ISI Admission Test 2024 Question Paper JRF Physics, JRF QROR, MS(QMBA), M.Tech(QROR) MMA

FOR ISI ADMISSION TEST EXAM PREPARATION

ISI Admission Test
2024
Question Paper · JRF
Physics, JRF QROR,
MS(QMBA),
M.Tech(QROR) MMA
EXAM YEAR TYPE

ISI Admission Test 2024 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 A1 and A2 be two arbitrarily chosen subsets of {1, 2, . . . , n}.

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The number of ways such that A1 ∩ A2 = ∅ is

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co m
(A) 2n (B) 4n − 3n (C) 3n (D) 3n − 1

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g a
l 2. The limit a
a
√ √ √
1+ 2 + 3 3 + ··· + n n
lim
n→∞ n

(A) is equal to 0 (B) is equal to 1
(C) is equal to 2 (D) does not exist

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and B = {(x, y) ∈ R : x + ym> 1}. Then
3. For x, y ≥ 0, consider the regions A = {(x, y) ∈ R2 : x2 +y 2 ≤ 1}

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a
2 4 4

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(A) A � B
ag (B) B � A

(C) A ∩ B = ∅ (D) A ∪ B = R2

4. The roots of the equation x4 + x3 + x2 + x + 1 = 0, when plotted
in the complex plane, are

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(A) the vertices of a square

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(B) some of the vertices of a regular pentagon

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(C) some of the vertices of a regular hexagon
(D) some of the vertices of a regular heptagon a
ag

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5. Let f, g : [0, 1] → (0, ∞) be two continuously differentiable
functions such that f � (x) = g � (x) for all x ∈ [0, 1]. Suppose that

f ( 12 ) f ( 14 )
= 1. Then is equal to
g( 12 ) g( 14 )

(A) 1 (B) 2

1 1
(C) (D) either 2 or
2 4

6. For w ∈ C, let Re w and Im w denote the real and imaginary
parts of w, respectively. Suppose z ∈ C is such that |z| = 1.
Which of the following is necessarily true?

(A) Re z is rational implies that Re z 2 is rational
(B) Re z 2 is rational implies that Re z is rational
(C) Im z is rational implies that Im z 2 is rational
(D) Im z 2 is rational implies that Im z is rational

7. The maximum value of the function
x 1
f (x) = −
2 1 + x2
on the set {x ∈ R : x(x − 1) ≤ 12} is
33 8
(A) 0 (B) (C) − (D) −1
17 5

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8. There are two coins in a box. One of the coins is a fair coin and

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for the other, the probability of head is 23 . A coin is chosen at

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random from the box and tossed 5 times. Given that at least 4

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. coin was chosen?
heads were obtained, what is the conditional probability that
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the fair

la (A) 2521 ag
g
1 2 243

a
(B) (C) (D)
2 3 275

9. Let m be a positive integer such that 2m + 1 is a prime. Then
which of the following statements is true?

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(A) m must be a prime

(C) m must be a power of 2m
(B) m must be an odd integer
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a
(D) m must be a power

10. Let f : [0, 1] → (0, ∞) be a continuous function. Define
� x
g(x) = f (t) dt for 0 ≤ x ≤ 1.

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0

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If g(1) = 1, then the equation g(x) = 12
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(A) may not have a solution

gla (B) may have more than one solution
a
a (C) will have at least two solutions
(D) will have a unique solution

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11. It is given that the equation (cos p − 1)x2 + (cos p)x + sin p = 0
has a real root. Then p must lie in the interval
� π π�
(A) (0, 2π) (B) (−π, π) (C) − , (D) (0, π)
2 2

12. Let �� √ � �
a b 5
S= √ : a, b ∈ Z .
−b 5 a
The number of A ∈ S such that A−1 is also in S is

(A) 2 (B) 4
(C) more than 4 but finite (D) infinite

13. The equations of three curves are given below:
1 1
y1 (x) = x + x5 ,
2 2
1 3 1 7
y2 (x) = x + x,
2 2
1 3 1 7
y3 (x) = 1 + x + x .
2 2
Which of the following is true?

(A) y1 and y2 meet at 2 points, y1 and y3 meet at 1 point
(B) y1 and y2 meet at 3 points, y1 and y3 meet at 2 points
(C) y1 and y2 meet at more than 3 points, y1 and y3 meet at 1
point
(D) y1 and y2 meet at 3 points, y1 and y3 meet at 1 point

� � � �
14. Let C = { 12 , 21 } be a basis of R2 and T : R2 → R2 be
� � � x+y �
defined by T xy = x−2y . Let T (C) represent the matrix of T
with respect to the basis C. Which of the following is true?
� � � �
(A) T (C) = −33 −21 (B) T (C) = −33 −21
� −3 −1 � � 3 −1
�
(C) T (C) = 3 2 (D) T (C) = −3 2

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15. Suppose that A is an n × n matrix where n ≥ 2. Then An = 0

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if and only if

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(B) . all the eigenvalues of A are zero e m
(A) A = 0

