Download JNU 2020 Mathematical Sciences (Math) Question Paper

Roll No:

Application No:

Name:

--- Content provided by FirstRanker.com ---

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

--- Content provided by​ FirstRanker.com ---

Question No.1 (Question Id - 6)

The value of the integral ¹/_2pi ?_|z|=1 ^z2/_z² dz is equal to :

(A) log 2

(B) (log 2)²

(C) 2(log 2)² (Correct Answer)

--- Content provided by⁠ FirstRanker.com ---

(D) ^{(log 2)2}/_2!

Question No.2 (Question Id - 1)

For each n ? N, consider the map g_n: [0, 3] ? R given by g_n(t) = ^tn/_n! for t ? [0, 3]. Which of the following assertions is correct?

(A) {g_n} converges to 0 uniformly on [0, 3]. (Correct Answer)

(B) {g_n} converges to 0 pointwise on [0, 3] but not uniformly.

--- Content provided by⁠ FirstRanker.com ---

(C) {g_n} converges to 0 pointwise on [0, 1] but not on [0, 3].

(D) {g_n} 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) d(x, y) := v(x² + y²) for x, y ? R.

--- Content provided by‌ FirstRanker.com ---

(B) d(x, y) := |x² - y²| for x, y ? R.

(C) d(x, y) := |x³ - y³| for x, y ? R. (Correct Answer)

(D) d(x, y) := |x⁴ - y⁴| for x, y ? R.

Question No.4 (Question Id - 8)

For 1 = p < 8, consider the normed spaces

--- Content provided by‍ FirstRanker.com ---

l^p := {(x_n): S|x_n|^p < 8} and c₀:= {(x_n): lim x_n = 0}

Which of the following assertions is correct?

(A) l¹ has a countable subset B such that span(B) = l¹

(B) l² ? l¹ and c₀ ? l¹ (Correct Answer)

(C) l² ? l¹ and c₀ ? l¹

--- Content provided by FirstRanker.com ---

(D) c₀ ? l¹

Question No.5 (Question Id - 3)

Let V be a finite dimensional vector space. A subspace W of V is said to be invariant under a linear transformation T: V ? V if T(W) ? W. A subspace W of V is said to be 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

--- Content provided by​ FirstRanker.com ---

(C) W₀ = {0} (Correct Answer)

(D) dim W₀ cannot be determined from the given information.

Question No.6 (Question Id - 2)

The set of limit points of the set A = {^k/_n : k ? N, ^k/_n < 4k} is:

(A) A itself

--- Content provided by‌ FirstRanker.com ---

(B) A ? {-1, 1}

(C) {-1, 1}

(D) [-1, 1] (Correct Answer)

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:

--- Content provided by FirstRanker.com ---

(A) 34

(B) 13

(C) 16 (Correct Answer)

(D) 24

Question No.8 (Question Id - 10)

--- Content provided by​ FirstRanker.com ---

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?

--- Content provided by​ FirstRanker.com ---

(A) A only

(B) B only (Correct Answer)

(C) A and C only

(D) B and C only

Question No.9 (Question Id - 5)

--- Content provided by FirstRanker.com ---

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?

--- Content provided by‍ FirstRanker.com ---

(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)

--- Content provided by FirstRanker.com ---

Let a, ß and ? be the eigenvalues of a matrix A ? M₃(R) such that A³ - A² + 2I = 0. Then the value of a² + ß² + ?² is:

(A) 5

(B) 3

(C) -5

(D) 1 (Correct Answer)

--- Content provided by FirstRanker.com ---

SECTION 2 - PART II

Question No.1 (Question Id - 12)

The set {z ? C: |e^z| = |z|} is:

(A) empty

(B) a non-empty finite set

--- Content provided by‍ FirstRanker.com ---

(C) a countably infinite set

(D) an uncountable set (Correct Answer)

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.

--- Content provided by‌ FirstRanker.com ---

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 addition and composition.

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

--- Content provided by​ FirstRanker.com ---

(B) G₂ and G₄ only

(C) G₁, G₂ and G₄ only

(D) G₂ only (Correct Answer)

Question No.3 (Question Id - 24)

What is the remainder when 28! is divided by 31?

--- Content provided by‌ FirstRanker.com ---

(A) 16

(B) 15 (Correct Answer)

(C) 30

(D) 1

Question No.4 (Question Id - 22)

--- Content provided by⁠ FirstRanker.com ---

Let (X, ||.||) be a Banach space and T: X ? X be a linear map. Define ||.||_T: X ? [0, 8) 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.

