ISI Admission Test 2019 Question Paper B.Stat B.Math UGA

FOR ISI EXAM PREPARATION

ISI 2019
Question Paper · B.Stat
B.Math UGA
EXAM YEAR TYPE SUBJECT

ISI 2019 Question Paper B.Stat B.Math UGA

Notes · Sample Papers · Previous Year Papers · Mock Tests

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2019

1. You are given a 4 × 4 chessboard, and asked to fill it with five 3 × 1
pieces and one 1 × 1 piece. Then, over all such fillings, the number of
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squares that can be occupied by the 1 × 1 piece is
(A) 4
c. o(B) 8 (C) 12 (D) 16. s e m
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2. A brand called Jogger’s Pride produces pairs of shoes in three dif-

g l a ferent units that are named U1 , U2 and U3 . These units produce
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10%, 30%, 60% of the total output of the brand with the chance that
a pair of shoes being defective is 20%, 40%, 10% respectively. If a ran-
domly selected pair of shoes from the combined output is found to be
defective, then what is the chance that the pair was manufactured in
the unit U3 ?

(A) 30% (B) 15% (C) 35 × 100% (D) Cannot be determined from
the given data.

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3. Consider a paper in the shape of an equilateral triangle ABC with

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circumcenter O and perimeter 9 units. If we fold the paper in such a
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way that each of the vertices A, B, C gets identified with O, then the
area of the resulting shape in square units is:
√
3 3 4
ag 3√3 √
(A) (B) √ (C) (D) 3 3.
4 3 2

4. Let P be a regular twelve-sided polygon. The number of right-angled
triangles formed by the vertices of P is

(A) 60 (B) 120 (C) 160 (D) 220.

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5. If the n terms a1 , a2 , . . . , an are in arithmetic progression with incre-

e
ment r, then the difference between the mean of their squares and the m
e m square of their mean is
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a n −1
g r2 ((n − 1)2 − 1) r2 2 2
r (n − 1) 2

a (A)
12
(B)
12
(C)
12
(D)
12

6. A father wants to distribute a certain sum of money between his daugh-
ter and son in such a way that if both of them invest their shares in
the scheme that offers compound interest at 25 3 % per annum, for t and
t + 2 years respectively, then the two shares grow to become equal. If

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the son’s share was rupees 4320, then the total money distributed by
the father was

(A) rupees 9360 (B) rupees 9390 (C) rupees 16, 590 (D) rupees
16, 640.

7. Let α denote a real number. The range of values of |α − 4| such that
|α − 1| + |α + 3| ≤ 8 is

(A) (0, 7) (B) (1, 8) (C) [1, 9] (D) [2, 5].

8. For each natural number k, choose a complex number zk with |zk | = 1
and denote by ak the area of the triangle formed by zk , izk , zk + izk .
Then, which of the following is true for the series below?
∞
�
(ak )k
k=1

(A) It converges only if every zk lies in the same quadrant. (B) It
always diverges. (C) It always converges. (D) none of the above.

9. The function y = ekx satisfies

d2 y dy dy dy
( 2
+ )( − y) = y
dx dx dx dx
for
(A) exactly one value of k. (B) two distinct values of k. (C) three
distinct values of k. (D) infinitely many values of k.

10. For a real number θ, consider the following simultaneous equations:

cos(θ)x − sin(θ)y = 1

sin(θ)x + cos(θ)y = 2
The number of solutions of these equations in x and y is

(A) 0 (B) 1 (C) infinite for some values of θ (D) finite only when
mπ
θ= for integers m, and n �= 0.
n

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11. In the range 0 ≤ x ≤ 2π, the equation cos(sin(x)) = has
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2

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(A) 0 solutions. (B) 2 solutions. (C) 4 solutions. (D) infinitely
many solutions.
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12. A particle is allowed to move in the XY -plane by choosing any one of
the two jumps:
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ag 1. move two units to right and one unit up, i.e., (a, b) �→ (a + 2, b + 1)
or
2. move two units up and one unit to right, i.e., (a, b) �→ (a + 1, b + 2).
Let P = (30, 63) and Q = (100, 100). If the particle starts at the
origin, then
(A) P is reachable but not Q.
(B) Q is reachable but not P .
(C) both P and Q are reachable.
(D) neither P nor Q is reachable.

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. P , let Z(P ) denote the locus of
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13. For a real polynomial in one variable
points (x, y) in the plane such e
a s that P (x) + P (y) = 0. Then,
l Q and Q such that Z(Q ) is a circle and
Z(Q ) is a parabola. ag
(A) there exist polynomials 1 2 1
2
(B) there does not exist any polynomial Q such that Z(Q) is a circle
or a parabola.
(C) there exists a polynomial Q such that Z(Q) is a circle but there
does not exist any polynomial P such that Z(P ) is a parabola.
(D) there exists a polynomial Q such that Z(Q) is a parabola but there
does not exist any polynomial P such that Z(P ) is a circle.

