ISI Admission Test 2024 Question Paper M.Math PMA

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ISI Admission Test
2024
Question Paper ·
M.Math PMA
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ISI Admission Test 2024 Question Paper M.Math PMA

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• N denotes the set of all positive integers.

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• Z denotes the set of all integers.

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• R denotes the set of all real numbers.
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l entries.
For n ∈ N, M (R) denotes the set of all n × n matrices with real
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n

a
• For A ∈ Mn (R), Ker(A) = {v ∈ Rn : Av = 0}.

• For n ∈ N, GLn (R) denotes the set of all n×n invertible matrices
with real entries.

• For n ∈ N, In denotes the n × n identity matrix.

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• dim(V ) denotes the dimension of a finite dimensional real vector
space V .
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• For a function f : X → Y , f (X) denotes the set {f (x) : x ∈ X}.

• For vector spaces V1 , V2 , V3 and linear maps S : V1 → V2 and
T : V2 → V3 , the symbol T S : V1 → V3 denotes the linear map
defined as T S(v) = T (S(v)) for all v ∈ V1 .

• For n ∈ N, Sn denotes the symmetric group on n elements.

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• For n ∈ N, An denotes the alternating group on n elements.

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1. Suppose A ∈ M3 (R) has eigenvalues 1, 2, 3. Then

a ( A ) A−1 = 61 (A2 − 6A − 11I3 ).
( B ) A−1 = 61 (A2 + 6A + 11I3 ).
( C ) A−1 = 16 (A2 + 6A − 11I3 ).
( D ) A−1 = 16 (A2 − 6A + 11I3 ).

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2. Let n and m be positive integers. Consider the vector spaces Rn
and Rm equipped with the respective Euclidean norms. Which
of the following statements must be true for every linear map
T : Rn → Rm satisfying the condition
� �
T {x ∈ Rn : �x� = 1} ⊆ {y ∈ Rm : �y� = 2}?

( A ) T is one to one.
( B ) T is onto.
( C ) Rank(T ) ≥ m − n.
( D ) �T (x)� ≤ 3 for all x ∈ Rn .

3. Let W be the subspace of R4 spanned by the vectors (1, 2, 3, 4)
and (4, 3, 2, 1). Let X denote the subspace of R4 defined as

X = {x ∈ R4 : �x, y� = 0 for all y ∈ W },
4
�
where �x, y� = xi yi , for x = (x1 , x2 , x3 , x4 ) and
i=1
y = (y1 , y2 , y3 , y4 ). Then the dimension of X is equal to

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

 
3 1 5
4. Consider the matrix A = 0 2 0 .
0 0 0
Which of the following statements is correct?

( A ) There exists B ∈ M3 (R) such that B �= 0 and BA = 0.
( B ) A3 = 0.
( C ) Rank(A) = 0.
( D ) dim Ker(A) = 2.

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5. Suppose A and B are similar 3 × 3 matrices with real entries.

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Which of the following statements is not necessarily correct?

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

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(eC ) Ker(A) = Ker(B). l a
( B ) Rank(A) = Rank(B).

a s
l ( D ) The characteristic polynomial of A and B are the same. a g
ag

6. For A ∈ Mk (R), At denotes the transpose of A. Let V� be the
�
T 0
subspace of M8 (R) consisting of matrices of the form
0 S
satisfying the following conditions:

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(i) T ∈ M3 (R), S ∈ M5 (R),
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s em
a
(ii) T t = −T,

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a
(iii) the sum of the diagonal entries of S is equal to 0.

Then the dimension of the vector space V is

( A ) 25. ( B ) 26. ( C ) 27. ( D ) 28.

7. Suppose T ∈ M2 (R) is such that T 2 = 0 but T �= 0. Which of
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the following statements is not correct?

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( A ) The only eigenvalue of T is zero.

a ( B ) For all v ∈ R2 \ {0}, either v is an eigenvector of T or
{v, T v} is a basis of R2 .
� �
0 0
( C ) T is similar to .
1 0
( D ) T is diagonalizable.

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8. Let V be the vector space of all polynomials with real coefficients
of degree less than or equal to 5. Define two linear maps S and
T from V to V by

S(f ) = f � , T (f ) = f �� .

Which of the following statements is not correct?

( A ) Rank(S) = 5.
( B ) ST = T S.
( C ) 0 is the only eigenvalue of ST.
( D ) There exists an invertible linear map U : V → V such that
U SU −1 = T.

9. The rank of a 4 × 4 real matrix with exactly 5 nonzero entries
cannot be

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

10. The number of cyclic subgroups of order 6 in S5 is equal to

( A ) 10. ( B ) 20. ( C ) 4. ( D ) 5.

11. Let R = R[X, Y ]/(XY − 1) and I be the ideal of R generated
by the image of the element X − Y in R. 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.

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( D ) R/I has infinitely many maximal ideals.

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c
. , 13X , 17X , 19X }. What is the minimum number em
12. Let I be the ideal of the polynomial ring Z[X] generated by the

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seof generators of I? l
11 7 5 3
set {11X

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

13. Let R be a finite commutative ring with unity. Which of the
following statements is necessarily correct?

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( A ) R is a field if it has a unique prime ideal.

( B ) R is a product of fields. .
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( C ) R is a field if the group of units of R is cyclic.

( D ) If any degree 2apolynomial f in R[X] has at most two roots
in R, then R is a field.

14. Let G be a group of order 2024 = 23 × 11 × 23. Which of the
following statements is necessarily correct?

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( A ) If G is abelian, then G is cyclic.

s emof G
em containing more than one element. Then K isla
( B ) Suppose K = G/H where H is a normal subgroup

las a g
cyclic.

ag ( C ) There does not exist any normal cyclic subgroup K of G
containing more than one element.

