ISI Admission Test 2023 Question Paper M.Math PMA

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ISI Admission Test
2023
Question Paper ·
M.Math PMA
EXAM YEAR TYPE SUBJECT

ISI Admission Test 2023 Question Paper M.Math PMA

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m
m .co
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e m
em l a s
s a g

• N denotes the set of all positive integers.

m
co
• Z denotes the set of all integers.

m .
.co
• R denotes the set of all real numbers.
s em
•se
m l a
l a M (R) denotes the set of all n × n matrices with real entries.
a g
g
n

a • I denotes the n × n identity matrix.
n

• dim (V ) denotes the dimension of a finite dimensional vector
space V .

• For a function f : X → Y , f (X) denotes the set {f (x) : x ∈ X}.

o m
c
.R consisting of all polynomials
m
e the subset W of all polynomials
s
1. Let V be the vector space over

l a
of degree 4. Which ofgthe following statements is correct?
with real coefficients. Consider

a
(A) W is not a subspace of V .
(B) W is a subspace of V having dimension 5.
(C) W is a subspace of V having dimension 4.
(D) W is a subspace of V having dimension 3.

o m
m .c
c. o s em
2. If A belongs to M (R), then which of the following statements

em la
3

s g
is not true in general?

gla a
a
2
(A) If A is diagonalizable, then A is diagonalizable.
(B) If A2 is diagonalizable, then A is diagonalizable.
(C) There exists a polynomial p of degree 3 such that p(A) = 0.
(D) A has a real eigenvalue.

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3. Let W denote the vector space over R consisting of all polyno-
mials of degree less than or equal to 6 with real coefficients. Let
D : W → W be the linear transformation defined by
dp
D(p) = .
dx
Which of the following statements is correct?
(A) D is invertible.
(B) Rank of D is equal to 6.
(C) Rank of D is equal to 5.
(D) Rank of D is equal to 1.

4. Let V be a finite dimensional vector space and P : V → V be a
non-zero linear transformation such that P 2 = P . Which of the
following statements is not true in general?
(A) Ker (P ) ∩ Ker (I − P ) = {0}.
(B) (I − P )2 = I − P .
(C) Ker (P ) = (I − P )(V ).
� � � �
(D) dim Ker (P ) = dim P (V ) .

5. Let A be a 2 × 2 real matrix such that (1, 0) is an eigenvector
of A with eigenvalue log 2, and (0, 1) is an eigenvector of A with
eigenvalue log 3. Let I denote the 2 × 2 identity matrix. Then
the limit in R2 of the vectors
� �n � �
A 1
lim I +
n→∞ n 1

(A) exists and equals (2, 3).
(B) does not exist.
(C) exists and equals (3, 5).
(D) exists and equals (2, 5).

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6. Consider the following vector spaces over R:

m
= {f : [0, 1] → R | f is a continuous function},

co
V

o m
W = {f : [0, 1] → R | f is a function}.
m .
c
If T .: V → W is the map defined by
s e
m a
lase a g l
g
(T f )(t) = tf (t), t ∈ [0, 1],
a then which of the following statements is not true?
(A) T (V ) ⊆ V .
(B) T is a linear map from V to V .
(C) T is one-to-one.
(D) T (V ) = V .

m
. co
7. Let W = {A ∈ M4 (R) : Rank (A) ≤ 3}. Which of the following

s em
statements is correct?

g l a
a
(A) W is not closed under scalar multiplication.
(B) If B, C belong to W, then B + C belongs to W .
(C) If B ∈ W , then I4 + B ∈ W .
(D) W is not a subspace of M4 (R).

8. For positive integers r and n with r < n, let P denote the n × n

o m
c
matrix � �
m .
c. o m
Ir 0r×(n−r)

e
,

denotes the zero matrix with k rows andalscolumns.
em
0 0 (n−r)×r (n−r)×(n−r)

s Let H denote the transpose of a matrix H andag
l
la
where 0 k×l

ag
t

V = {H ∈ Mn (R) : H = HP + P H = H t }.

