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Roll No:
Application No:
Name:
Exam Date: 05-Oct-2020
Exam Time: 15:00-18:00
Examination: 1. Course Code - Ph.D.
2. Field of Study - Mathematical Sciences (MATH)
SECTION 1 - PART I

Question No.1 (Question Id - 6)
(A)
log 2
(B)
(log 2)2
(C)
(D)
Question No.2 (Question Id - 1)
(A)
{gn} converges to 0 uniformly on [0, 3]. (Correct Answer)
(B)
{gn} converges to 0 pointwise on [0, 3] but not uniformly.
(C)
{gn} converges to 0 pointwise on [0, 1] but not on [0, 3].
(D)
{gn} does not converge pointwise to 0 on [0, 1].
Question No.3 (Question Id - 7)
Which of the following expressions defines a metric on R ?
(A)
(B)
(C)
(D)
Question No.4 (Question Id - 8)
(A)
l1 has a countable subset B such that span(B) = l1
(B)
l10 l20 and c0 l1 (Correct Answer)
(C)
l20 l10 and c0 l1
(D)
c0 l1
Question No.5 (Question Id - 3)

Let V be a finite dimensional vector space over R. A subspace W of V is said to be invariant under a
linear transformation T : V V if T(W) W. Suppose W0 is a subspace of V such that W0 is
invariant under every linear transformation from V to itself. Which of the following is true ?
(A)
dim W0 = 1
(B)
dim W0 = dim V - 1
(C)
(D)
dim W0 cannot be determined from the given information.
Question No.6 (Question Id - 2)
(A)
A itself
(B)
A {-1, 1}
(C)
(-1, 1)
(D)
Question No.7 (Question Id - 9)
In a class of 60 students, 55 students register for Mathematics, 47 register for Physics and 34
students register for Chemistry. The minimum number of students who must have registered for all
the three subjects is :
(A)
34
(B)
13
(C)
(D)
24
Question No.8 (Question Id - 10)
Consider the following statements :
A. There are 20 primitive roots modulo 25.
B. There are 8 primitive roots modulo 25.
C. There are 16 primitive roots modulo 100.
Which of the above statements is/are correct ?
(A)
A only
(B)
(C)
A and C only
(D)
B and C only
Question No.9 (Question Id - 5)
Let F be a field having k 4 elements. Consider the following statements :
A. F contains more than 2 roots of 1.
B. F is isomorphic to Z/pnZ for some prime number p and n N.
C. F contains Z/pZ for some prime number p.
Which of the above statements is/are necessarily true ?
(A)
A and C only (Correct Answer)
(B)
All A, B and C
(C)
B and C only
(D)
B only

Question No.10 (Question Id - 4)
Let , and be the eigenvalues of a matrix A M3(R) such that A3 - A2 + 2I = 0. Then the value of
2 + 2 + 2 is :
(A)
5
(B)
3
(C)
-5
(D)
SECTION 2 - PART II
Question No.1 (Question Id - 12)
The set {z C : |ez| = |z|} is :
(A)
empty
(B)
a non-empty finite set
(C)
a countably infinite set
(D)
Question No.2 (Question Id - 14)
Consider sets and operations :
G1 = {f : R R| f is continuous} with respect to composition of maps and pointwise multiplication.
G2 = {f : R R| f is continuous} with respect to pointwise addition and multiplication.
G3 = {f : R2 R2| f is a linear projection onto a one-dimensional subspace of R2} with respect to
G4 = {f : R2 R2| f is linear} with respect to addition and composition.
Which of the above is/are commutative ring(s) with unity ?
(A)
G3 only
(B)
G2 and G4 only
(C)
G1, G2 and G4 only
(D)
Question No.3 (Question Id - 24)
What is the remainder when 28! is divided by 31 ?
(A)
16
(B)
(C)
30
(D)
1
Question No.4 (Question Id - 22)
Let (X, || . ||) be a Banach space and T : X X be a linear map. Define || . ||T : X [0, ) by ||x||T =
||T(x)|| for x X. Consider the following assertions :
A. || . ||T is a norm on X if and only if T is surjective.
B. || . ||T is a norm on X if and only if T is injective.
C. || . ||T is a norm on X if and only if T is continuous.
D. (X, || . ||T) is Banach space if T is bijective.
Which of the above assertions is/are always true ?
(A)
A and D only
(B)
B and C only
(C)
B only
(D)
B and D only (Correct Answer)

