ISI Admission Test 2022 Question Paper JRF Physics, JRF QROR, MS(QMBA), M.Tech(QROR) MMA

FOR ISI ADMISSION TEST EXAM PREPARATION

ISI Admission Test
2022
Question Paper · JRF
Physics, JRF QROR,
MS(QMBA),
M.Tech(QROR) MMA
EXAM YEAR TYPE

ISI Admission Test 2022 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. Let C be the circle in the xy-plane which passes through the

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origin and touches the line y = −2. If the centre of C lies on

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the straight line x + y = 1, then the radius of C is

c. o √12 . (B) 5 ± √12 . (C) 4 ± √12 . (D) √12 ± 3 . s e m
em a
(A) 6 ±

a s g l
g l 2. The set of positive real numbers x for which the series a
a � ∞
1
n=1
xn + x−n

diverges

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. c
(A) is the empty set.

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ateleast 2 elements.
(B) is a set having exactly 1 element.
(C) is a finite set having s
(D) is an infinite set.gl
a
a
3. Let X be a non-empty set and let P(X) be its power set. Define
two operations ‘+’ and ‘∗’ on P(X) as follows. For A, B ∈ P(X),
let
A+B =A∪B, A∗B =A∩B.

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If G1 = (P(X), +) and G2 = (P(X), ∗), then,

c. o se
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(A) G1 and G2 are groups.

ag (B) G1 is a group but G2 is not.
(C) G2 is a group but G1 is not.
(D) neither G1 nor G2 is a group.

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4. The system of equations

x + 2y + 3z = c
2x − y + z = 2
ax + y + 4z = b
has no solution if

(A) a �= 3. (B) b = c + 2.
(C) a = 3, b �= c + 2. (D) a = 3, b = c + 2.

5. Let 
4x − 1, if x ≤ 2 ,
f (x) =
 x2 , if x > 2 .
At x = 2,

(A) f is differentiable.
(B) the left and right derivatives of f exist and are unequal.
(C) the left derivative of f exists and the right derivative does
not.
(D) the right derivative of f exists and the left derivative does
not.

6. What is the sum of the maximum and the minimum values of
1
1 + (2 cos x − 4 sin x)2

over x ∈ R ?

19 22 23 25
(A) 21
(B) 21
(C) 22
(D) 23

2

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7. Suppose that f : A → B is a function, where A and B are sets

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having 10 and 20 elements, respectively. Let us denote for all

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subsets U of A,

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eand for all subsets V of B, a
f (U ) = {f (a) : a ∈ U } ,

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g l a
a −1
f (V ) = {a ∈ A : f (a) ∈ V } .

Which of the following is true?

(A) f −1 (f (A)) = A and f (f −1 (B)) = B

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(B) f −1 (f (A)) = A and f (f −1 (B)) �= B

em (B)) �= B
(C) f −1 (f (A)) �= A and f (f −1 (B)) = B
(D) f (f (A)) �= A and fs(f
la
−1 −1

a g
8. Let n
�
P (x) = ai x i , x ∈ R ,
i=0

where n ≥ 5 and a0 , . . . , an ∈ R. If

P (x) − ex

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lim = 0,

co
x→0 x4

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then

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1

la se
s
(A) a4 = 0 . (B) a4 = 24 .

la
(C) a5 = 0 . 1
(D) a5 = 120 .
a g
ag

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9. For all x ∈ R, let
d−1
� d−1
�
d i d
P (x) = x + ai x , and Q(x) = x + bi x i ,
i=0 i=0

where d ≥ 1 and a0 , . . . , ad−1 , b0 , . . . , bd−1 ∈ R. Suppose that
n ≥ d and x1 , . . . , xn are distinct real numbers. Let C(P, Q) be
the number of elements in the set

{xi : P (xi ) �= Q(xi )} .

Which of the following conditions ensures that P and Q are
identical?

(A) C(P, Q) ≤ 1
(B) C(P, Q) ≤ d − 1
(C) C(P, Q) ≤ n − d
(D) None of the above

10. For any positive integer n, the value of
�n � �
j 2n
(−1)
j=0
2j

is

(A) 0 . (B) 2n sin(nπ/2) .
(C) 2n/2 cos(nπ/2) . (D) 2n cos(nπ/2) .

11. The set of all solutions of the equation

z 3 + 8i = 0 ,
√
where i = −1 is
√
(A) {2i}. (B) {2 2i}.
√ √ √ √
(C) {2i, ( 5−1)i, (− 5−1)i}. (D) {2i, 3 − i, − 3 − i}.

4

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12. A lie detector determines correctly whether a person is lying or

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telling the truth with probability 0.8, independently of whether

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the person is actually lying or not. Assume that people tell

c o
the .lie detector says that the person is telling the truth, what
the truth 90% of the times. Given that for a particular person
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seis the conditional probability that he is actually telling the truth?
la ag l
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a (A) 0.9 (B) 0.8 (C) 36
37
(D) 76
77

13. Let f (x, y) = ax2 + by 2 − c, for (x, y) ∈ R2 , where a, b, c ∈ R
with bc > 0 and a �= 0. Let

S = {(x, y) ∈ R2 : f (x, y) = 0} .

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Consider the map p : R2 → R defined by p(x, y) = x. If

s emy) : (x, y) ∈ S} ,
l a
p(S) = {p(x,

g
a can never be true?
then which of the following

(A) p(S) = R.
(B) p(S) is a closed bounded interval.
(C) p(S) is an open bounded interval.

