CMI Entrance Exam 2019 Question Paper for M.Sc or Ph.D Mathematics

CHENNAI MATHEMATICAL INSTITUTE
Postgraduate Programme in Mathematics
MSc/PhD Entrance Examination
15 May 2019
Instructions:
M − −
• Enter your Admit Card Number here:

• The allowed time is 3 hours.
• This examination has two parts. You may use the blank pages at the end for your
rough-work.
• Part A is worth 40 marks and 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. Record your answers to
Part A in the attached bubble-sheet.
• Part B is worth 60 marks. You should answer six (6) questions in Part B. In order to
qualify for the PhD Mathematics interview, you must obtain at least fifteen (15) marks
from among the starred questions (17∗ )–(20∗ ). Indicate the six questions to be
marked in the boxes in the bubble-sheet. Write your solutions to Part B in
the page assigned to each question.
• Please read the further instructions given before Part A and inside each part
carefully.

For office use only

Part B Part B (ctd.)

No. Marks Remarks No. Marks Remarks

11 17∗

12 18∗

13 19∗

14 20∗

15

16

Part A

Further remarks: Part B

Total

CHENNAI MATHEMATICAL INSTITUTE
Postgraduate Programme in Mathematics
MSc/PhD Entrance Examination
15 May 2019

M − −
Admit Card Number:

Bubble Sheet
Part A
Part B
In the boxes below, clearly indicate the
1. A B C D
SIX (6) solutions that should be marked.
In order to qualify for the PhD Mathe-
2. A B C D matics interview, you must obtain at least
fifteen (15) marks from among the starred
questions (17∗ )–(20∗ ).
3. A B C D

4. A B C D

5. A B C D

6. A B C D

7. A B C D

8. A B C D

9. A B C D

10.

For office use only:

Number of correct answers in Part A:

Marks in Part A:

CHENNAI MATHEMATICAL INSTITUTE
Postgraduate Programme in Mathematics
MSc/PhD Entrance Examination
15 May 2019
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 field F , Mn (F ) stands for the set of n×n matrices over
F and GLn (F ) is the set of invertible n×n matrices over F . The symbol ı denotes a square-root
of −1. When considered as topological spaces, Rn or C are taken with the euclidean topology.
2
When Mn (R) is considered as a topological space, it is identified with Rn .

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) For a field F , F × denotes the multiplicative group (F \ {0}, ×). Choose the correct
statement(s) from below:
(A) Every finite subgroup of R× is cyclic;
(B) The order of every non-trivial finite subgroup of R× is a prime number;
(C) There are infinitely many non-isomorphic non-trivial finite subgroups of R× ;
(D) The order of every non-trivial finite subgroup of C× is a prime number.

(2) Let R be a commutative ring with 1 and I and J ideals of R. Choose the correct
statement(s) from below:
(A) If I or J is maximal then IJ = I ∩ J;
(B) If IJ = I ∩ J, then I or J is maximal;
(C) If IJ = I ∩ J, then 1 ∈ I + J;
(D) If 1 ∈ I + J then IJ = I ∩ J.

(3) Let (X, d) and (Y, ρ) be metric spaces and f : X −→ Y a homeomorphism. Choose the
correct statement(s) from below:
(A) If B ⊆ Y is compact, then f −1 (B) is compact;
(B) If B ⊆ Y is bounded, then f −1 (B) is bounded;
(C) If B ⊆ Y is connected, then f −1 (B) is connected;
(D) If {yn } is Cauchy in Y , then {f −1 (yn )} is Cauchy in X.

(4) Let a, b ∈ R, and consider the R-linear map f : C −→ C, z 7→ az + bz. Choose the
correct statement(s) from below:
(A) f is onto (i.e., surjective) if ab 6= 0;
(B) f is one-one (i.e., injective) if ab 6= 0;
(C) f is onto if a2 6= b2 ;
(D) if a2 = b2 , f is not one-one.

(5) Let
x3 y 3
(
x2 +y 2
, (x, y) 6= (0, 0)
f (x, y) =
0, (x, y) = (0, 0)
Choose the correct statement(s) from below:
(A) f is continuous on R2 ;
(B) f is continuous at every point of R2 \ {(0, 0)};
(C) f is differentiable at every point of R2 \ {(0, 0)};
(D) f is not differentiable at (0, 0).

(6) Let K be the smallest subfield of C containing all the roots of unity. Choose the correct
statement(s) from below:
(A) C is algebraic over K;
(B) K has countably many elements;
(C) Irreducible polynomials in K[X] do not have multiple roots;
(D) The characteristic of K is zero.

