ISI Admission Test 2021 Question Paper JRF Physics, JRF QROR, MS(QMBA), M.Tech(QROR) MMA

FOR ISI ADMISSION TEST EXAM PREPARATION

ISI Admission Test
2021
Question Paper · JRF
Physics, JRF QROR,
MS(QMBA),
M.Tech(QROR) MMA
EXAM YEAR TYPE

ISI Admission Test 2021 Question Paper
SUBJECT

JRF Physics, JRF QROR, MS(QMBA), M.Tech(QROR) MMA

Notes · Sample Papers · Previous Year Papers · Mock Tests

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1. Suppose f : R → R is a continuous function such that f (x) = 2−sin x+4
2x
for all x 6= 0. Then the value of f (0) is

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(A) − 81 (B) 18 (D) − 14
.
(C) 0

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a
l dice is a perfect square, then the probability that the number 3 a
2. A person throws a pair of fair dice. If the sum of the numbers on

g
a appeared on at least one of the dice is
the

(A) 1/9 (B) 4/7 (C) 1/18 (D) 7/36

3. Consider the system of linear equations: x+y+z = 5, 2x+2y+3z = 4.
Then

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. c
(A) the system is inconsistent

(C) the system has infinitelyse
m
(B) the system has a unique solution

a
(D) none of the above isgltrue
many solutions

a
4. If g 0 (x) = f (x) then
R
x3 f (x2 )dx is given by

1 2
R R
(A) x2 g(x2 ) − xg(x2 )dx + C (B) 2
x g(x2 ) − xg(x2 )dx + C
x2 g(x2 ) − 21 xg(x2 )dx + C
R R
(C) 2x2 g(x2 ) − xg(x2 )dx + C (D)

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5. If (n C0 +n C1 )(n C1 +n C2 ) · · · (n Cn−1 +n Cn ) = k n C0 n C1 · · · n Cn−1 ,

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then k is equal to
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(n+1)n

la
(n+1)n
s(D)e (n+1)n+1

s
nn

g
(A) (B) (C)

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n! n! nn! n!

ag 6. Let {f } be a sequence of functions defined as follows:
n

fn (x) = xn cos(2πnx), x ∈ [−1, 1].

Then limn→∞ fn (x) exists if and only if x belongs to the interval

(A) (−1, 1) (B) [−1, 1) (C) [0, 1] (D) (−1, 1]

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7. Let S be a set of n elements. The number of ways in which n distinct
non-empty subsets X1 , ..., Xn of S can be chosen such that X1 ⊆
X2 · · · ⊆ Xn , is

n

n

· · · nn


(A) 1 2
(B) 1 (C) n! (D) 2n

8. Let A be a 4 × 4 matrix such that both A and Adj(A) are non-null.
If det A = 0, then the rank of A is

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

9. The set of all a satisfying the inequality
Z a
1 3√ 1 
√ x + 1 − √ dx < 4
a 1 2 x

is equal to the interval

(A) (−5, −2) (B) (1, 4) (C) (0, 2) (D) (0, 4)

10. Let C0 be the set of all continuous functions f : [0, 1] → R and C1
be the set of all differentiable functions g : [0, 1] → R such that the
derivative g 0 is continuous. (Here, differentiability at 0 means right
differentiability and differentiability at 1 means left differentiability.)
If T : C1 → C0 is defined by T (g) = g 0 , then

(A) T is one-to-one and onto
(B) T is one-to-one but not onto
(C) T is onto but not one-to-one
(D) T is neither one-to-one nor onto.

11. Suppose a, b, c are in A.P. and a2 , b2 , c2 are in G.P. If a < b < c and
a + b + c = 32 , then the value of a is

1
(A) √
2 2
(B) − 2√1 2 (C) 1
2
− √13 (D) 1
2
− √12

2

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12. The number of distinct even divisors of

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5

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Y
k!

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k=1

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is

a s (D) 72 a g l
g l
a13. Let D be the triangular region in the xy-plane with vertices at
(A) (B) 32 (C) 64

(0, 0), (0, 1) and (1, 1). Then the value of
Z Z
2
2
dx dy
D 1+x

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is

c. o (C) 2 ln 2
em
(A) π2 (B) π2 − ln 2 (D) ln 2

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ag
14. Given a real number α ∈ (0, 1), define a sequence {xn }n≥0 by the
following recurrence relation:

xn+1 = αxn + (1 − α)xn−1 , n ≥ 1.