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(D) For all z ∈ p(S), −z also belongs to p(S).

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14. For α ∈ R with α �= 0,

las � ∞
a g
ag
αx−2|αx|
xe dx
−∞

equals
10 −2
(A) 9
α . (B) 10
9
(α|α|)−1 .
8 −2 8
(C) 9
α . (D) 9
(α|α|)−1 .

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15. Suppose that f : R → R is a differentiable non-decreasing
function whose derivative f � is continuous. Fix a < b, and let

S = {x ∈ (a, b) : f � (x) = 0} .

If f (a) < f (b), then which of the following necessarily holds?

(A) The set S is empty.
(B) The set S is finite and non-empty.
(C) The set S is countably infinite.
(D) None of the above.

16. Let S be the set of all continuous functions f : [0, ∞) → R
satisfying the equation
� x
f (t) dt = f (x) − 2 , for all x > 0 .
0

Then,

(A) S is an infinite set.
(B) S has exactly one element.
(C) S is a finite set containing more than one element.
(D) S is the empty set.

17. Let A = (3, 1), B = (4, 3) and C = (4, 4) be three points in the
coordinate plane with origin O. Let D be an arbitrary point
on the line segment OC. What is the largest possible area of
�ABD?

1 5
(A) 2
(B) 1 (C) 2
(D) 5

6

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18. Suppose that f : R → R is a differentiable function satisfying

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f (x) + f � (x) = sin x + cos x for all x ∈ R ,

o m m .
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where f � is the derivative of f . Then, f is a bounded function
ifm s e
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la (A) f (0) = 0. (B) f (0) ≤ 0. (C) f (0) ≥ 0. (D) f (0) > 0. a
and only if

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19. If A is a 4 × 4 real matrix with at least 2 non-zero entries, then
which of the following necessarily holds?

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(A) At least one eigenvalue of A is non-zero.

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(B) At least one eigenvalue of A is real.
(C) Rank of A is at least 2. m
(D) None of the above. as
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20. If α and β are the two roots of the quadratic equation

x2 + x + 1 = 0 ,

then the value of α2022 + β 2022 is

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(A) −1 . (B) 0 . (C) 1 . (D) 2 .

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21. Let

gla a
Z100 = {0, 1, 2, . . . , 99}

a be the additive group of integers modulo 100. How many
subgroups does Z100 have which do not contain 10?

(A) 5 (B) 4 (C) 3 (D) 2

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√ √
22. If 3 x + 5 − 3 x − 5 = 1, then what is the value of x2 ?

(A) 50 (B) 51 (C) 52 (D) 53

23. How many natural numbers n are there such that 100 ≤ n ≤ 999
and the sum of the digits of n is divisible by 5?

(A) 180 (B) 120 (C) 200 (D) 90

24. In how many ways can 12 identical balls be thrown into 5
distinguishable boxes such that no box gets an odd number of
balls?
�10� �11�
(A) 4
(B) 5
(C) 56 (D) 65

25. The range of the function
� �
2 1 −x2
f (x) = x + e , x ∈ (−∞, ∞) ,
2

is
� � � �
(A) 0, e−1/2 . (B) 0, e−1/2 .
� � � �
(C) 12 , e−1/2 . (D) 0, 34 .

8

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26. An examination centre has 50 rooms. In each room, 10

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candidates have been assigned. Each candidate is absent with

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probability p, where 0 < p < 1, independently of the other

c o
have. full attendance?
candidates. What is the probability that at least one room will
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a (A) (1 − p) 500

50
(B) (1 − p10 )
50
(C) [1 − (1 − p)10 ]
50
(D) 1 − [1 − (1 − p)10 ]

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27. Suppose Δ is the bounded region in the positive quadrant of the

mintegral
xy-plane enclosed by the x-axis, the straight line y = x and the
curve x + y = π/2. Then e
�� a s
2 2
the
l
ag sin(x + y ) dx dy
Δ
2 2

equals
� 2
�
π π
(A) 8
. (B) 8
1 − cos π4 .
√
π 2−1
(C) 4
. (D) 4
.

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28. A sequence of real numbers {an } has a property P if and only
if for every ε > 0 there exists N ∈ N such that |an | < ε for all
n ≥ N . Then, {an } does not have the property P if and only if

(A) there exists ε > 0 such that for all N ∈ N there exists
n ≥ N satisfying |an | ≥ ε.
(B) for all ε > 0 and N ∈ N there exists n ≥ N such that
|an | ≥ ε.
(C) there exists ε > 0 and N ∈ N such that |an | ≥ ε for all
n ≥ N.
(D) for all ε > 0 there exists N ∈ N such that |an | ≥ ε for some
n ≥ N.

29. How many ordered pairs (x, y) of positive integers satisfy the
equation
x2 − 2y 2 = 1 ,

where y is a prime number?

(A) 0 (B) 1 (C) 2 (D) 4

30. Let � �
4
�
S= (x1 , x2 , x3 , x4 ) ∈ R4 : x2i = 1 ,
i=1

and
4
� 4
�
m= min |xi | , M= max |xi | .
(x1 ,...,x4 )∈S (x1 ,...,x4 )∈S
i=1 i=1

Then, m/M equals

1 1 3
(A) 0 . (B) 4
. (C) 2
. (D) 4
.

10

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