(7) The power series
∞
X n2 xn
n!
n=1
equals
(A) x2 ex ;
(B) xex ;
(C) (x2 + x)ex ;
(D) (x2 − x)ex ;

(8) Let f : R −→ R be twice continuously differentiable. Suppose further that f 00 (x) ≥ 0
for every x ∈ R. Choose the correct statement(s) from below:
(A) f is bounded;
(B) f is constant;
(C) RIf f is bounded, then it is infinitely differentiable;
x
(D) 0 f (t)dt is infinitely differentiable with respect to x.
2

(9) Let f (z) be a power-series (with complex coefficients) centred at 0 ∈ C and with a radius
of convergence 2. Suppose that f (0) = 0. Choose the correct statement(s) from below:
(A) f −1 (0) = {0};
(B) If f is a non-constant function on {|z| < 2}, then f −1 (0) = {0};
(C) If f is a non-constant function, then for all ζ ∈ C with sufficiently small |ζ|, the
equation f (z) = ζ has a solution;
(D) Z
f (n) (z)dz = 0
γ
for every n ≥ 1, where γ is a unit circle centred at 0, oriented clockwise, and f (n)
is the nth derivative of f (z).

Instructions: The answer to Question 10 is an integer. You are required to write the
answer in decimal form in the attached bubble-sheet. The question is worth four (4)
marks. )

(10) Let z be a complex variable, and write x = <(z) and y = =(z) for the real and the
imaginary parts, respectively. Let f (z) be a complex polynomial. Let R > 0 be a real
number and γ the circle in C of radius R and centre at 0, oriented in the counter-
clockwise direction. What is the value of
Z
1
(<(f (z))dx + =(f (z))dy)
2πıR γ

3

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) Fix a non-negative integer d. Let
Ad := {A ⊆ C : A is the zero-set of a polynomial of degree ≤ d in C[X]}.
Let T be the coarsest topology on C in which A is closed for every A ∈ Ad .
(A) Determine whether T is Hausdorff.
(B) Show that for every polynomial f (X) ∈ C[X], the function C −→ C defined by
z 7→ f (z) is continuous, where C (on both the sides) is given the topology T .

(12) Let an , n ≥ 0 be complexPnumbers such that limn an = 0.
(A) Show that F (z) := n≥0 an z n is a holomorphic function on {z ∈ C : |z| < 1}.
(B) Let G(z) be a meromorphic function on {z ∈ C : |z| < 2}, with a pole at 1. Show
that G 6= F on {z ∈ C : |z| < 1}. (Hint: consider the function (1 − z)F (z) as
z −→ 1.)

(13) Let | · | : R −→ R≥0 be a function such that for every x, y ∈ R, (i) |x| = 0 if and only if
x = 0; (ii) |x + y| ≤ |x| + |y|; (iii) |xy| = |x||y|. Show that the following are equivalent:
(A) The set {|n| : n ∈ Z} is bounded;
(B) |x + y| ≤ max{|x|, |y|} for every x, y ∈ R.

(14) Let f : [0, 1] −→ R be a continuous function. Show that the sequence
Z 1  n1
|f (x)|n dx
0
is convergent.

(15) Let V be a subspace of the complex vector space Mn (C). Suppose that every non-zero
element of V is an invertible matrix. Show that dimC V ≤ 1.
4

(16) Let n be a positive integer such that every group of order n is cyclic. Show the following.
(A) For all prime numbers p, p2 does not divide n.
(B) If p and q are prime divisors of n, then p does not divide q − 1. (Hint: Consider
2 × 2 matrices  
x y
0 1
p
with x, y ∈ Z/qZ and x = 1. )
(C) Show that (n, φ(n)) = 1, where φ(n) is the number of integers m such that 1 ≤
m ≤ n with gcd(n, m) = 1.

(17∗ ) Let F be a field and G = GLn (F ). For g ∈ G, write Cg = {hgh−1 | h ∈ G}. Let
X = {Cg | g ∈ G, the order of g is 2}. Determine |X|.

(18∗ ) A compactification of a topological space X is a compact topological space Y which
contains a dense subspace homeomorphic to X. Let X = (0, 1], in the subspace topology
of R and f : X −→ R, x 7→ sin x1 . Show the following:
(A) Y := [0, 1] is a compactification of X, but f does not extend to a continuous
function Y −→ R, i.e., there does not exist a continuous function g : Y −→ R such
that g|X = f .
(B) X is homeomorphic to the set X1 := {(t, sin 1t ) | t ∈ X} ⊆ R2 .
(C) The closure Y1 of X1 in R2 is a compactification of X
(D) f extends to a continuous function Y1 −→ R.

(19∗ ) Let f (X) ∈ Z[X] be a monic polynomial. Suppose that α ∈ C and 3α are roots of f .
(A) Show that f (0) 6= 1. (Hint: if ζ and ζ 0 are complex numbers satisfying monic
polynomials in Z[X], then ζζ 0 satisfies a monic polynomial in Z[X].)
(B) Assume that f is irreducible. Let K be the smallest subfield of C containing all
the roots of f . Let σ be a field automorphism of K such that σ(α) = 3α. Show
that σ has finite order and that α = 0.

(20∗ ) Let f : [0, 1] −→ R be a continuous function. Define g(0) = f (0) and g(x) = max{f (y) |
0 ≤ y ≤ x} for 0 < x ≤ 1. Show that g is well-defined and that g is monotone continuous
function.

5

Solution to Question (11)

6

Solution to Question (12)

7

Solution to Question (13)

8

Solution to Question (14)

9

Solution to Question (15)

10

Solution to Question (16)

11

Solution to Question (17*)

12

Solution to Question (18*)

13

Solution to Question (19*)

14

Solution to Question (20*)

15