Which of the following statements is true?

(A) dim (V ) = r2 . (B) dim (V ) = (n − r)2 .
(C) dim (V ) = r(n − r). (D) dim (V ) = 2r(n − r).

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9. Let φ : G → H be a group homomorphism between finite groups.
Which of the following statements is not true in general?

(A) Image of any subgroup of G under φ is a subgroup of H.

(B) For every g ∈ G, order of φ(g) divides the order of g.

(C) φ(G) is a subgroup of H.

(D) Image of any normal subgroup of G under φ is a normal
subgroup of H.

10. Consider the group G = Z6 × Z15 × Z20 . The order of the
element (1, 2, 3) in G is

(A) 30. (B) 40. (C) 60. (D) 120.

11. Consider the ring

√ √
Z[ −2022] = {a + b −2022 : a, b ∈ Z}.

The number of elements in this ring having multiplicative
inverse is

(A) 2. (B) 3. (C) 4. (D) infinite.

12. Let p be a prime number. Consider the ring R = Z[x]/(x2 − px)
with multiplicative identity denoted by 1R . Then, the number
of ring homomorphisms φ : R → Z such that φ(1R ) = 1 is

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

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13. Let R be a commutative ring with unity and a, b ∈ R. Consider

m
the ideal

m I = �a, b� = {xa + yb : x ∈ R, y ∈ R}. .co
.co s em
m
e l a
a s
l If a g
ag ��
n �
I2 = ai bi : ai , bi ∈ I, n ∈ N ,
i=1

then which of the following statements is not true in general?

(A) I = �a − 5b, b�. (B) I 2 = �a2 , b2 �.

m
(C) I 2 ⊆ I. (D) I + I = I.

. co
s em
l a
g . Consider the following system of linear
14. For a prime number p and k ∈ N, let k denote the equivalence
class of k in the fieldaZ p

equations in Zp :

10X + 5Y = 8
3X + 12Y = 11.

m
m .co
For which of the following values of p does the above system

c. o have a unique solution?
se
m
s em g la p = 11.
la a
(A) p = 3. (B) p = 5. (C) p = 7. (D)

ag
15. Let G be a group of order 75 which has an element of order 25.
Then the number of elements of order 5 in G is

(A) 4. (B) 5. (C) 12. (D) 24.

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16. Let G be a group of order 30. Which of the following statements
is necessarily true?

(A) G is abelian but may not be cyclic.
(B) G is cyclic.
(C) G cannot have more than 7 subgroups of order 5.
(D) G can simultaneously have one subgroup of order 2, five
subgroups of order 3, and six subgroups of order 5.

17. The total number of subgroups of Z2 × Z2 × Z2 is

(A) 4. (B) 12. (C) 16. (D) 22.

18. If
� �n
3 + (−1)n
an = ,
4n
∞
�
then the power series an xn converges if and only if
n=0

(A) x = 0. (B) x ∈ [−1, 1).
(C) x ∈ (−1, 1]. (D) x ∈ (−1, 1).

19. If
� 2n �1/n
2 + 32n
lim = L,
n→∞ 6n

then L is equal to
4 13 7 3
(A) . (B) . (C) . (D) .
3 12 6 2
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20. For any a ∈ [0, 1], the limit of the sequence {xn }∞
n=1 , where

m
co
1 + 2 a + . . . + na

m .
xn =

o m
na+1

is . c s e
m a
se(A) 1 − a .
la
1
a g l
g
a (C) 1 + a .
(B) .
2 1 + a2
1 1
(D) 2 .
2a − a + 1
� π π�
21. Consider the sequence {fn }∞
n=1 of functions on − 2 , 2 given by

fn (x) = (sin x)n cos x.

m
co
The sequence {fn } converges uniformly

.
em
� � � �

s
(A) on − π4 , π4 but not on any strictly larger subset of − π2 , π2 .
� �
g l a � �

a
(B) on − π4 , π4 but not on any strictly larger subset of − π2 , π2 .
� � � �
(C) on − π2 , π2 but not on any strictly larger subset of − π2 , π2 .
� �
(D) on − π2 , π2 .