Question No.5 (Question Id - 20)
Let X = Z and be the smallest topology on X containing all sets of the form {n, n + 3} for all n Z.
Consider the following assertions :
A. is same as the smallest topology on X containing all sets of the form {n, n + 1} for all n Z.
B. is same as the smallest topology on X containing all sets of the form {n, n + 2} for all n Z.
C. is same as the smallest topology on X containing all sets of the form {n, n + 1, n + 2} for all n Z.
D. is a countable collection.
Which of the above assertions is/are correct ?
(A)
C only
(B)
A and D only
(C)
A, B and C only (Correct Answer)
(D)
A, B and D only
Question No.6 (Question Id - 23)
For any fixed n N, the number of ordered triplets (X1, X2, X3) of subsets of N such that X1 X2
X3 = {1, 2, . . ., n} is equal to :
(A)
(B)
8n
(C)
n8
(D)
n3
Question No.7 (Question Id - 19)
The set {z C : ez = z} is :
(A)
empty
(B)
a non-empty finite set
(C)
a countably infinite set (Correct Answer)
(D)
an uncountable set
Question No.8 (Question Id - 18)
Let S = {A M2(R)|A2 = I}. Which of the following assertions is true ?
(A)
S is not a group. (Correct Answer)
(B)
S is a finite abelian group.
(C)
S is an infinite abelian group.
(D)
S is an infinite non-abelian group.
Question No.9 (Question Id - 13)
(A)
B and D only (Correct Answer)
(B)
A, C and D only

(C)
A and D only
(D)
B, C and D only
Question No.10 (Question Id - 21)
Let {An : n N} be a countable collection of non-empty subsets of R2 such that An+1 An for all n
N. Consider the following assertions :
A. If An is connected for every n N, then nAn is connected.
B. If An is compact for every n N, then nAn is compact.
C. If An is uncountable for every n N, then nAn is uncountable.
D. If An is countable for every n N, then nAn is non-empty.
Which of the above assertions is/are always true ?
(A)
(B)
A and D only
(C)
C and D only
(D)
A and B only
Question No.11 (Question Id - 11)
(A)
S1 and S2 both converge.
(B)
S1 diverges and S2 converges. (Correct Answer)
(C)
S1 converges and S2 diverges.
(D)
S1 and S2 both diverge.
Question No.12 (Question Id - 16)
Let V be a finite-dimensional vector space over R. Let {v1, v2, . . ., vn} be a basis for V and let {w1,
w2, . . ., wn} V. Consider the following statements :
A. There exists a unique linear map T : V V such that T(vi) = wi for 1 i n.
B. If there exists a linear map T : V V such that T(vi) = wi for 1 i n, then T is injective.
C. If there exists an injective linear map T : V V such that T(vi) = wi for 1 i n, then {w1, w2, . . .,
wn} is a basis for V.
D. There exists a unique linear map T : V V such that T(wi) = vi for 1 i n.
Which of the above statements are correct ?
(A)
A and D only
(B)
A, B and C only
(C)
A and C only (Correct Answer)
(D)
A, B and D only
Question No.13 (Question Id - 15)

(A)
(B)
A and C only
(C)
B and D only
(D)
B only
Question No.14 (Question Id - 17)
Consider the following statements :
A. There exists a finitely generated group containing some element of infinite order.
B. There exists an infinite group which is not finitely generated but all whose elements have finite
order.
C. There exists a finitely generated infinite group no element of which has infinite order.
Which of the above statements are correct ?
(A)
All A, B and C (Correct Answer)
(B)
B and C only
(C)
A and C only
(D)
A and B only
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