CMI Entrance Exam 2020 Question Paper for M.Sc or Ph.D Mathematics

CHENNAI MATHEMATICAL INSTITUTE
Postgraduate Programme in Mathematics
MSc/PhD Entrance Examination
4th October 2020
Important: Questions in Part A will be used for screening. There will be a cut-off for Part A,
which will not be more than twenty (20) marks (out of 40). Your solutions to the questions in
Part B will be marked only if your score in Part A places you over the cut-off. (In particular, if
your score in Part A is at least 20 then your solutions to the questions in Part B will be marked.)
However, note that the scores in both the sections will be taken into account while making the
final decision. In order to qualify for the PhD Mathematics interview, you must
obtain at least fifteen (15) marks from among the starred questions (17∗ )–(20∗ ).

Notation: N, Z, Q, R, R≥0 , R+ and C stand, respectively, for the sets of non-negative integers,
of integers, of rational numbers, of real numbers, of non-negative real numbers, of positive real
numbers, and of complex numbers. For a prime power q, Fq is the field with q elements. For
a field F , Mn (F ) stands for the set of n × n matrices over F . When considered as topological
spaces, Rn or C are taken with the euclidean topology.

Part A
Instructions: Each of the questions 1–9 has one or more correct answers. Record your
answers on the attached bubble-sheet by filling in the appropriate circles. Every question is
worth four (4) marks. A solution receives credit if and only if all the correct answers are chosen,
and no incorrect answer is chosen.
(1) Let G be a group and N be a proper normal subgroup. Pick the true statement(s) from
below.
(A) If N and the quotient G/N is finite, then G is finite.
(B) If the complement G \ N of N in G is finite, then G is finite.
(C) If both N and the quotient G/N are cyclic, then G is cyclic.
(D) G is isomorphic to N × G/N .
(2) Let R denote the ring of all continuous functions from R to R, where addition and
multiplication are given, respectively, by (f +g)(x) = f (x)+g(x) and (f g)(x) = f (x)g(x)
for every f, g ∈ R and x ∈ R. A zero-divisor in R is a non-zero f ∈ R such that f g = 0
for some non-zero g ∈ R. Pick the true statement(s) from below:
(A) R has zero-divisors.
(B) If f is a zero-divisor, then f 2 = 0.
(C) If f is a non-constant function and f −1 (0) contains a non-empty open set, then f
is a zero-divisor.
(D) R is an integral domain.
(3) Let U = {(x, y) ∈ R2 | x < y 2 < 4} and V = {(x, y) ∈ R2 | 0 < xy < 4}, both taken
with the subspace topology from R2 . Which of the following statement(s) is/are true?
(A) There exists a non-constant continuous map V −→ R whose image is not an inter-
val.
(B) Image of U under any continuous map U −→ R is bounded.
(C) There exists an  > 0 such that given any p ∈ V the open ball B (p) with centre p
and radius  is contained in V .
(D) If C is a closed subset of R2 which is contained in U , then C is compact.
(4) Let A and B be 5 × 5 real matrices with A2 = B 2 . Which of the following statements
is/are correct?
(A) Either A = B or A = −B.
(B) A and B have the same eigen spaces.
(C) A and B have the same eigen values.
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(D) A13 B 3 = A3 B 13 .
(5) Consider the function f : R2 −→ R given by
x2 p 2
 
