ISI Admission Test 2026 Question Paper M.Math PMA

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ISI Admission Test
2026
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M.Math PMA
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ISI Admission Test 2026 Question Paper M.Math PMA

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The symbol R will denote the set of all real numbers while C will denote the set of all complex numbers.
For a natural number m, Mm (R) will denote the vector space of all m × m matrices with real entries and
GLm (R) will denote the set of all m × m invertible matrices with real entries.

1. Let A ∈ M3 (R) with eigenvalues 1, 2, 3. Then

m
(a) A−1 = 61 (A2 − 6A + 11I3 ).

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(b) A−1 = 16 (A2 + 6A + 11I3 ).

.co e m
(c) A−1 = 16 (A2 + 6A − 11I3 ).

m a s
e be positive integers. Consider the vector spaces R and R equipped with the Euclidean
(d) A−1 = 16 (A2 − 6A − 11I3 ).

l
norms. a
l s
2. Let n and m n m

a g
a g
n m
For any linear map T : R → R satisfying
n m
T ({x ∈ R : ||x|| = 1}) ⊆ {y ∈ R : ||y|| = 2},

which of the following statements must be true?
(a) T is one-one.
(b) T is onto.
(c) Rank(T ) ≥ m − n.
(d) ||T (x)|| ≤ 3 for all x ∈ Rn .

m
c. o
 
3 1 5

em
3. Consider the matrix A = 0 2 0 .

s
0 0 0

g laB 6= 0 and BA = 0.
Which of the following statements is correct?

3
(b) A = 0.
a
(a) There exists B ∈ M (R) such that
3

(c) Rank(A) = 0.
(d) dim Ker(A) = 2.

4. Let A and B be similar 3 × 3 matrices with real entries. Which of the following statements is not
necessarily correct?

m
(a) If A is invertible, then B is invertible.

m .c o
(b) Rank(A) = Rank(B).

c. o (d) The characteristic polynomials of A and B are the same. m
e following statements is
(c) Ker(A) = Ker(B).

em l a s
las a g
2 2 2
5. Let T : R → R be a linear map such that T = 0 but T 6
= 0. Which of the

ag
not correct?
(a) The only eigenvalue of T is zero.
2 2
(b) For all v ∈ R , either v is an eigenvector of T or {v, T v} is a basis of R .
 
0 0
(c) T is similar to .
1 0
(d) T is diagonalizable.

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6. For a k × k matrix A, At will denote the transpose of A and the (i, j)-th entry of A will be denoted by
aij . Let V be the subspace of M8 (R) defined as
n T 0  o
V = : T ∈ M3 (R), S ∈ M5 (R), T t = −T, s11 + s22 + s33 + s44 + s55 = 0 .
0 S

Then the dimension of the vector space V is
(a) 25.
(b) 26.
(c) 27.
(d) 28.
7. Let V denote the vector space of all polynomials with real coefficients of degree less than or equal to
5. Moreover, we define two linear maps S and T from V to V by

S(f ) = f 0 , T (f ) = f 00 .

Which of the following statements is not correct?
(a) Rank(S) = 5.
(b) S and T are similar linear maps, i.e, there exists an invertible linear map U : V → V such that
U ◦ S ◦ U −1 = T.
(c) 0 is the only eigenvalue of S ◦ T.
(d) S ◦ T = T ◦ S.
8. The rank of a 4 × 4 real matrix with exactly 5 nonzero entries cannot be
(a) 1.
(b) 2.
(c) 3.
(d) 4.
9. Let us denote the subspace of R4 spanned by the vectors (1, 0, 0, 0) and (0, 1, 0, 0) by W. Moreover, let
X denote the subspace of R4 defined as

X = {x ∈ R4 : hx, yi = 0 for all y ∈ W }.

Then
(a) dimension of X is 1.
(b) dimension of X is 2.
(c) dimension of X is 3.
(d) dimension of X is 4.
10. Let R = R[X, Y ]/(XY − 1) and I be an ideal of R generated by the image of the element X − Y in R.
Examine which of the following statements is correct.
(a) R/I is an integral domain with exactly two maximal ideals.
(b) R/I is a field.
(c) R/I is not an integral domain and R/I has exactly two maximal ideals.
(d) R/I has infinitely many maximal ideals.
11. Let I be an ideal of the polynomial ring Z[X] generated by the set {11X 11 , 13X 7 , 17X 5 , 19X 3 }. What
is the minimum number of generators of I?

