
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(for example, BA extends to BA� with BA� = 2BA) to form a bigger

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regular hexagon A� B � C � D� E � F � as in the figure.
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Then, the ratio of the areas of m
(C) 2 s3e (D) π.
√ the bigger to the smaller hexagon is:
(A) 2 (B) 3

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27. Between 12 noon and 1 PM, there are two instants when the hour hand
and the minute hand of a clock are at right angles. The difference in
minutes between these two instants is:

8 8 5 5
(A) 32 11 (B) 30 11 (C) 32 11 (D) 30 11 .

28. For which values of θ, with 0 < θ < π/2, does the quadratic polynomial
in t given by t2 + 4t cos θ + cot θ have repeated roots?

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5π 5π 5π 5π
(A) 6 or 18
π
(B) 6 or 12
π
(C) 12 or 18
π
(D) 12 or 12
π

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29. Let α, β, γ be complex numbers which are the vertices of an equilateral

las triangle. Then, we must have:
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(C) α +β +γ +αβ+βγ+γα = 0 (D) (α−β)2 +(β−γ)2 +(γ−α)2 = 0
2 2 2

30. Assume that n copies of unit cubes are glued together side by side to form
a rectangular solid block. If the number of unit cubes that are completely
invisible is 30, then the minimum possible value of n is:

(A) 204 (B) 180 (C) 140 (D) 84.

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