CMI MSc/PhD Mathematics Entrance 2022 Question Paper

Organisation : MSc/PhD Mathematics
Exam : MSc/PhD Mathematics Entrance Exam
Document Type : Question Paper
Year : 2022
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 2021 Question Paper

MSc/PhD Mathematics Question Paper

(1) By a simple group, we mean a group 𝐺 in which the only normal subgroups are {1𝐺 } and 𝐺. Pick the correct statement(s) from below.
(A) No group of order 625 is simple.
(B) GL(2, R) is simple.
(C) Let 𝐺 be a simple group of order 60. en 𝐺 has exactly six subgroups of order 5.
(D) Let 𝐺 be a group of order 60. en 𝐺 has exactly seven subgroups of order 3.

(2) Let 𝑓 : R −→ (0, ∞) be an infinitely diferentiable function with ∫ ∞ −∞ 𝑓 (𝑡)𝑑𝑡 = 1. Pick the correct statement(s) from below.
(A) 𝑓 (𝑡) is bounded.
(B) lim|𝑡 |−→∞ 𝑓 ′ (𝑡) = 0.
(C) ere exists 𝑡0 ∈ R such that 𝑓 (𝑡0) ≥ 𝑓 (𝑡) for all 𝑡 ∈ R.
(D) 𝑓 ′′ (𝑎) = 0 for some 𝑎 ∈ R.

(3) Let P𝑛 = {𝑓 (𝑥) ∈ R[𝑥] | deg 𝑓 (𝑥) ≤ 𝑛}, considered as an (𝑛 + 1)-dimensional real vector space. Let 𝑇 be the linear operator 𝑓 ↦ → 𝑓 + d𝑓 d𝑥 on P𝑛. Pick the correct statement(s) from below.
(A) 𝑇 is invertible.
(B) 𝑇 is diagonalizable.
(C) 𝑇 is nilpotent.
(D) (𝑇 − 𝐼 )2 = (𝑇 − 𝐼 ) where 𝐼 is the identity map.

(4) Pick the correct statement(s) from below.
(A) ere exists a finite commutative ring 𝑅 of cardinality 100 such that 𝑟 2 = 𝑟 for all 𝑟 ∈ 𝑅.
(B) ere is a field 𝐾 such that the additive group (𝐾, +) is isomorphic to the multiplicative group (𝐾×, ·). 1
(C) An irreducible polynomial in Q[𝑥] is irreducible in Z[𝑥].
(D) A monic polynomial of degree 𝑛 over a commutative ring 𝑅 has at most 𝑛 roots in 𝑅.

(5) Pick the correct statement(s) from below.
(A) if 𝑓 is continuous and bounded on (0, 1), then 𝑓 is uniformly continuous on (0, 1).
(B) If 𝑓 is uniformly continuous on (0, 1), then 𝑓 is bounded on (0, 1).
(C) If 𝑓 is continuous on (0, 1) and lim𝑥−→0+ 𝑓 (𝑥) and lim𝑥−→1− 𝑓 (𝑥) exists, then 𝑓 is uniformly continuous on (0, 1).
(D) Product of a continuous and a uniformly continuous function on [0, 1] is uniformly continuous.

(6) Let 𝑋 be the metric space of real-valued continuous functions on the interval [0, 1] with the “supremum distance”: 𝑑 (𝑓 , 𝑔) = sup{|𝑓 (𝑥) − 𝑔(𝑥)| : 𝑥 ∈ [0, 1]} for all 𝑓 , 𝑔 ∈ 𝑋 . Let 𝑌 = {𝑓 ∈ 𝑋 : 𝑓 ( [0, 1]) ⊂ [0, 1]} and 𝑍 = {𝑓 ∈ 𝑋 : 𝑓 ( [0, 1]) ⊂ [0, 1 2 ) ∪ ( 1 2 , 1]}. Pick the correct statement(s) from below.
(A) 𝑌 is compact.
(B) 𝑋 and 𝑌 are connected.
(C) 𝑍 is not compact.
(D) 𝑍 is path-connected.

(7) Let 𝑋 := {(𝑥, 𝑦, 𝑧) ∈ R3 | 𝑧 ≤ 0, or 𝑥, 𝑦 ∈ Q} with subspace topology. Pick the correct statement(s) from below.
(A) 𝑋 is not locally connected but path connected.
(B) ere exists a surjective continuous function 𝑋 −→ Q≥0 (the set of non-negative rational numbers, with the subspace topology of R).
(C) Let 𝑆 be the set of all points 𝑝 ∈ 𝑋 having a compact neighbourhood (i.e. there exists a compact 𝐾 ⊂ 𝑋 containing 𝑝 in its interior). en 𝑆 is open.
(D) e closed and bounded subsets of 𝑋 are compact.

(8) Consider the complex polynomial 𝑃 (𝑥) = 𝑥6 + 𝑖𝑥4 + 1. (Here 𝑖 denotes a square-root of −1.) Pick the correct statement(s) from below.
(A) 𝑃 has at least one real zero.
(B) P has no real zeros.
(C) 𝑃 has at least three zeros of the form 𝛼 + 𝑖𝛽 with 𝛽 < 0.
(D) 𝑃 has exactly three zeros 𝛼 + 𝑖𝛽 with 𝛽 > 0.

(9) Let 𝑣 a (fixed) unit vector in R3. (We think of elements of R𝑛 as column vectors.) Let 𝑀 = 𝐼3 −2𝑣𝑣𝑡 . Pick the correct statement(s) from below.
(A) 0 is an eigenvalue of 𝑀.
(B) 𝑀2 = 𝐼3.
(C) 1 is an eigenvalue of 𝑀.
(D) e eigenspace for the eigenvalue −1 is 2-dimensional.

(10) Let 𝑓 (𝑧) = Í𝑛 ≥0 𝑎𝑛𝑧𝑛 be an analytic function on the open unit disc 𝐷 around 0 with 𝑎1 ≠ 0. Suppose that Í𝑛 ≥2 |𝑛𝑎𝑛 | < |𝑎1 |. en which of the following are true?
(A) ere are only finitely many such 𝑓 .
(B) |𝑓 ′ (𝑧)| > 0 for all 𝑧 ∈ 𝐷.
(C) If 𝑧, 𝑤 ∈ 𝐷 are such that 𝑧 ≠ 𝑤 and 𝑓 (𝑧) = 𝑓 (𝑤), then 𝑎1 = − Í𝑛 ≥2 𝑎𝑛 (𝑧𝑛−1 + 𝑧𝑛−2𝑤 + · · · + 𝑤𝑛−1).
(D) 𝑓 is one-one on 𝐷

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