14. Let P (X) = X 4 + a3 X 3 + a2 X 2 + a1 X + a0 be a polynomial in X with
m
m real coefficients. Assume that
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P (0) = 1, P (1) = 2, P (2) = 3, and P (3) = 4.

s e g la
la a the given data.
Then, the value of P (4) is

ag (A) 5 (B) 24 (C) 29 (D) not determinable from

15. Let f be a real-valued differentiable function defined on the real line R
such that its derivative f � is zero at exactly two distinct real numbers
α and β. Then,
(A) α and β are points of local maxima of the function f .
(B) α and β are points of local minima of the function f .

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(C) one must be a point of local maximum and the other must be a
point of local minimum of f .
(D) given data is insufficient to conclude about either of them being
local extrema points.

16. A school allowed the students of a class to go to swim during the days
March 11th to March 15, 2019. The minimum number of students the
class should have had that ensures that at least two of them went to
swim on the same set of dates is :

(A) 6 (B) 32 (C) 33 (D) 121.

17. Let a1 < a2 < a3 < a4 be positive integers such that
4
� 1 11
= .
ai 6
i=1

Then, a4 − a2 equals

(A) 11 (B) 10 (C) 9 (D) 8.

18. Three children and two adults want to cross a river using a rowing
boat. The boat can carry no more than a single adult or, in case no
adult is in the boat, a maximum of two children. The least number of
times the boat needs to cross the river to transport all five people is:

(A) 9 (B) 11 (C) 13 (D) 15.

19. Let M be a 3×3 matrix with all entries being 0 or 1. Then, all possible
values for det(M ) are

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

20. In the following picture, ABC is an isosceles triangle with an inscribed
circle with center O. Let P be the mid-point of BC. If AB = AC = 15
and BC = 10, then OP equals:

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A

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R Q

O

B C
P

o√m
c
√
(A) √
5
(B) √
5
m
(C) 2 5 (D) . 5 2.
√
2 2
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21. For every real number x �=a−1, let f (x) =
l
x
g
. Write f (x) = f (x) 1
and for n ≥ 2, f (x) =af (f
x+1
n (x)). Then,
n−1

f1 (−2) · f2 (−2) · · · · · fn (−2)

must equal

2n 1 �2n� �2n�
(A) 1·3·5····(2n−1) (B) 1 (C) (D) .
2 n n

22. Let the integers ai for 0 ≤ i ≤ 54 be defined by the equation
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(1 + X + X 2 )27 = a0 + a1 X + a2 X 2 + · · · + a54 X 54 .
m
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s e m
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g la (A) 326 (B) 327 (C) 328 (D) 329 .
a
a 23. An examination has 20 questions. For each question the marks that
can be obtained are either −1 or 0 or 4. Let S be the set of possible
total marks that a student can score in the examination. Then, the
number of elements in S is

(A) 93 (B) 94 (C) 95 (D) 96.

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24. Chords AB and CD of a circle intersect at right angle at the point P .
If the lengths of AP, P B, CP, P D are 2, 6, 3, 4 units respectively, then
the radius of the circle is:
√ √ √
65 66 67
(A) 4 (B) (C) (D)
2 2 2
25. The locus of points (x, y) in the plane satisfying sin2 (x) + sin2 (y) = 1
consists of

(A) A circle that is centered at the origin.
(B) infinitely many circles that are all centered at the origin.
(C) infinitely many lines with slope ±1.
(D) finitely many lines with slope ±1.
� n � �n+1�
26. The number of integers n ≥ 10 such that the product 10 · 10 is a
perfect square is

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

27. Let a ≥ b ≥ c ≥ 0 be integers such that 2a + 2b − 2c = 144. Then,
a + b − c equals:
(A) 7 (B) 8 (C) 9 (D) 10.

28. The number of integers n for which the cubic equation X 3 − X + n = 0
has 3 distinct integer solutions is:
(A) 0 (B) 1 (C) 2 (D) infinite.

29. The number of real solutions of the equation x2 = ex is:

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

30. The number of distinct real roots of the equation x sin(x)+cos(x) = x2
is
(A) 0 (B) 2 (C) 24 (D) none of the above.

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Solution Key

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1. (A)
m
2. (A)
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3. (C)

g la4. (A)
a 5. (C)
6. (B)

7. (C)

8. (C)

9. (C)

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10. (B)

11. (A)
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12. (D)

13. (C)

14. (C)

15. (D)

16. (C)

17. (B)

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18. (B)

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19. (B)

m s e
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20. (B)

las ag
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21. (A)
a 22. (A)

23. (C)

24. (B)

25. (C)

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26. (B)

27. (B)

28. (B)

29. (B)

30. (B)

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