CMI Entrance Exam 2018 Question Paper for M.Sc or Ph.D Mathematics

CHENNAI MATHEMATICAL INSTITUTE
Postgraduate Programme in Mathematics
MSc/PhD Entrance Examination
15 May 2018
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 2018

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 2018
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+ and C stand, respectively, for the sets of non-negative integers,
of integers, of rational numbers, of real numbers, of positive real numbers, and of complex
numbers. For a prime number p, Fp is the field with p elements. 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. When Mn (R) is considered as a topological space, it is
2
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) Let G be a group of order 6. Let C1 , C2 , . . . , Ck be the distinct conjugacy classes of G.
Which of the following sequences of integers are possible values of (|C1 |, |C2 |, . . . , |Ck |)?
(A) (1, 1, 1, 1, 1, 1);
(B) (1, 5);
(C) (3, 3);
(D) (1, 2, 3).

(2) Let R = F2 [X]. Choose the correct statement(s) from below:
(A) R has uncountably many maximal ideals;
(B) Every maximal ideal of R has infinitely many elements;
(C) For all maximal ideals m of R, R/m is a finite field;
(D) For every integer n, every ideal of R has only finitely many elements of degree ≤ n.

(3) Which of the following spaces are connected?
(A) {(x, y) ∈ R2 | xy = 1} as a subspace of R2 ;
(B) The set of upper triangular matrices as a subspace of Mn (R);
(C) The set of invertible diagonal matrices as a subspace of Mn (R);
(D) {(x, y, z) ∈ R3 | z ≥ 0, z 2 ≥ x2 + y 2 } as a subspace of R3 .

(4) Let A be an n × n nilpotent real matrix A. Define
1 1
eA = In + A + A2 + A3 + · · · .
2! 3!
Choose the correct statement(s) from below:
(A) For every real number t, etA is invertible;
(B) There exists a basis of Rn such that eA is upper-triangular;
(C) There exist B, P ∈ GLn (R) such that B = P eA P −1 and trace(B) = 0;
(D) There exists a basis of Rn such that A is lower-triangular.

(5) Let f (w, x, y, z) = wz − xy. Choose the correct statement(s) from below:
(A) The directional derivative at (1, 0, 0, 1) in the direction (a, b, c, d) is 0 if a + d = 0;
(B) The directional derivative at (1, 0, 0, 1) in the direction (a, b, c, d) is 0 only if a+d =
0;
(C) The vector (0, −1, −1, 0) is normal to f −1 (1) at the point (1, 0, 0, 1);
(D) The set of points (a, b, c, d) where the total derivative of f is zero is finite.

(6) Choose the correct statement(s) from below:
(A) There exists a subfield F of C such that F * R and F ≃ Q[X]/(2X 3 − 3X 2 + 6);
(B) For every irreducible cubic polynomial f (X) ∈ Q[X], there exists a subfield F of
C such that F * R and F ≃ Q[X]/f (X);
(C) There exists a subfield F of R such that F ≃ Q[X]/(2X 3 − 3X 2 + 6);
(D) For every irreducible cubic polynomial f (X) ∈ Q[X], there exists a subfield F of
R such that F ≃ Q[X]/f (X).

R1
(7) For a continuous function f : [0, 1] −→ R, define an (f ) = 0 xn f (x)dx. Choose the
correct statement(s) from below:
(A) The sequence {an (f )} is bounded for every continuous function f : [0, 1] −→ R;
(B) The sequence {an (f )} is Cauchy for every continuous function f : [0, 1] −→ R;
(C) The sequence {an (f )} converges to 0 for every continuous function f : [0, 1] −→ R;
(D) There exists a continuous function f : [0, 1] −→ R such that the sequence {an (f )}
is divergent.

2

(8) Let f : C −→ C be a holomorphic function. Choose the correct statement(s) from
below:
(A) f (z) is holomorphic;
(B) Suppose that f (R) ⊆ R. Then f (R) is open in R;
(C) the map z 7→ ef (z) is holomorphic;
(D) Suppose that f (C) ⊂ R. Then f (A) is closed in C for every closed subset A of C.

