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CMI MSc/PhD Mathematics Entrance 2021 Question Paper

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 โˆžโˆซ 2 1 ๐‘ฅ (log ๐‘ฅ)2 ๐‘‘๐‘ฅ and the infinite series โˆžร ๐‘˜=2 1 ๐‘˜ (log ๐‘˜)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 ๐ด โˆˆ ๐‘€2 (R) be a nonzero matrix. Pick the correct statement(s) from below.
(A) If ๐ด2 = 0, then (๐ผ2 โˆ’ ๐ด)5 = 0.
(B) If ๐ด2 = 0, then (๐ผ2 โˆ’ ๐ด) is invertible.
(C) If ๐ด3 = 0, then ๐ด2 = 0.
(D) If ๐ด2 = ๐ด3 โ‰  0, then ๐ด is invertible.

(4) Let ๐‘“ : [0, 1] โˆ’โ†’ [0, 1] be a continuous function. Which of the following is/are true?
(A) For every continuous ๐‘” : [0, 1] โˆ’โ†’ R with ๐‘”(0) = 0 and ๐‘”(1) = 1 there exists ๐‘ฅ โˆˆ [0, 1] with ๐‘“ (๐‘ฅ) = ๐‘”(๐‘ฅ). 1
(B) For every continuous ๐‘” : [0, 1] โˆ’โ†’ R with ๐‘”(0) < 0 and ๐‘”(1) > 1 there exists ๐‘ฅ โˆˆ [0, 1] with ๐‘“ (๐‘ฅ) = ๐‘”(๐‘ฅ).
(C) For every continuous ๐‘” : [0, 1] โˆ’โ†’ R with 0 < ๐‘”(0) < 1 and 0 < ๐‘”(1) < 1 there exists ๐‘ฅ โˆˆ [0, 1] with ๐‘“ (๐‘ฅ) = ๐‘”(๐‘ฅ).
(D) For every continuous ๐‘” : [0, 1] โˆ’โ†’ [0, 1] there exists ๐‘ฅ โˆˆ [0, 1] with ๐‘“ (๐‘ฅ) = ๐‘”(๐‘ฅ).

(5) Let ๐ผ, ๐ฝ be nonempty open intervals in R. Let ๐‘“ : ๐ผ โˆ’โ†’ ๐ฝ and ๐‘” : ๐ฝ โˆ’โ†’ R be functions. Let โ„Ž : ๐ผ โˆ’โ†’ R be the composite function ๐‘” โ—ฆ ๐‘“ . Pick the correct statement(s) from below.
(A) If ๐‘“ , ๐‘” are continuous, then โ„Ž is continuous.
(B) If ๐‘“ , ๐‘” are uniformly continuous, then โ„Ž is uniformly continuous.
(C) If โ„Ž is continuous, then ๐‘“ is continuous.
(D) If โ„Ž is continuous, then ๐‘” is continuous.

(6) Let ๐ด, ๐ต be non-empty subsets of R2. Pick the correct statement(s) from below:
(A) If ๐ด is compact, ๐ต is open and ๐ด โˆช ๐ต is compact, then ๐ด โˆฉ ๐ต โ‰  ล“.
(B) If ๐ด and ๐ต are path-connected and ๐ด โˆฉ ๐ต โ‰  ล“ then ๐ด โˆช ๐ต is path-connected.
(C) If ๐ด and ๐ต are connected and open and ๐ด โˆฉ ๐ต โ‰  ล“, then ๐ด โˆฉ ๐ต is connected.
(D) If ๐ด is countable with |๐ด| โ‰ฅ 2, then ๐ด is not connected.

(7) Pick the correct statement(s) from below.
(A) ๐‘‹ = รŽโˆž ๐‘›=1 ๐‘‹๐‘› where ๐‘‹๐‘› = {1, 2, . . . , 2๐‘› } for ๐‘› โ‰ฅ 1 is not compact in the product topology.
(B) ๐‘Œ = รŽโˆž ๐‘›=1 ๐‘Œ๐‘› where ๐‘Œ๐‘› = [0, 2๐‘›] โŠ† R for ๐‘› โ‰ฅ 1 is path-connected in the product topology.
(C) ๐‘ = รŽโˆž ๐‘›=1 ๐‘๐‘› where ๐‘๐‘› = (0, 1 ๐‘› ) โŠ† R for ๐‘› โ‰ฅ 1 is compact in the product topology.
(D) ๐‘ƒ = รŽโˆž ๐‘›=1 ๐‘ƒ๐‘› where ๐‘ƒ๐‘› = {0, 1} for ๐‘› โ‰ฅ 1 (with product topology) is homeomorphic to (0, 1).

(8) Let ๐‘“ (๐‘ง) = ๐‘’๐‘ง โˆ’1 ๐‘ง (๐‘งโˆ’1) be defined on the extended complex plane C โˆช {โˆž}. Which of the following is/are true?
(A) ๐‘ง = 0, ๐‘ง = 1, ๐‘ง = โˆž are poles.
(B) ๐‘ง = 1 is a simple pole.
(C) ๐‘ง = 0 is a removable singularity.
(D) ๐‘ง = โˆž is an essential singularity

(9) For ๐ด โˆˆ ๐‘€3 (C), let ๐‘Š๐ด = {๐ต โˆˆ ๐‘€3 (C) | ๐ด๐ต = ๐ต๐ด}. Which of the following is/are true?
(A) For all diagonal ๐ด โˆˆ ๐‘€3 (C), ๐‘Š๐ด is a linear subspace of ๐‘€3 (C) with dimC ๐‘Š๐ด โ‰ฅ 3.
(B) For all ๐ด โˆˆ ๐‘€3 (C), ๐‘Š๐ด is a linear subspace of ๐‘€3 (C) with dimC ๐‘Š๐ด > 3.
(C) ere exists ๐ด โˆˆ ๐‘€3 (C) such that ๐‘Š๐ด is a linear subspace of ๐‘€3 (C) with dimC ๐‘Š๐ด = 3.
(D) If ๐ด โˆˆ ๐‘€3 (C) is diagonalizable, then every element of ๐‘Š๐ด is diagonalizable.

(10) Let ๐พ be a field of order 243 and let ๐น be a subfield of ๐พ of order 3. Pick the correct statement(s) from below.
(A) ere exists ๐›ผ โˆˆ ๐พ such that ๐พ = ๐น (๐›ผ).
(B) e polynomial ๐‘ฅ242 = 1 has exactly 242 solutions in ๐พ.
(C) e polynomial ๐‘ฅ26 = 1 has exactly 26 roots in ๐พ.
(D) Let ๐‘“ (๐‘ฅ) โˆˆ ๐น [๐‘ฅ] be an irreducible polynomial of degree 5. en ๐‘“ (๐‘ฅ) has a root in ๐พ.

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