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Organisation : MSc/PhD Mathematics
Exam : MSc/PhD Mathematics Entrance Exam
Document Type : Question Paper
Year : 2021
Website : https://www.cmi.ac.in/admissions/syllabus.php

CMI MSc/PhD Mathematics Question Paper

The entrance examination is a test of aptitude for Mathematics featuring both multiple choice questions and problems requiring detailed solutions drawn mostly from the following topics: algebra, real analysis, complex analysis, calculus.

Related / Similar Question Paper : CMI MSc/PhD Mathematics Entrance 2022 Question Paper

MSc/PhD Mathematics Question Paper

(1) Which of the following can not be the class equation for a group of appropriate order?
(A) 14 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 7.
(B) 18 = 1 + 1 + 1 + 1 + 2 + 3 + 9.
(C) 6 = 1 + 2 + 3.
(D) 31 = 1 + 3 + 6 + 6 + 7 + 8.

(2) Consider the improper integral ∞∫ 2 1 𝑥 (log 𝑥)2 𝑑𝑥 and the infinite series ∞Í 𝑘=2 1 𝑘 (log 𝑘)2 . Which of the following is/are true?
(A) e integral converges but the series does not converge.
(B) e integral does not converge but the series converges.
(C) Both the integral and the series converge.
(D) e integral and the series both fail to converge.

(3) Let 𝐴 ∈ 𝑀2 (R) be a nonzero matrix. Pick the correct statement(s) from below.
(A) If 𝐴2 = 0, then (𝐼2 − 𝐴)5 = 0.
(B) If 𝐴2 = 0, then (𝐼2 − 𝐴) is invertible.
(C) If 𝐴3 = 0, then 𝐴2 = 0.
(D) If 𝐴2 = 𝐴3 ≠ 0, then 𝐴 is invertible.

(4) Let 𝑓 : [0, 1] −→ [0, 1] be a continuous function. Which of the following is/are true?
(A) For every continuous 𝑔 : [0, 1] −→ R with 𝑔(0) = 0 and 𝑔(1) = 1 there exists 𝑥 ∈ [0, 1] with 𝑓 (𝑥) = 𝑔(𝑥). 1
(B) For every continuous 𝑔 : [0, 1] −→ R with 𝑔(0) < 0 and 𝑔(1) > 1 there exists 𝑥 ∈ [0, 1] with 𝑓 (𝑥) = 𝑔(𝑥).
(C) For every continuous 𝑔 : [0, 1] −→ R with 0 < 𝑔(0) < 1 and 0 < 𝑔(1) < 1 there exists 𝑥 ∈ [0, 1] with 𝑓 (𝑥) = 𝑔(𝑥).
(D) For every continuous 𝑔 : [0, 1] −→ [0, 1] there exists 𝑥 ∈ [0, 1] with 𝑓 (𝑥) = 𝑔(𝑥).

(5) Let 𝐼, 𝐽 be nonempty open intervals in R. Let 𝑓 : 𝐼 −→ 𝐽 and 𝑔 : 𝐽 −→ R be functions. Let ℎ : 𝐼 −→ R be the composite function 𝑔 ◦ 𝑓 . Pick the correct statement(s) from below.
(A) If 𝑓 , 𝑔 are continuous, then ℎ is continuous.
(B) If 𝑓 , 𝑔 are uniformly continuous, then ℎ is uniformly continuous.
(C) If ℎ is continuous, then 𝑓 is continuous.
(D) If ℎ is continuous, then 𝑔 is continuous.

(6) Let 𝐴, 𝐵 be non-empty subsets of R2. Pick the correct statement(s) from below:
(A) If 𝐴 is compact, 𝐵 is open and 𝐴 ∪ 𝐵 is compact, then 𝐴 ∩ 𝐵 ≠ œ.
(B) If 𝐴 and 𝐵 are path-connected and 𝐴 ∩ 𝐵 ≠ œ then 𝐴 ∪ 𝐵 is path-connected.
(C) If 𝐴 and 𝐵 are connected and open and 𝐴 ∩ 𝐵 ≠ œ, then 𝐴 ∩ 𝐵 is connected.
(D) If 𝐴 is countable with |𝐴| ≥ 2, then 𝐴 is not connected.

(7) Pick the correct statement(s) from below.
(A) 𝑋 = Î∞ 𝑛=1 𝑋𝑛 where 𝑋𝑛 = {1, 2, . . . , 2𝑛 } for 𝑛 ≥ 1 is not compact in the product topology.
(B) 𝑌 = Î∞ 𝑛=1 𝑌𝑛 where 𝑌𝑛 = [0, 2𝑛] ⊆ R for 𝑛 ≥ 1 is path-connected in the product topology.
(C) 𝑍 = Î∞ 𝑛=1 𝑍𝑛 where 𝑍𝑛 = (0, 1 𝑛 ) ⊆ R for 𝑛 ≥ 1 is compact in the product topology.
(D) 𝑃 = Î∞ 𝑛=1 𝑃𝑛 where 𝑃𝑛 = {0, 1} for 𝑛 ≥ 1 (with product topology) is homeomorphic to (0, 1).

(8) Let 𝑓 (𝑧) = 𝑒𝑧 −1 𝑧 (𝑧−1) be defined on the extended complex plane C ∪ {∞}. Which of the following is/are true?
(A) 𝑧 = 0, 𝑧 = 1, 𝑧 = ∞ are poles.
(B) 𝑧 = 1 is a simple pole.
(C) 𝑧 = 0 is a removable singularity.
(D) 𝑧 = ∞ is an essential singularity

(9) For 𝐴 ∈ 𝑀3 (C), let 𝑊𝐴 = {𝐵 ∈ 𝑀3 (C) | 𝐴𝐵 = 𝐵𝐴}. Which of the following is/are true?
(A) For all diagonal 𝐴 ∈ 𝑀3 (C), 𝑊𝐴 is a linear subspace of 𝑀3 (C) with dimC 𝑊𝐴 ≥ 3.
(B) For all 𝐴 ∈ 𝑀3 (C), 𝑊𝐴 is a linear subspace of 𝑀3 (C) with dimC 𝑊𝐴 > 3.
(C) ere exists 𝐴 ∈ 𝑀3 (C) such that 𝑊𝐴 is a linear subspace of 𝑀3 (C) with dimC 𝑊𝐴 = 3.
(D) If 𝐴 ∈ 𝑀3 (C) is diagonalizable, then every element of 𝑊𝐴 is diagonalizable.

(10) Let 𝐾 be a field of order 243 and let 𝐹 be a subfield of 𝐾 of order 3. Pick the correct statement(s) from below.
(A) ere exists 𝛼 ∈ 𝐾 such that 𝐾 = 𝐹 (𝛼).
(B) e polynomial 𝑥242 = 1 has exactly 242 solutions in 𝐾.
(C) e polynomial 𝑥26 = 1 has exactly 26 roots in 𝐾.
(D) Let 𝑓 (𝑥) ∈ 𝐹 [𝑥] be an irreducible polynomial of degree 5. en 𝑓 (𝑥) has a root in 𝐾.

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