Notesale: Turn your study into money

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CHENNAI MATHEMATICAL INSTITUTE
Postgraduate Programme in Mathematics
MSc/PhD Entrance Examination
18 May 2015
Instructions:
• Enter your Registration Number here: CMI–PG–
• Enter the name of the city where you write this test:
• The allowed time is 3 hours
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You may use the blank pages at the end for your
rough-work
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There will be a cut-off for
Part A, which will not be more than twenty (20) marks (out of 40)
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(In particular, if your score in Part A is at least 20 then your solutions to the
questions in Part B will be marked
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Record your answers to
Part A in the attached bubble-sheet
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You should answer six (6) questions in Part B
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Indicate the six questions to be
marked in the boxes in the bubble-sheet
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• Please read the further instructions given before Part A and inside each part
carefully
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Marks

Part B (ctd
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11

17∗

12

18∗

13

19∗

14

20∗

15
16

Part A
Further remarks:

Part B

Total

Marks

Remarks

This page is intentionally left blank
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A

B

C

2
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A

B

C

D

4
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A

B

C

D

6
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A

B

C

Part B
In the boxes below, clearly indicate the
SIX (6) solutions that should be marked
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D

8
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10
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CHENNAI MATHEMATICAL INSTITUTE
Postgraduate Programme in Mathematics
MSc/PhD Entrance Examination
18 May 2015
Important: Questions in Part A will be used for screening
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Your solutions to the questions in
Part B will be marked only if your score in Part A places you over the cut-off
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)
However, note that the scores in both the sections will be taken into account while making the
final decision
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Notation: N, Z, Q, R and C stand, respectively, for the sets of the natural numbers, of the
integers, of the rational numbers, of the real numbers, and of the complex numbers
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For a field F , Mm×n (F ) stands for the set of
m × n matrices over F and GLn (F ) is the set of invertible n × n matrices over F
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Part A
Instructions: Each of the questions 1–7 has one or more correct answers
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Every question is
worth four (4) marks
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(1) Which of the following topological spaces is/are connected?
(A) GL1 (R)
(B) GL1 (C)
(C) GL2 (R)
x −y
(D)
: x, y ∈ R, x2 + y 2 = 1
y x
1
(2) Consider f : {z ∈ C : |z| > 1} −→ C, f (z) = z
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(B) There does not exist an entire function g such that g(z) = f (z) for every z ∈ C
with |z| > 1
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(3) Let
a b
1 b
:b∈R
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(B) G/N is isomorphic to {a ∈ R : a > 0} under multiplication
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(D) N is isomorphic to R under addition
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(5) Which of the following complex numbers has/have a prime number as the degree of its
minimal polynomial over Q?
(A) ζ7 , a primitive 7th root of unity;
√
√
(B) √2 + 3;
(C) √−1;
(D) 3 2
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Choose the correct statement(s):
(A) R is a field;
(B) R contains Z as a subring;
(C) Every ideal in R[X] is principal;
(D) R contains Fp as a subring for some prime number p
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Define F : R −→ R by F (x) =
x
−∞ f (t)dt
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Instructions: The answers to questions 8–10 are integers
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Every question is worth four
(4) marks
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How many distinct possible images of ω
are there under all the field homomorphisms Q(ω) −→ C
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What is value of M such that
1
dz?
2πıM =
2 − 5z + 6
C z

(10) Consider the set R[X] of polynomials in X with real coefficients as a real vector space
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2
dX
dX

What is the nullity of f ?
2

Part B
Instructions: Answer six (6) questions from below
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Write your solutions on the page assigned to each question
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In order to qualify for the PhD Mathematics interview, you
must obtain at least fifteen (15) marks from among the starred questions
(17∗ )–(20∗ )
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If the boxes are unfilled, we will mark the first six solutions
that appear in your answer-sheet
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(11) Let f ∈ R[x, y] be such that there exists a non-empty open set U ⊆ R2 such that
f (x, y) = 0 for every (x, y) ∈ U
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(12) Let A ∈ Mn×n (C)
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Show that λ is an eigenvalue of (In + A) if and only if
λ = 1
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)
(b) Suppose that A2 = −1
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(13) Let f be a non-constant entire function satisfying the following conditions:
(a) f (0) = 0;
(b) For every positive real number M , the set {z : |f (z) < M } is connected
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(14) Let (amn )m≥1,n≥1 be a double sequence of real numbers such that
(a) For every n, bn := lim amn exists;
m→∞

(b) For all strictly increasing sequences (mk )k≥1 and (nk )k≥1 of positive integers,
lim amk nk = 1
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(15) Let f ∈ C[x, y] be such that f (x, y) = f (y, x)
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(16) Let X be a topological space and f : X −→ [0, 1] be a closed continuous surjective map
such that f −1 (a) is compact for every 0 ≤ a ≤ 1
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(A
map is said to be closed if it takes closed sets to closed sets
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(Hint: For a set P of positive prime
1
numbers, consider the smallest subring of Q that contains { p | p ∈ P }
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n

Show that f is continuous
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(19∗ ) Let m and n be positive integers and p a prime number
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Let U ⊆ GLm (Fp ) be the subgroup that consists of all the matrices with
1’s on the diagonal and 0’s below the diagonal
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(20∗ ) Let m and n be positive integers and 0 ≤ k ≤ min{m, n} an integer
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(You may use the following fact: For t ≥ 2, GLt (C) is connected
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Solution to Question (11)

4

Please indicate in the bubble-sheet the questions in Part B to be marked
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Solution to Question (13)

6

Please indicate in the bubble-sheet the questions in Part B to be marked
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Solution to Question (15)

8

Please indicate in the bubble-sheet the questions in Part B to be marked
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Solution to Question (17*)

10

Please indicate in the bubble-sheet the questions in Part B to be marked
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Solution to Question (19*)

12

Please indicate in the bubble-sheet the questions in Part B to be marked

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