m
co
� 2π �
m .
n

c. o m
22. Let an = sin n , n ∈ N. Which of the following

se
2n + 1 3

em la
statements is true?

las 1
a g
g
(A) lim an = .
a
n→∞ 2 √
3
(B) lim sup an = .
n→∞ 4
√
3
(C) lim sup an = .
n→∞ 2
(D) lim sup an = 0.
n→∞

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23. Let f and g be polynomials with real coefficients of degree m
and n, respectively. Suppose g(x) �= 0 for all x ∈ [1, ∞). The
integral
� ∞
|f (x)|
dx
1 |g(x)|
converges if and only if

(A) n − m ≥ 2. (B) m − n ≤ 2.
(C) n − m ≥ 1. (D) m ≥ 1 and n ≥ 1.

24. Fix x0 , x1 ∈ R with x0 �= x1 . Define

xn+1 = axn + (1 − a)xn−1 , n ∈ N.

The sequence {xn }∞
n=1 is convergent if and only if

(A) a ∈ (0, 2). (B) a ∈ [0, 2).
(C) a ∈ [0, 2]. (D) a ∈ (0, 1).

25. Let f : R → R be a differentiable function such that f � is
continuous and satisfies

|f � (x) − e2x | ≤ 3 for all x ∈ R.

Then the limit
f (x)
lim
x→+∞ e2x

(A) may not exist.
(B) exists and equals 12 .
(C) exists and equals 34 .
(D) exists and equals 14 .

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26. If � 2� 2
e(x−1)

m
lim dx = L,

co
�→0+ x

m .
�

c o m
then L is equal to

(A) .2. e
m a s
se(C) log(e + 2). l
(B) e log 2.

la
(D) e . 2

a g
ag

27. The following series of functions
�∞
x2
, x ∈ R,
n=1
(1 + x2 )n
converges uniformly on
o m
(A) R \ (a, b) for all a ≥ 0. m.
c
s e
l a
(C) R \ (a, b) for allag
(B) intervals of the form (−∞, a) and (a, ∞) for all a ∈ R.
a < 0 and b > 0.
(D) all compact subsets of R.

28. Let (X, d) be a metric space and x0 ∈ X. Let γ : [0, ∞) → X
be such that
� �

m
d γ(s), γ(t) = |s − t| for all s, t ∈ [0, ∞).

m .co
c. o t ∈ [0, ∞). em
Define a function f : [0, ∞) → R by

em s
� � � �

l a
f (t) = d x0 , γ(t) − d γ(0), γ(t) ,

las ag
g
Which of the following statements is necessarily true?
a (A) The function f is monotone increasing and bounded.
(B) The function f is monotone increasing and unbounded.
(C) The function f is monotone decreasing and bounded.
(D) The function f is monotone decreasing and unbounded.

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29. Let (X, d) be a compact metric space. Let x ∈ X be such that
X \ {x} = {y ∈ X : y �= x} is complete. Which of the following
statements is necessarily true?

(A) There is a sequence {xn }∞
n=1 in X converging to x such that
d(xn , x) > 0 for all n ∈ N.
(B) There exists p ∈ X such that 0 < d(p, x) = inf d(q, x).
q∈X\{x}
(C) Given a point p ∈ X \ {x}, there is a point q �= p such that
d(p, x) = d(q, x).
(D) For every p ∈ X \ {x}, there is a point q ∈ X such that
0 < d(q, x) < d(p, x).

30. The probability that a person tests positive for Covid by
RTPCR, given that the person is infected with Covid, is 0.95.
Also, the probability that a person tests positive for Covid
by RTPCR, given that the person is not infected with Covid,
is 0.15. Assume that in a country, 10% of the population is
infected with Covid. A person is randomly chosen and the
RTPCR test yields positive result. The probability that the
person is actually infected with Covid is
19 17 19 9
(A) . (B) . (C) . (D) .
20 46 46 10

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