ISI Admission Test 2020 Question Paper B.Stat B.Math UGA

FOR ISI EXAM PREPARATION

ISI 2020
Question Paper · B.Stat
B.Math UGA
EXAM YEAR TYPE SUBJECT

ISI 2020 Question Paper B.Stat B.Math UGA

Notes · Sample Papers · Previous Year Papers · Mock Tests

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1. The number of subsets of {1, 2, 3, . . . , 10} having an odd number of ele-
ments is
m
(A) 1024
om (B) 512 (C) 256 (D) 50. . co
. c e m
m
e l as
a s
l which of the following is true ?
2. For the function on the real line R given by f (x) = |x| + |x + 1| + e ,
ag x

ag
(A) It is differentiable everywhere.
(B) It is differentiable everywhere except at x = 0 and x = −1.
(C) It is differentiable everywhere except at x = 1/2.
(D) It is differentiable everywhere except at x = −1/2.

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3. If f, g are real-valued differentiable functions on the real line R such that

s e
f (g(x)) = x and f 0 (x) = 1 + (f (x))2 , then g 0 (x) equals

l a
+g
(B) 1 a
1 2 1
(A) x (C) (D) 1 + x4 .
1 + x2 1+x 4

4. The number of real solutions of ex = sin(x) is

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

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n

.co
e−k/n
m
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5. What is the limit of as n tends to ∞ ?

m
n
s e
la
k=1

s e g
g la (A) The limit does not exist.
a
a (B) ∞
(C) 1 − e−1
(D) e−0.5

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6. A group of 64 players in a chess tournament needs to be divided into 32
groups of 2 players each. In how many ways can this be done ?
     
64! 64 62 4 2
(A) (B) ···
32!232 2 2 2 2
64! 64!
(C) (D)
32!32! 264
9999
X 1
7. The integral part of √ equals
n=2
n

(A) 196 (B) 197 (C) 198 (D) 199.

8. Let an be the number of subsets of {1, 2, . . . , n} that do not contain any
two consecutive numbers. Then

(A) an = an−1 + an−2 (B) an = 2an−1
(C) an = an−1 − an−2 (D) an = an−1 + 2an−2 .

9. There are 128 numbers 1, 2, . . . , 128 which are arranged in a circular
pattern in clockwise order. We start deleting numbers from this set
in a clockwise fashion as follows. First delete the number 2, then skip
the next available number (which is 3) and delete 4. Continue in this
manner, that is, after deleting a number, skip the next available number
clockwise and delete the number available after that, till only one number
remains. What is the last number left ?

(A) 1 (B) 63 (C) 127 (D) None of the above.

10. Let z and w be complex numbers lying on the circles of radii 2 and 3
respectively, with centre (0, 0). If the angle between the corresponding
vectors is 60 degrees, then the value of |z + w|/|z − w| is:
√ √ √ √
19 7 12 7
(A) √ (B) √ (C) √ (D) √ .
7 19 7 12

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11. Two vertices of a square lie on a circle of radius r and the other two
vertices lie on a tangent to this circle. Then the length of the side of the
square is
o m
3r o
m . c
(A) .c
4r 6r 8r
e m
em
2
(B)
3
(C)
5
(D)
5
.

l as
s
l12.a For a real number x, let [x] denote the greatest integer less than or equal ag
g
a to x. Then the number of real solutions of 2x − [x] = 4 is
(A) 4 (B) 3 (C) 2 (D) 1.

13. Let f, g be differentiable functions on the real line R with f (0) > g(0).
Assume that the set M = {t ∈ R | f (t) = g(t)} is non-empty and that
f 0 (t) ≥ g 0 (t) for all t ∈ M . Then which of the following is necessarily
true ?
m
(A) If t ∈ M , then t < 0.
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s e
l a
(B) For any t ∈ M , f 0 (t) > g 0 (t).
/ M , f (t)g> g(t).
(C) For any t ∈
(D) None of the above.
a

14. Consider the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, . . . obtained by
writing one 1, two 2’s, three 3’s and so on. What is the 2020th term in
the sequence ?

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(A) 62 (B) 63 (C) 64 (D) 65
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15. Let A = {x1 , x2 , . . . , x50 } and B = {y1 , y2 , . . . , y20 } be two sets of real
g
g la numbers. What is the total number of functions f : A → B such that f
a
a
is onto and f (x1 ) ≤ f (x2 ) ≤ · · · ≤ f (x50 ) ?

49 49 50 50





(A) 19 (B) 20 (C) 19 (D) 20

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16. The number of complex roots of the polynomial z 5 − z 4 − 1 which have
modulus 1 is

(A) 0 (B) 1 (C) 2 (D) more than 2.