(9) Let f : R −→ R be a twice-differentiable function such that f ( n1 ) = 0 for every positive
integer n. Choose the correct statement(s) from below:
(A) f (0) = 0;
(B) f 0 (0) = 0;
(C) f 00 (0) = 0;
(D) f is a nonzero polynomial.

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 A be a non-zero 4×4 complex matrix such that A2 = 0. What is the largest possible
rank of A?

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) A subspace Y of R is said to be a retract of R if there exists a continuous map r : R −→ Y
such that r(y) = y for every y ∈ Y .
(A) Show that [0, 1] is a retract of R.
(B) Determine (with appropriate justification) whether every closed subset of R is a
retract of R.
(C) Show that (0, 1) is not a retract of R.

(12) Let N be a positive integer and an be a complex number for every −N ≤ n ≤ N .
Consider the holomorphic function on {z ∈ C|z 6= 0} given by
n=N
X
F (z) = an z n .
n=−N
Consider the function f defined on the open unit disc {z ∈ C : |z| < 1} by
Z
1 F (ξ)
f (z) = dξ,
2πi Γ ξ − z
where Γ is the boundary of the disc, oriented counterclockwise. Write down an expres-
sion for f in terms of the coefficients an of F .

(13) Let φ : [0, 1] −→ R be a continuous function such that
Z 1
φ(t)e−at dt = 0
0
for every a ∈ R+ . Show that for every non-negative integer n,
Z 1
φ(t)tn dt = 0.
0

(14) Let U be a non-empty open subset of R. Suppose that there exists a uniformly contin-
uous homeomorphism h : U −→ R. Show that U = R.

4

(15) Let A be 2 × 2 orthogonal matrix such that det(A) = −1. Show that A represents
reflection about a line in R2 .

(16) A subgroup H of a group G is said to be a characteristic subgroup if σ(H) = H for
every group isomorphism σ : G −→ G of G.
(A) Determine all the characteristic subgroups of (Q, +) (the additive group).
(B) Show that every characteristic subgroup of G is normal in G. Determine whether
the converse is true.

(17∗ ) Write V for the space of 3 × 3 skew-symmetric real matrices.
(A) Show that for A ∈ SO3 (R) and M ∈ V , AM At ∈ V . Write A · M for this action.
(B) Let Φ : R3 −→ V be the map
   
u 0 w −v
 v  7→ −w 0 u .
w v −u 0
With the usual action of SO3 (R) on R3 and the above action on V , show that
Φ(Av) = A · Φ(v) for every A ∈ SO3 (R) and v ∈ R3 .
(C) Show that there does not exist M ∈ V , M 6= 0 such that for every A ∈ SO3 (R),
A · M belongs to the span of M .

(18∗ ) Let m > 1 be an integer and consider the following equivalence relation on C r {0}:
2πıa
z1 ∼ z2 if z1 = z2 e m for some a ∈ Z. Write X for the set of equivalence classes and
π : C r {0} −→ X for the map that takes z to its equivalence class. Define a topology on
X by setting U ⊆ X to be open if and only if π −1 (U ) is open in the euclidean topology
of C r {0}. Determine (with appropriate justification) whether X is compact.

(19∗ ) Let k be a field, n a positive integer and G a finite subgroup of GLn (k) such that |G| > 1.
Further assume that every g ∈ G is upper-triangular and all the diagonal entries of g
are 1.
(A) Show that char k > 0. (Hint: consider the minimal polynomials of elements of G.)
(B) Show that the order of g is a power of char k, for every g ∈ G.
(C) Show that the centre of G has at least two elements.

(20∗ ) Let f : [0, 1] −→ R be a continuous function. Determine (with appropriate justification)
the following limit: Z 1
lim nxn f (x)dx.
n−→∞ 0

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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*)

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