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 πΊ in which the only normal subgroups are {1πΊ } and πΊ. Pick the correct statement(s) from below.
(A) No group of order 625 is simple.
(B) GL(2, R) is simple.
(C) Let πΊ be a simple group of order 60. en πΊ has exactly six subgroups of order 5.
(D) Let πΊ be a group of order 60. en πΊ has exactly seven subgroups of order 3.
(2) Let π : R ββ (0, β) be an infinitely diferentiable function with β« β ββ π (π‘)ππ‘ = 1. Pick the correct statement(s) from below.
(A) π (π‘) is bounded.
(B) lim|π‘ |βββ π β² (π‘) = 0.
(C) ere exists π‘0 β R such that π (π‘0) β₯ π (π‘) for all π‘ β R.
(D) π β²β² (π) = 0 for some π β R.
(3) Let Pπ = {π (π₯) β R[π₯] | deg π (π₯) β€ π}, considered as an (π + 1)-dimensional real vector space. Let π be the linear operator π β¦ β π + dπ dπ₯ on Pπ. Pick the correct statement(s) from below.
(A) π is invertible.
(B) π is diagonalizable.
(C) π is nilpotent.
(D) (π β πΌ )2 = (π β πΌ ) where πΌ is the identity map.
(4) Pick the correct statement(s) from below.
(A) ere exists a finite commutative ring π
of cardinality 100 such that π 2 = π for all π β π
.
(B) ere is a field πΎ such that the additive group (πΎ, +) is isomorphic to the multiplicative group (πΎΓ, Β·). 1
(C) An irreducible polynomial in Q[π₯] is irreducible in Z[π₯].
(D) A monic polynomial of degree π over a commutative ring π
has at most π roots in π
.
(5) Pick the correct statement(s) from below.
(A) if π is continuous and bounded on (0, 1), then π is uniformly continuous on (0, 1).
(B) If π is uniformly continuous on (0, 1), then π is bounded on (0, 1).
(C) If π is continuous on (0, 1) and limπ₯ββ0+ π (π₯) and limπ₯ββ1β π (π₯) exists, then π is uniformly continuous on (0, 1).
(D) Product of a continuous and a uniformly continuous function on [0, 1] is uniformly continuous.
(6) Let π be the metric space of real-valued continuous functions on the interval [0, 1] with the βsupremum distanceβ: π (π , π) = sup{|π (π₯) β π(π₯)| : π₯ β [0, 1]} for all π , π β π . Let π = {π β π : π ( [0, 1]) β [0, 1]} and π = {π β π : π ( [0, 1]) β [0, 1 2 ) βͺ ( 1 2 , 1]}. Pick the correct statement(s) from below.
(A) π is compact.
(B) π and π are connected.
(C) π is not compact.
(D) π is path-connected.
(7) Let π := {(π₯, π¦, π§) β R3 | π§ β€ 0, or π₯, π¦ β Q} with subspace topology. Pick the correct statement(s) from below.
(A) π is not locally connected but path connected.
(B) ere exists a surjective continuous function π ββ Qβ₯0 (the set of non-negative rational numbers, with the subspace topology of R).
(C) Let π be the set of all points π β π having a compact neighbourhood (i.e. there exists a compact πΎ β π containing π in its interior). en π is open.
(D) e closed and bounded subsets of π are compact.
(8) Consider the complex polynomial π (π₯) = π₯6 + ππ₯4 + 1. (Here π denotes a square-root of β1.) Pick the correct statement(s) from below.
(A) π has at least one real zero.
(B) P has no real zeros.
(C) π has at least three zeros of the form πΌ + ππ½ with π½ < 0.
(D) π has exactly three zeros πΌ + ππ½ with π½ > 0.
(9) Let π£ a (fixed) unit vector in R3. (We think of elements of Rπ as column vectors.) Let π = πΌ3 β2π£π£π‘ . Pick the correct statement(s) from below.
(A) 0 is an eigenvalue of π.
(B) π2 = πΌ3.
(C) 1 is an eigenvalue of π.
(D) e eigenspace for the eigenvalue β1 is 2-dimensional.
(10) Let π (π§) = Γπ β₯0 πππ§π be an analytic function on the open unit disc π· around 0 with π1 β 0. Suppose that Γπ β₯2 |πππ | < |π1 |. en which of the following are true?
(A) ere are only finitely many such π .
(B) |π β² (π§)| > 0 for all π§ β π·.
(C) If π§, π€ β π· are such that π§ β π€ and π (π§) = π (π€), then π1 = β Γπ β₯2 ππ (π§πβ1 + π§πβ2π€ + Β· Β· Β· + π€πβ1).
(D) π is one-one on π·
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