ISI Admission Test 2026 Question Paper JRF Physics, JRF QROR, MS(QMBA), M.Tech(QROR) MMA

FOR ISI ADMISSION TEST EXAM PREPARATION

ISI Admission Test
2026
Question Paper · JRF
Physics, JRF QROR,
MS(QMBA),
M.Tech(QROR) MMA
EXAM YEAR TYPE

ISI Admission Test 2026 Question Paper
SUBJECT

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

Notes · Sample Papers · Previous Year Papers · Mock Tests

m
m .co
c. o

e m
em l a s
s a g

• The symbol R denotes the set of all real numbers.

m
m .co
• The symbol Q denotes the set of all rationals numbers.

c o
. Z denotes the set of all integers. e m
e m
• The symbol
l a s
a s ag
•gl
a
The symbol N denotes the set of all natural numbers.

−−−−−

1. The number of factors of 2025 (including 1 and itself) equals

m
(A) 15 (B) 25 (C) 35 (D) 45

. co
"se
m#
la a 1
2. If the matrix

Ag=
a 2 3
has 1 as an eigenvalue, then det(A) equals

(A) 1 (B) 2 (C) 4 (D) 10

m
co
3. Suppose sec x + tan x = 3 for some x ∈ (0, π/2). Then sin 2x

m .
c. o 24 em
equals

em
1 3 4
l a s
s ag
(A) (B) (C) (D)

la
25 5 5 25

ag 2
4. Let V := {a + a x + a x | a , a , a ∈ R}. Define a linear
0 1 2 0 1 2

transformation T : V → V by T (f ) = f ′ . Then the rank of T 3
is

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

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5. Consider the equation x7 + 4x5 + 2x3 + x + 1 = 0. The number
of real root(s) of this equation is

(A) 1 (B) 3 (C) 5 (D) 7

√
6. Let i = −1 and ω be a non-real root of the equation x3 = 1.
If (iω)n is an integer, then n is necessarily

(A) a multiple of 3 but need not be even
(B) even but need not be a multiple of 3
(C) a multiple of 12
(D) a multiple of 6 but need not be a multiple of 12

7. Let G be a cyclic group of order 30 and let g be a generator of
G. Then the order of g 24 is

(A) 3 (B) 5 (C) 6 (D) 15

1000
X
8. The remainder obtained when m! is divided by 18 is
m=1

(A) 7 (B) 9 (C) 12 (D) 16

9. There are two types of newly launched phones A and B. The
probability that type A has good battery life is 0.9 and the
probability that type B has good battery life is 0.7. Suppose
from a collection of phones containing equal number of phones
of each type, a phone is selected at random. The probability
that the selected phone has good battery life equals

(A) 0.63 (B) 0.75 (C) 0.80 (D) 0.85

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em
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10. Let R be the region defined by

m
R = {(x, y) ∈ R2 : x2 + y 2 ≤ 50, x + y ≥ 0}.

o m .co
c m
The area of R equals

m . s e
s e
(A) 10π (B) 15π (C) 20π (D) 25π
g l a
l a a
agFor functions f, g : Nf→(n)R, we say that f is o(g) (in words, f is
11.
little-oh of g) if lim = 0. Then
n→∞ g(n)

(A) n is o n1/2 (B) n−1/2 is o n−1/4




(C) n−1/4 is o(n−1/2 ) (D) 1 is o n−1/2



m
12. If α, β, γ are roots of x3 + ax + 1 = 0, then
. co
s em
α β β γ α γ

l a
+ + + + +

g
β α γ β γ α
equals
a
(A) a (B) 3a (C) −1 (D) −3

13. Let A be a 4 × 4 upper triangular matrix whose diagonal entries
1 1
are −1, 1, − , . Then A−1 equals
2 2

m
co
(A) 5A − 4A3 (B) 5A + 4A3
m (C) −5A + 4A .
c. o se
m
3 3

em
(D) −5A − 4A

s g l a
gla a
◦
14. Let α = sin 10 . Then α is a root of the equation
a 3
(A) 8x − 6x − 1 = 0
(B) 8x3 − 6x + 1 = 0
(C) 8x3 + 6x − 1 = 0
(D) 8x3 + 6x + 1 = 0

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15. Suppose f is a real-valued function defined on the set of real
numbers. Let
sin x
f (x) = for x ̸= 0, f (0) = a,
x
where a is a real number. Let f be differentiable at x = 0. Let
f ′ (0) = b. Then a + b equals

(A) −1 (B) 0 (C) 1 (D) 2

16. The equation

(x − 2)2 + (y − 3)2 + (x − 2)(y − 3) = 0

represents

(A) a parabola (B) an ellipse

(C) a pair of straight lines (D) a point

17. Suppose P is a moving point in the Euclidean plane whose
distance from the point A : (0, 0) is twice the distance from the
point B : (3, 0). The locus of P is