If limn→∞ xn = ` then the value of ` is

αx0 +x1 (1−α)x0 +x1
(A) (B)
m
1−α 2−α

c o
αx0 +x1 (1−α)x1 +x0

m .
(C) (D)

c. o 15. A straight line passes through the intersection of the lines
2−α 2−α

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e given by
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a of the
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same sign on the coordinate axes. The equation ofathe straight line is
3x − 4y + 1 = 0 and 5x + y = 1 and makes equal intercepts

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(A) 23x − 23y + 11 = 0 (B) 23x − 23y − 11 = 0
(C) 23x + 23y + 11 = 0 (D) 23x + 23y − 11 = 0

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16. The series
X 3 · 6 · 9 · · · 3n
xn , x>0
n
7 · 10 · 13 · · · (3n + 4)

(A) converges for 0 < x ≤ 1 and diverges for x > 1
(B) converges for all x > 0
(C) converges for 0 < x < 21 and diverges for x ≥ 12
(D) converges for 21 < x < 1 and diverges for 0 < x ≤ 12 , x ≥ 1.

17. Suppose A and B are two square matrices such that the largest eigen-
value of (AB − BA) is positive. Then the smallest eigen value of
(AB − BA)

(A) must be positive (B) must be negative
(C) must be 0 (D) is none of the above

18. The number of saddle points of the function f (x, y) = 2x4 − x2 + 3y 2
is

(A) 1 (B) 0 (C) 2 (D) none of the above

19. Suppose G is a cyclic group and a, b ∈ G. There does not exist any
x ∈ G such that x2 = a. Also, there does not exist an y ∈ G such
that y 2 = b. Then,

(A) there exists an element g ∈ G such that g 2 = ab.
(B) there exists an element g ∈ G such that g 3 = ab.
(C) the smallest exponent k > 1 such that g k = ab for some g ∈ G is
4.
(D) none of the above is true.

20. The number of real roots of the polynomial x3 − 2x + 7 is

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

4

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21. Suppose that a 3 × 3 matrix A has an eigen value −1. If the matrix
A + I is equal to

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 
1 0 −2

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 0 0 0 
.
 

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e the eigen vectors of A corresponding to the eigenvalue −1 are in a
0 0 0

a s g l
l the form, a
then

ag    
2t 2t
(A)  0  , t ∈ R (B)  s  , s, t ∈ R
   

t t

   
t t

m
(C)  0  , t ∈ R (D)  s  , s, t ∈ R
   

m
−2t 2t

22. Consider the function f : R s →e R defined by f (x, y) = x (y − 1). For
a
~u = ( , ) and ~v = (3, 4),glthe value of the limit
2 2

a
1 1
2 2

f (~u + t~v ) − f (~u)
lim
t→0 t
is

(A) 34 6
(B) 13 (C) − 21 (D) none of the above

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23. Suppose φ is a solution of the differential equation y 00 − y 0 − 2y = 0

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such that φ(0) = 1 and φ0 (0) = 5. Then

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(A) φ(x) → ∞ as |x| → ∞ (B) φ(x) → −∞ as |x| → ∞
a (C) φ(x) → −∞ as x → −∞ (D) φ(x) → ∞ as x → −∞

24. A fair die is rolled five times. What is the probability that the largest
number rolled is 5?

(A) 5/6 (B) 1/6 (C) 1 − (1/6)6 (D) (5/6)5 − (2/3)5

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25. Two rows of n chairs, facing each other, are laid out. The number of
different ways that n couples can sit on these chairs such that each
person sits directly opposite to his/her partner is

(A) n! (B) n!/2 (C) 2n n! (D) 2n!.

26. Consider the function f : C → C defined on the complex plane C by
f (z) = ez . For a real number c > 0, let A = {f (z)| Re z = c} and
B = {f (z)| Im z = c}. Then

(A) both A and B are straight line segments
(B) A is a circle and B is a straight line segment
(C) A is a straight line segment and B is a circle
(D) both A and B are circles

27. Consider two real valued functions f and g given by

x
f (x) = x−1 for x > 1, and g(x) = 7 − x3 for x ∈ R.

Which of the following statements about inverse functions is true?

(A) Neither f −1 nor g −1 exists
(B) f −1 exists, but not g −1
(C) f −1 does not exist, but g −1 does
(D) Both f −1 and g −1 exist.

28. A circle is drawn with centre at (−1, 1) touching x2 +y 2 −4x+6y−3 = 0
externally. Then the circle touches

(A) both the axes (B) only the x-axis
(C) none of the two axes (D) only the y-axis

6

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29. Let f (x − y) = ff (x)
(y)
for all x, y ∈ R and f 0 (0) = p, f 0 (5) = q. Then the
value of f 0 (−5) is

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2
(C) pq (D) pq
.
(B) −q
e
(A) q

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30. Let
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ag
 
a 1 1
A =  b a 1 .
 

1 1 1
The number of elements in the set
n o
2
(a, b) ∈ Z : 0 ≤ a, b ≤ 2021, rank(A) = 2

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(A) 2021 (B) 2020 (C) 20212 − 1 (D) 2020 × 2021

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