f (x, y) = 1 − cos x + y2
y
for y 6= 0 and f (x, 0) = 0. (The square root is chosen to be non-negative). Pick the
correct statement(s) from below:
(A) f is continuous at (0, 0).
(B) f is an open map.
(C) f is differentiable at (0, 0).
(D) f is a bounded function. P
(6) Which P of the following is/arePtrue for a series of real numbers an ?
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(A) If P an converges then P an converges;
(B) If P a2n converges then P an converges;
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(C) if a2 converges then a converges;
P n Pn 1 n
(D) If |an | converges then n an converges;
(7) Which of the following functions are uniformly continuous on R?
(A) f (x) = x;
(B) f (x) = x2 ;
(C) f (x) = (sin x)2 ;
(D) f (x) = e−|x| .
(8) Let U and V be non-empty open connected subsets of C and f : U −→ V a analytic
function. Which of the following statement(s) is/are true?
(A) f 0 (z) 6= 0 for every z ∈ U .
(B) If f is bijective, then f 0 (z) 6= 0 for every z ∈ U .
(C) If f 0 (z) 6= 0 for every z ∈ U , then f is bijective.
(D) If f 0 (z) 6= 0 for every z ∈ U , then f is injective.
(9) Let U denote the unit open disc centred at 0. Let f : U \ {0} −→ C be an analytic
function. Assume that limz−→0 zf (z) = 0.
(A) limz−→0 |f (z)| exists and is in R.
(B) f has a pole of order 1 at 0.
(C) zf (z) has a zero of order 1 at 0.
(D) There exists an analytic function g : U −→ C such that g(z) = f (z) for every
z ∈ U \ {0}.
Instructions: The answers to Question 10 is an integer. Please write the answer in
decimal form in the attached bubble-sheet. The question is worth four (4) marks.
(10) Let f (x) = x2 + ax + b ∈ F3 [X]. What is the number of non-isomorphic quotient rings
F3 [X]/(f (X))?

Part B
Instructions: Answer six (6) questions from below. Provide sufficient justification.
Write your solutions on the page assigned to each question. Each of the questions is worth
ten (10) marks. In order to qualify for the PhD Mathematics interview, you
must obtain at least fifteen (15) marks from among the starred questions
(17∗ )–(20∗ ). Clearly indicate which six questions you would like us to mark in the six
boxes in the bubble sheet. If the boxes are unfilled, we will mark the first six solutions
that appear in your answer-sheet. If you do not want a solution to be considered, clearly
strike it out.
(11) Let (X, d) be a compact metric space. For x ∈ X and  > 0, define B (x) := {y ∈
X | d(x, y) < }. For C ⊆ X and  > 0, define B (C) := ∪x∈C B (x). Let K be the
set of non-empty compact subsets of X. For C, C 0 ∈ K, define δ(C, C 0 ) = inf{ | C ⊆
B (C 0 ) and; C 0 ⊆ B (C)}. Show that (K, δ) is a compact metric space.
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(12) Let f be a non-constant entire function with f (z) 6= 0 for all z ∈ C. Consider the set
U = {z : |f (z)| < 1}. Show that all connected components of U are unbounded.
(13) Let F ⊆ R3 be a non-empty finite set, and X = R3 \F , taken with the subspace topology
of R3 . Show that X is homeomorphic to a complete metric space. (Hint: Look for a
suitable continuous function from X to R.)
(14) Show that there is no differentiable function f : R −→ R such that f (0) = 1 and
f 0 (x) ≥ (f (x))2 for every x ∈ R.
(15) Let a1 , . . . , an be distinct complex numbers. Show that the functions ea1 z , . . . , aan z are
linearly independent over C.
(16) The Frattini subgroup of a finite group G is the intersection of all its proper maximal
subgroups. Let p be a prime number. Show that the Frattini subgroup of Z/pn , n ≥ 2,
is generated by p.
(17∗ ) Let M ∈ Mn (C). Show that M is diagonalizable if and only if for every polynomial
P (X) ∈ C[X] such that P (M ) is nilpotent, P (M ) = 0.
(18∗ ) X is said to have the universal extension property if for every normal space Y and
every closed subset A ⊂ Y and every continuous function f : A −→ X, f extends to
a continuous function from Y to X. You may assume, without proof, that R2 has the
universal extension property.
(A) Prove or find a counterexample: If X has the universal extension property, then X
is connected.
(B) Give an example (with justification) of a compact subset X of R2 that does not
have the universal extension property.
(C) Let X = {(x, sin x) | x ∈ R}. Then show that X has the universal extension
property.
∗
(19 ) Let p be a prime number and q a power of p. Let K be an algebraic closure of Fq . Say
that a polynomial f (X) ∈ K[X] is a q-polynomial if it is of the form
n
i
X
f (X) = ai X q
i=0
Let f (X) be a q-polynomial of degree q n , with a 0 6= 0. Show that the set of zeros of
f (X) is an n-dimensional vector-space over Fq .
(20∗ ) Let an , n ≥ 1 be a sequence of real numbers. If an → a, show that
a1 + 2a2 + 3a3 + · · · + nan a
bn = 2
→ .
n 2

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