--- Content provided by​ FirstRanker.com ---

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)

--- Content provided by⁠ FirstRanker.com ---

Question No.5 (Question Id - 20)

Let X = Z and t be the smallest topology on X containing all sets of the form {n, n + 1} for all n ? Z. Consider the following assertions:

A. t is same as the smallest topology on X containing all sets of the form {n, n + 1} for all n ? Z.

B. t is same as the smallest topology on X containing all sets of the form {n, n + 2} for all n ? Z.

C. t is same as the smallest topology on X containing all sets of the form {n, n + 1, n + 2} for all n ? Z.

--- Content provided by FirstRanker.com ---

D. t 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)

--- Content provided by⁠ FirstRanker.com ---

(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) 7ⁿ (Correct Answer)

(B) 8ⁿ

--- Content provided by FirstRanker.com ---

(C) n⁸

(D) n³

Question No.7 (Question Id - 19)

The set {z ? C: e^z = z} is:

(A) empty

--- Content provided by​ FirstRanker.com ---

(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?

--- Content provided by⁠ FirstRanker.com ---

(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)

--- Content provided by FirstRanker.com ---

Let m denote the Lebesgue measure on R. Consider the following assertions:

A. If U is an open set in R containing Q, then m(U) = 8.

B. There exists an open set U in R containing Q with m(U) < ¹/₂₀₂₀.

C. If U is an open set in R containing Q with m(U) = 8, then m(R\U) = 0.

D. If G is a closed set in R containing Q, then m(G) = 8.

--- Content provided by‌ FirstRanker.com ---

Which of the above assertions are always true?

(A) B and D only (Correct Answer)

(B) A, C and D only

Question No.10 (Question Id - 21)

Let {A_n : n ? N} be a countable collection of non-empty subsets of R² such that A_n+1 ? A_n for all n ? N. Consider the following assertions:

--- Content provided by​ FirstRanker.com ---

A. If A_n is connected for every n ? N, then nA_n is connected.

B. If A_n is compact for every n ? N, then nA_n is compact.

C. If A_n is uncountable for every n ? N, then nA_n is uncountable.

D. If A_n is countable for every n ? N, then nA_n is non-empty.

Which of the above assertions is/are always true?

--- Content provided by⁠ FirstRanker.com ---

(A) B only (Correct Answer)

(B) A and D only

(C) C and D only

(D) A and B only

Question No.11 (Question Id - 11)

--- Content provided by⁠ FirstRanker.com ---

Let S₁ and S₂ be the series S₁ = S ¹/_n=1 ¹/_{2 log n} and S₂ = S ⁸/_n=1 ^{(-1)n sin n}/_vn

(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.

--- Content provided by‍ FirstRanker.com ---

Question No.12 (Question Id - 16)

Let V be a finite-dimensional vector space over R. Let {v₁, v₂, ..., v_n} be a basis for V and let {w₁, w₂, ..., w_n} ? V. Consider the following statements:

A. There exists a unique linear map T: V ? V such that T(v_i) = w_i for 1 = i = n.

B. If there exists a linear map T: V ? V such that T(v_i) = w_i for 1 = i = n, then T is injective.

C. If there exists an injective linear map T: V ? V such that T(v_i) = w_i for 1 = i = n, then {w₁, w₂, ..., w_n} is a basis for V.

--- Content provided by FirstRanker.com ---

D. There exists a unique linear map T: V ? V such that T(w_i) = v_i 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)

--- Content provided by⁠ FirstRanker.com ---

(D) A, B and D only

Question No.13 (Question Id - 15)

Let V be a finite dimensional vector space and T, S: V ? V be linear transformations. Let Z = Im(S).

Following statements:

A. If T.S is invertible then so are T and S.

--- Content provided by⁠ FirstRanker.com ---

B. If S and T are both injective then dim Z = dim V.

C. If dim W > dim V, then T.S cannot be surjective.

D. If dim W < dim Z, then T.S cannot be surjective.

Which of the above statements is/are always true?

(A) D only (Correct Answer)

--- Content provided by FirstRanker.com ---

(B) A and C only

(C) B and D only

(D) B only

Question No.14 (Question Id - 17)

Consider the following statements:

--- Content provided by‍ FirstRanker.com ---

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)

--- Content provided by‍ FirstRanker.com ---

(B) B and C only

(C) A and C only

(D) A and B only

FirstRanker.com

--- Content provided by‌ FirstRanker.com ---