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(a) 1.
(b) 2.
(c) 3.
(d) 4.
12. Let M be a fixed element of GL3 (R) and S3 the symmetric group on 3 elements. Suppose φ : S3 →
GL3 (R) is a group homomorphism such that M φ(σ)M −1 is a diagonal matrix for all σ ∈ S3 . Which

m
of the following statements is necessarily true?

o m .co
(a) Ker(φ) = A3 .

c
(c) φ((12)) = I.. em
2πi

s
(b) e is an eigenvalue of φ((123)).

em= I .
3

s g l a
3

la of cyclic subgroups of order 6 in S is equal to a
(d) φ((123))

g
3

a
13. The number 5

(a) 10.
(b) 20.
(c) 4.
(d) 5.
14. Let G be a group of order 2024 = 23 .11.23. Which of the following statements is necessarily correct?

m
(a) If G is abelian, then G is cyclic.

c o
. K of G containing more than one element.
(b) Every subgroup H of G whose order is divisible by 23 has a normal cyclic subgroup K having

m
more than one element.

s e subgroup of G containing more than one element. Then
(c) There does not exist any normal cyclic subgroup

g l
(d) Suppose K = G/H where H is a normal
a
a
K is cyclic.
15. Two ideals I and J in a ring R are called comaximal if there exist x ∈ I and y ∈ J such that x + y = 1.
Suppose I and J are two comaximal proper ideals in a ring R. Which of the following statements is
not necessarily correct?
(a) I 2 and I 2 + J 2 are comaximal.
(b) I and J 2 are comaximal.
(c) I + J 5 and I are comaximal.

m
(d) I 2 + JI and I 2 J + I 3 are comaximal.

m (a) R is a field if it has a unique prime ideal. .co
16. Let R be a finite commutative ring. Which of the following statements is necessarily correct?

c. o (b) se
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R is a product of fields.

la (d) If any degree 2 polynomial f in R[x] has at most two roots ina
(c) R is a field if the group of units of R is cyclic.
g
ag R, then R is a field.
17. Let N denote the number of elements of order 2 in a group of order 98. Which of the following
statements is necessarily correct?
(a) N ≤ 2.
(b) N can be more than 2 but N is less than or equal to 7.
(c) N can be more than 7 but N ≤ 49.

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(d) N can be greater than 49.

18. The value of the infinite sum
∞ n
X 1  1 X
k

lim m
k n→∞ nk+1 m=1
k=1

is equal to
(a) 1.
(b) 2.
(c) e.
(d) 3.
∞
xn
P
19. The series n
n=1

(a) converges uniformly on [−1, 1) to a function f and f is a continuous function on [−1, 1).
(b) converges pointwise on [−1, 1) to a function f but the convergence is not uniform.
(c) does not converge pointwise everywhere in [−1, 1).
(d) converges uniformly on (−1, 1) to a function f but f is not a continuous function on [−1, 1).
∞
P
20. Suppose an is a convergent series of real numbers. Which of the following statements must be
n=1
correct?
∞
P
(a) |an | converges.
n=1
∞ ∞
P an P |an |
(b) Both n and n2 converges.
n=1 n=1
∞ ∞
P |an | P an
(c) Both √
n
and n converges.
n=1 n=1
∞
P |an |
(d) n converges.
n=1

21. Let
∞
n X |x|n o
S= x∈R: is convergent .
n=1
1 + x2n
Then
(a) S = R.
(b) S = R \ {−1, 1}.
(c) S = {x ∈ R : |x| < 1}.
(d) S = {x ∈ R : |x| > 1}.
22. For all continuous functions f : R → R and bounded sequences {xn }∞
n=1 , which of the following
statements is correct?

(a) lim inf f (xn ) = f (lim inf xn ).
n→∞ n→∞
(b) lim inf f (xn ) ≥ f (lim inf xn ).
n→∞ n→∞
(c) lim inf f (xn ) ≤ f (lim inf xn ).
n→∞ n→∞
(d) lim sup f (xn ) ≤ f (lim sup xn ).
n→∞ n→∞

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23. Let f : [0, 1] → R be a monotonically increasing function such that f (0) = 0 and f (1) = 100. Let
n 1o
A = a ∈ (0, 1) : lim+ f (x) − lim− f (x) ≥ .
x→a x→a 4
Which of the following may possibly be the cardinality of the set A?
(a) 500.
(b) 200.

o m
c
(c) 1000.

m
o set and f : U → R be a differentiable function such that f (x) = 0 for all x ∈ U.em .
c
(d) ∞.