17. The number of real roots of the polynomial

p(x) = (x2020 + 2020x2 + 2020)(x3 − 2020)(x2 − 2020)

is

(A) 2 (B) 3 (C) 2023 (D) 2025.

18. Which of the following is the sum of an infinite geometric sequence whose
terms come from the set {1, 21 , 14 , . . . , 21n , . . .} ?

1 1 1 1
(A) (B) (C) (D)
5 7 9 11

19. If a, b, c are distinct odd natural numbers, then the number of rational
roots of the polynomial ax2 + bx + c

(A) must be 0.

(B) must be 1.

(C) must be 2.

(D) cannot be determined from the given data.

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20. Let A, B, C be finite subsets of the plane such that A ∩ B, B ∩ C and
C ∩ A are all empty. Let S = A ∪ B ∪ C. Assume that no three points
m
co
of S are collinear and also assume that each of A, B and C has at least

om .
3 points. Which of the following statements is always true ?

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(A) There exists a triangle having a vertex from each of A, B, C that
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as
l (B) Any triangle having a vertex from each of A, B, C must contain a a
does not contain any point of S in its interior.
g
g
a point of S in its interior.
(C) There exists a triangle having a vertex from each of A, B, C that
contains all the remaining points of S in its interior.

(D) There exist 2 triangles, both having a vertex from each of A, B, C
such that the two triangles do not intersect.

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21. Shubhaangi thinks she may be allergic to Bengal gram and takes a test
m
e
that is known to give the following results:
s the allergy, the test says “Yes” 90% of
l a
• For people who really do have
g
the time.
a
• For people who do not have the allergy, the test says “Yes” 15% of the
time.
If 2% of the population has the allergy and Shubhaangi’s test says “Yes”,
then the chances that Shubhaangi does really have the allergy are

(A) 1/9

m
(B) 6/55

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(C) 1/11

e m
e m (D) cannot be determined from the given data.
las
las ag
ag 22. If sin(tan−1 (x)) = cot(sin−1 (
q
13
17 )) then x is

4 2 q
172 −132
q
172 −132
(A) (B) (C) 172 +132
(D) 17×13 .
17 3

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23. If the word PERMUTE is permuted in all possible ways and the different
resulting words are written down in alphabetical order (also known as
dictionary order), irrespective of whether the word has meaning or not,
then the 720th word would be:

(A) EEMPRTU (B) EUTRPME (C) UTRPMEE (D) MEET-
PUR.

24. The points (4, 7, −1), (1, 2, −1), (−1, −2, −1) and (2, 3, −1) in R3 are the
vertices of a

(A) rectangle which is not a square.

(B) rhombus.

(C) parallelogram which is not a rectangle.

(D) trapezium which is not a parallelogram.

25. Let f (x), g(x) be functions on the real line R such that both f (x) + g(x)
and f (x)g(x) are differentiable. Which of the following is FALSE ?

(A) f (x)2 + g(x)2 is necessarily differentiable.

(B) f (x) is differentiable if and only if g(x) is differentiable.

(C) f (x) and g(x) are necessarily continuous.

(D) If f (x) > g(x) for all x ∈ R, then f (x) is differentiable.

26. Let S be the set consisting of all those real numbers that can be written
as p − 2a where p and a are the perimeter and area of a right-angled
triangle having base length 1. Then S is

(A) (2, ∞) (B) (1, ∞) (C) (0, ∞) (D) the real line R.

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27. Let S = {1, 2, . . . , n}. For any non-empty
P subset A of S, let l(A) denote
the largest number in A. If f (n) = A⊆S l(A), that is, f (n) is the sum
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co
of the numbers l(A) while A ranges over all the nonempty subsets of S,

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then f (n) is

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em
(A) 2n (n + 1) (B) 2n (n + 1) − 1
l as
l as n (D) 2n (n − 1) + 1. ag
g
(C) 2 (n − 1)
a
28. The area of the region in the plane R2 given by points (x, y) satisfying
|y| ≤ 1 and x2 + y 2 ≤ 2 is

(A) π + 1 (B) 2π − 2 (C) π + 2 (D) 2π − 1.

o
29. Let n be a positive integer and t ∈ c(0,
m n
n  

.
X
r n−r
1). Then r t (1 − t)

e m r
r=0
equals
l as
(A) nt (B) (n − 1)(1a −gt) (C) nt + (n − 1)(1 − t) (D) (n − 2n + 2)t. 2

30. For any real number x, let [x] be the greatest integer m such that m ≤ x.
Then the number of points of discontinuity of the function g(x) = [x2 −2]
on the interval (−3, 3) is

(A) 5 (B) 9 (C) 13 (D) 16.
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