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

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15. 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.
( D ) N can be greater than 49.

16. Let R be a commutative ring with unity. Two ideals I and J in
R are called comaximal if there exist x ∈ I and y ∈ J such that
x + y = 1. For two comaximal proper ideals I and J in R, which
of the following statements is not 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.
( D ) I 2 + JI and I 2 J + I 3 are comaximal.

17. Let M be a fixed element of GL3 (R) and φ : S3 → GL3 (R) is a
group homomorphism such that M φ(σ)M −1 is a diagonal matrix
for all σ ∈ S3 . Which of the following statements is necessarily
true?

( A ) Ker(φ) = A3 .
2πi
( B ) e 3 is an eigenvalue of φ((123)).
( C ) φ((12)) = I3 .
( D ) φ((123)) = I3 .

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18. Which of the following statements is correct for every continuous

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function f : R → R and every bounded sequence {xn } of real

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numbers?

(eA ) lim inf f (x ) = f (lim inf x ).
s g l
g a
l ( B ) lim inf f (x ) ≥ f (lim inf x ). a
a
n n
n→∞ n→∞

n n
n→∞ n→∞
( C ) lim inf f (xn ) ≤ f (lim inf xn ).
n→∞ n→∞
( D ) lim sup f (xn ) ≤ f (lim sup xn ).
n→∞ n→∞

19. Let

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∞
�

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|x|n
S = {x ∈ R :
.
is convergent}.
1 + x2n

em
n=1

a s
Then

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( A ) S = R.
( B ) S = R \ {−1, 1}.
a
( C ) S = {x ∈ R : |x| < 1}.
( D ) S = {x ∈ R : |x| > 1}.

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�

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1

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20. Let Sk = , k ≥ 1. Define a sequence {xn }, n ≥ 2 as

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m

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m=1

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n
�

s
1

la g
xn = .
a
kSk−1 Sk

ag
k=2

Then lim xn equals
n→∞

1
( A ) 2. ( B ) 1. (C) . ( D ) ∞.
2

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21. The value of the infinite sum
∞
� 1 � 1 � n �
k
lim m
k=1
k n→∞ nk+1 m=1

is equal to

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

∞
�
22. Suppose an is a convergent series of real numbers. Which of
n=1
the following statements must be correct?
�∞
(A) n=1 |an | converges.
�∞ an
�∞ |an |
( B ) Both n=1 n and n=1 n2 converges.
� � ∞ an
( C ) Both ∞ |an |
n=1 n and
√
n=1 n converges.
�∞ |an |
(D) n=1 n converges.

23. The series ∞
� xn
n=1
n

( A ) converges uniformly on [−1, 1) to a function f which is
continuous 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).

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24. Let U ⊆ R be an open set and f : U → R be a differentiable

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function such that f � (x) = 0 for all x ∈ U. Let A = f (U ). Which

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of the following statements is necessarily correct?

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( A ) A must contain exactly one element.

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a
l ( C ) A is either finite or countably infinite. a
( B ) A may contain more than one but finitely many elements.

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a ( D ) A can be any subset of R.

25. Let f : [0, 1] → R be a monotonically increasing function such
that f (0) = 0 and f (1) = 100. Let
� �
1

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A = a ∈ (0, 1) : lim f (x) − lim f (x) ≥ .

c
. cardinality of the set A?
x→a+ x→a− 4

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Which of the following can be the

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( A ) 500. ( B ) 200. ( C ) 1000. ( 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 � (x) = 0}.

Let C be the set of all differentiable functions f : R → R such
that
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f (0) = f (1) = 0 and the sets A(f ), B(f ) are disjoint.

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

a ( B ) There exists f ∈ C such that A(f ) is infinite.
( C ) For all f ∈ C, B(f ) is a finite set.
( D ) There exists f ∈ C such that B(f ) is empty.

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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
� x � 1
f (t)dt − f (t)dt = 0.
−1 x

Which of the following statements is correct?

( 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. 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 } is a convergent sequence in X, then {f (xn )} is a
convergent sequence in Y.
( B ) If {xn } is a Cauchy sequence in X, then {f (xn )} is a Cauchy
sequence in Y.
( C ) If {xn } has a convergent subsequence in X, then {f (xn )}
has a convergent subsequence in Y.
( D ) If {xn } is dense in a compact subset of X, then {f (xn )} is
dense in a compact subset of Y.

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29. Consider R2 equipped with the metric defined by

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�
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� �

s
d((x, y), (x , y )) = (x − x� )2 + (y − y � )2 .
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eLet X and Y be subsets of R defined as l a
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ag X = {(x, y) ∈ R2 : x2 + y 2 = 1} and
� � � �
Y = {−1, 1} × [−1, 1] ∪ [−1, 1] × {−1, 1} .

Which of the following statements is correct?

( A ) There is a bijective continuous function f : X → Y such

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that f −1 is also continuous.

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( B ) There is a function f : X → Y satisfying d(f (x), f (y)) =

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s
d(x, y) for all x, y ∈ X.

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is not onto. ag
( C ) There is a continuous one to one function f : X → Y which

( D ) There is no non-constant continuous function from X to Y.

30. An urn contains 50 red balls, 50 black balls and 100 white balls.
If 80 balls are drawn from the urn without replacement, what is

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the probability that at least one ball of each color is drawn?

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2 150

se
80
( A ) 1 − �200 �.

s em la
80

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�150�

gla a
80
( B ) 1 − �200 �.

a 2 80
80
�150�
�100�
80
( C ) 1 − �200� − �200 �.
80 80
�150� �100�
2 80 80
( D ) 1 − �200 � + �200 �.
80 80

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