(A) a circle (B) a hyperbola

(C) an ellipse (D) a straight line

18. There are 10 points on a plane in such a way that 6 of these
points are collinear and other than these 6 points, no set of 3 or
more points are collinear. The number of triangles which can
be formed with vertices among these 10 points equals

(A) 10 (B) 60 (C) 100 (D) 120

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19. Consider the functions R −→ R defined by

m
o m
f (x) = 2x, g(x) = x + 1 and h(x) = 2x + 1.
.co
. c e m
Whichm
e l a s
a s g
of the following statements is correct?

g l(A) f is a group homomorphism but g, h are not a
a (B) g is a group homomorphism but f, h are not
(C) f and g are group homomorphisms but h is not
(D) f, g, h are all group homomorphisms

20. Let f : [0, 1] → [0, 1] be defined as follows. For x ∈ [0, 1],

o m
c
if x. is rational,
m
 x

se


la
2




g
f (x) =

a

 x+1

 if x is irrational.
2

(A) f is a bijection
(B) f is neither injective, nor surjective
(C) f is injective, but not surjective

m
(D) f is surjective, but not injective

m .c o
. co 21. In the decimal system with digits in the set {0, . . . , 9}, s ema
s em notation like 234 stands for 2 × 10 + 3 × 10 + 4; gbut l a in a
la a notation
2

ag different base r with digits in the set {0, . . . , r − 1}, the
2
234 will stand for the number 2 × r + 3 × r + 4. If the relation
430 + 240 = 1000 holds when numbers are represented in a
certain base r, then r equals

(A) 5 (B) 6 (C) 7 (D) 8

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22. Consider the following two subsets of the group Q of rational
numbers under addition:
m
(1) H = { : m, n ∈ Z, n ̸= 0, n divides 2025},
n
m
(2) K = { : m, n ∈ Z, n ̸= 0, n is co-prime to 2025}.
n
Which of the following statements is correct?

(A) Both H and K are subgroups of Q under addition
(B) H is subgroup of Q under addition but K is not
(C) K is subgroup of Q under addition but H is not
(D) Neither H nor K is a subgroup of Q under addition

23. The equation of the curve satisfying

y dx − x dy + log x dx = 0, x > 0,

and passing through (1, −1) is

(A) y + log x + 1 = 0
(B) −y 2 + log x + 1 = 0
(C) y 3 + (log x)2 + 1 = 0
(D) none of the above

24. Let D be the triangular region whose vertices are
(0, 0), (0, 2), (2, 0). The integral
ZZ
xy dx dy
D

equals

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

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25. Let f : R2 → R be a function defined by

m
 y p

co
 x2 + y 2 , x ̸= 0,

m .
f (x, y) = |x|

Then . c o 0, x = 0.
e m


e m l a s
l a s ag
agnot exist at (0, 0)
(A) f is not continuous, and the directional derivatives do

(B) f is not continuous, but the directional derivatives
exist at (0, 0)
(C) f is continuous, but the directional derivatives do not
exist at (0, 0)

m
c. o
(D) f is continuous, and the directional derivatives exist
at (0, 0)

s em
g la 1
26. Consider the real-valued function

a = x + , x > 0.
f (x)
x
The tangents to the graph of f at the points (1/2, f (1/2)) and
(2, f (2)) intersect at the point (4/5, 8/5). The area of the region
enclosed by the graph of f and these two tangents equals
15 6
(A) 2 log 2 + (B) 2 log 2 −

m
8 5

m .co
c. o
15 3

m
se
(C) 2 log 2 + (D) 2 log 2 −
4 5

s em g l a
la tossed until at least one head and at least one tail a
27. A coin with probability of head equal to 1/3 is repeatedly

ag is obtained.
The random experiment is stopped as soon as this condition
is satisfied. The probability that the experiment stops after
exactly 8 tosses equals
128 130 256 128
(A) (B) (C) (D)
2187 6561 2187 6561

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28. Five marbles of different sizes are to be coloured using the three
colours red, yellow, and blue, where each marble is coloured
with exactly one colour. The number of possible colourings of
all the marbles such that each colour is used to colour at least
one marble is

(A) 32 (B) 64 (C) 75 (D) 150

29. Consider the set of all possible polynomials

x3 + a 2 x2 + a 1 x + a 0 ,

with a0 , a1 , a2 integers and |a0 | ∈ {0, . . . , 9}, such that for
each polynomial, its roots are distinct positive integers in some
geometric progression. The number of such polynomials is equal
to

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

30. Let
∞
1/k
X
A = {x ∈ [0, ∞) : the series (x)k converges}.
k=1

Then

(A) A = {0, 1/2} (B) A = [0, 1)

(C) A = {0} (D) A = [0, 1/2)

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