.
mWhich of the following statements is necessarily correct? s
a
0

e
24. Let U ⊆ R be an open

(a) A a s g l
Let A = f (U ).

l
(b)gA may contain more than one but finitely many elements.
a
a
must contain exactly one element.

(c) A is either finite or countably infinite.
(d) A can be any subset of R.
k
1
P
25. Let Sk = m . Define a sequence {xn }n≥1 as
m=1

n
X 1
xn = .

m
kSk−1 Sk

co
k=2

.
Then lim xn equals

em
n→∞

a s
(a) 0.
(b) 1.
g l
a
1
(c) 2.
(d) ∞.
26. For a function f : R → R, we define two sets as follows:

A(f ) = {x ∈ [0, 1] : f (x) = 0} and B(f ) = {x ∈ [0, 1] : f 0 (x) = 0}}.

Let C denote the set of all differentiable functions f : R → R such that

f (0) = f (1) = 0 and A(f ) ∩ B(f ) = φ.

m
m (a) For all f ∈ C, A(f ) is a finite set. .co
Which of the following statements is correct?

c. o (b) There exists f ∈ C such that A(f ) is infinite. se
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(c) For all f ∈ C, B(f ) is a finite set.

ag
(d) There exists f ∈ C such that B(f ) = φ.
27. A set A ⊆ [−1, 1] is said to have property P if for all Riemann integrable function f : [−1, 1] → R,
there exists x ∈ A (x may depend on f ) such that
Z x Z 1
f (t)dt − f (t)dt = 0.
−1 x

Which of the following statements is correct?

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(a) [−1, 1] does not have property P.
(b) [−1, 1) does not have property P but [−1, 1] has property P.
(c) (−1, 1) does not have property P but [−1, 1) has property P.
(d) For all a, b such that −1 < a < b < 1, the set (a, b) does not have property P but (−1, 1) has
property P.
28. A function f : R → Mn (R) can be expressed as f (t) = (aij (t)) ∈ Mn (R), where aij (t) is the (i, j)-th
entry of f (t). The function f is said to be differentiable if each coordinate function aij is a (real valued)
differentiable function on R. Then its derivative f 0 : R → Mn (R) is defined by f 0 (t) = (a0ij (t)).

Let f : R → M2 (R) be defined by
 
cos θ sin θ
f (θ) = .
− sin θ cos θ
Which of the following is false?
(a) f 0 (θ)f 0 (θ)t = I2 .
(b) f (θ + τ ) = f (θ)f (τ ).
(c) f 0 (θ)f 0 (τ ) = −f (θ)f (τ ).
(d) f 0 (θ)f (τ ) = −f 0 (τ )f (θ).
29. Let X, Y be metric spaces and f : X → Y be a continuous function. Which of the following statements
is not necessarily correct?
(a) If {xn }n is a convergent sequence in X, then {f (xn )}n is a convergent sequence in Y.
(b) If {xn }n is a Cauchy sequence in X, then {f (xn )}n is a Cauchy sequence in Y.
(c) If {xn }n has a convergent subsequence in X, then {f (xn )}n has a convergent subsequence in Y.
(d) If {xn }n is dense in a compact subset of X, then {f (xn )}n is dense in a compact subset of Y.
30. Let X and Y be subsets of R2 defined as
X = {(x, y) ∈ R2 : x2 + y 2 = 1} and
Y = ({−1, 1} × [−1, 1]) ∪ ([−1, 1] × {−1, 1}).
Then X and Y are equipped with the metric d defined by
p
d((x, y), (x0 , y 0 )) = (x − x0 )2 + (y − y 0 )2 .
Which of the following statements is correct?
(a) There is a continuous one-one function f : X → Y which is not surjective.
(b) There is a function f : X → Y satisfying d(f (x), f (y)) = d(x, y) for all x, y ∈ X.
(c) There is a bijective continuous function f : X → Y such that f −1 is also continuous.
(d) There is no non-constant continuous function from X to Y.
31. A subset A of R2 is said to be path-connected if given any two points x, y in A, there exists a path
γ : [0, 1] → A such that γ(0) = x and γ(1) = y.
Consider a subset S of R2 which is the union of 4 circles of radius 1 having centres at
(1, 0), (−1, 0), (0, 1) and (0, −1).
Then
(a) S \ {(0, 0)} is not path-connected.
(b) S \ {(2, 0), (0, 0)} is not path-connected.
(c) S \ {(2, 0), (−2, 0)} is not path-connected.
(d) None of the above is true.

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