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se(C) A is singular
la (D) Rank A ≤ 1 ag l
ag

16. For fixed a1 , a2 , . . . , a5 ∈ R, the graph of

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a1 x2 + a2 x + a3 y 2 + a4 y + a5 = 0

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on the xy-plane can never be

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(A) a circle
(B) an ellipse a
(C) a parabola with axis parallel to either the x-axis or the
y-axis
(D) a parabola with axis parallel to neither the x-axis nor the
y-axis

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17. If

las 20212024 = 100 m + n,
a g
ag for some m ∈ {0, 1, 2, . . . } and n ∈ {0, 1, . . . , 99}, then the value
of n is

(A) 21 (B) 31 (C) 61 (D) 81

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18. Let A be an m × n matrix with each entry equal to +1, −1 or
0 such that every column has exactly one +1 and exactly one
−1. Then

(A) Rank A ≤ m − 1 (B) Rank A = n
(C) m ≤ n (D) m − 1 ≤ n

19. Let f : R → R be a bounded differentiable function such that

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

where f � is the derivative of f . Then the value of f (0) is
1 1
(A) 0 (B) 1 (C) (D) −
2 2

20. Let f (θ) = 2(1 − cos θ) − θ sin θ = 4 sin 2θ (sin 2θ − 2θ cos 2θ ). Then
which of the following assertions is true?

(A) f (θ) < 0 for θ ∈ (0, π)
(B) f (θ) < 0 for θ ∈ (π, 2π)
(C) f (θ) > 0 for θ ∈ (0, 2π)
(D) f (θ) takes both negative and positive values in (0, 2π)

21. The value of the infinite sum
∞
� � nπ �
2−n cos
n=0
3

equals
√ √ √
(A) 1 (B) 4 + 2 3 (C) 2 + 3 (D) 2 − 3

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22. Consider the parabola y 2 = 4ax, a > 0. From any point

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A(x1 , y1 ) on the parabola a line is drawn which is perpendicular

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to the line x + a = 0. Let B be the foot of the perpendicular.

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. implies
If C is the focus of the parabola, then the triangle ABC is
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se(A) y = ±3a l
equilateral

la (C) y = ±4a
√
ag
ag 1 (B) y1 = ±2 3a
1 (D) x1 = 2a

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� 1
23. Let Sn = . Then
k=1
k

(A) S2n ≥ n2 for every n ≥ 1

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(B) {Sn }∞
n=1 is a bounded sequence

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(C) |S2n − S2n−1 | → 0 as n → ∞

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(D) Snn → 1 as n → ∞

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a
24. Let p(x) be a polynomial in x of degree k ≥ 0. For n = 1, 2, . . . ,
define an = p(n + 1) − p(n). If {an }∞
n=1 is in arithmetic
progression, then we must have

(A) k ≤ 1 (B) k ≤ 2
(C) k ≥ 3 (D) k ≥ 4

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25. Consider the circular disc on the xy-plane with centre (0, 1)

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s point A
em a
and radius 1 unit. Let A = (0, 2). The disc is rolled

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la ag of B are
the x-axis so that its centre is now at (1, 1). The

ag
has now moved to a point, say, B. Then the coordinates
� 1 1
� � �
(A) 1 + sin 2π , 1 + cos 2π (B) 1 + sin 1, 1 + cos 1
� 1 1
�
(C) 1 + sin 2π , 1 − cos 2π (D) (1, 2)

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26. For x ∈ R, let [x] denote the greatest integer less than or equal
to x. Let e2 < a < e3 . Then
� 1
dx
[loge x]
1/a x2

is equal to

(A) e2 + e + 1 − a (B) e2 + e + 1 − 2a
(C) e2 + e + 1 − 3a (D) e2 + e + 1 − 4a

27. Let n ≥ 2. Suppose α1 , α2 , . . . , αn are real numbers such that
α1 < α2 < · · · < αn . Let

P (x) = (x − α1 )2 (x − α2 ) . . . (x − αn ), x ∈ R.

If P � is the derivative of P , then P � (x) has

(A) no real root
(B) n real roots, the smallest of which is α1
(C) n − 1 real roots, the smallest of which is α1
(D) n real roots, the smallest of which is strictly larger than α1

28. Let A be a real matrix of order 3 such that AAT = I3 . Here
AT denotes the transpose of the matrix A and I3 is the identity
matrix of order 3. Assume that det A = −1 and AT �= A.
Let Tr A denote the trace of the matrix A. Then the quantity
(Tr A)2 − Tr A2 simplifies to

(A) Tr A (B) 2(Tr A)

(C) 3(Tr A) (D) 4(Tr A)

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29. The 26 letters of the English alphabet are randomly permuted.

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What is the probability that at least two of the three letters I,

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N, D come next to each other?

c. 325 e m
s
72 3 3 3

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(A) (B) (C) (D)

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26 13 25 × 26

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30. The value of � 1� 1
3/2
ey dy dx
0 x2
is
2 2e 2e 3
(A) (e − 1) (B) e − 1 (C) (D) −

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3 3 3 2

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a

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a

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