CMI Entrance Exam 2026 Question Paper for M.Sc or Ph.D Mathematics

FOR CMI EXAM PREPARATION

CMI 2026
Question Paper · M.Sc
or Ph.D Mathematics
EXAM YEAR TYPE SUBJECT

CMI 2026 Question Paper M.Sc or Ph.D Mathematics

Notes · Sample Papers · Previous Year Papers · Mock Tests

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CHENNAI MATHEMATICALaINSTITUTE
Postgraduate Programme in Mathematics
MSc/PhD Entrance Examination
2nd May 2026
Important information and instructions:
(1) Questions in Part A (Questions 1 – 10) will be used for screening. There will be a
cut-off for Part A, which will not be more than 20 marks (out of 40).
(2) Each question in Part A has one or more correct answers. Enter your answers to
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these questions into the computer as instructed. Every question is worth 4 marks. A solu-
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tion will receive credit if and only if all the correct answers are chosen, and no incorrect answer
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is chosen.
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(3) Your solutions to the questions in Part B (Questions 11– 20∗ ) will be marked only
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if your score in Part A places you over the cut-off. (In particular, if your score in Part A
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is at least 20 then your solutions to the questions in Part B will be marked.)
(4) Answer 6 questions from Part B, on the pages assigned to them, with sufficient
justification. Each question is worth 10 marks. Clearly indicate which six questions you
would like us to mark in the six boxes on the front 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.
(5) The scores in both the sections will be taken into account while making the final
decision. You are advised to spend at least 90 minutes on Part B. In order to qualify
for the PhD Mathematics interview, you must obtain at least 15 marks from among the
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starred questions 17∗ –20∗ .
(6) Time: 3 hours.

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Notation: N, Z, Q, R, and C stand, lrespectively, for the sets of non-negative integers,
of integers, of rational numbers, of a
real numbers, and of complex numbers. For a field F
and a positive integer n, M (F ) stands for the set of n × n matrices over F and GL (F )
n n
stands for the set of invertible n×n matrices over F . The n×n identity matrix is denoted
by In ; the field will be clear from context. When considered as topological spaces, Rn or
Cn are taken with the euclidean topology, unless otherwise stated.

Part A
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(1) Let X be a metric space. A subset A ⊂ X is said to be totally bounded if for every
e
 > 0, there exist a1 , . . . , an ∈ A such that A is contained in the union of the balls m
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of radius  centered at a1 , . . . , an . Pick the correct statement(s) from below.

las (A) Every bounded set in Rn is totally bounded.
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ag (B) Suppose that every sequence in X has a Cauchy subsequence. Then X is
totally bounded.
(C) Let f : X −→ Y be a continuous map of metric spaces. If A ⊂ X is bounded,
then f (A) is bounded in Y .
(D) Let f : X −→ Y be a uniformly continuous map of metric spaces. If A ⊂ X
is totally bounded, then f (A) is totally bounded in Y .

(2) Let V be a complex vector space. A linear operator P : V −→ V is called a
projection if P 2 = P . Pick the correct statement(s) from below.
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(A) Let P be a projection operator on V . Then the rank of P is equal to the
trace of P .
(B) Let P be a projection operator on V . Then 0 is an eigenvalue of P .
(C) Let T be a linear operator on V which commutes with every projection
operator. Then every non-zero vector v ∈ V is an eigenvector of T .
(D) Let T be a linear operator on V which commutes with every projection
operator. Then T is a scalar multiple of the identity operator.

(3) Consider the function f : R2 \ {(0, 0)} −→ R defined by
xy
f (x, y) = .
x2 + xy + y 2
Pick the correct statement(s) from below.
(A) f is everywhere differentiable and extends differentiably to the whole of R2 .
(B) f is everywhere differentiable and extends continuously but not differentiably
to the whole of R2 .
(C) f is everywhere differentiable but does not extend continuously to the whole
of R2 .
(D) f is everywhere differentiable and extends to a function g on the whole of R2
such that both partial derivatives of g exist at (0, 0).

(4) Let F be a field of order 212 .
Pick the correct statement(s) from below.
(A) F has exactly 6 subfields, including F .
(B) Every element of F is a square (that is, for every x ∈ F , there exists an
element y ∈ F such that y 2 = x).
(C) If K and L are subfields of F of order 8 and 64 respectively, then K ∪ L is a
subfield of F .
(D) F contains a subfield of order 128.

(5) Let G be a finite group of order p2 q, where p < q are primes. Which of the
following statements are always true?
(A) If p is odd, then G has a normal Sylow q-subgroup.
(B) If p is odd, then G has a normal and abelian subgroup N such that G/N is
abelian.
(C) If p divides q − 1, then G has at least two Sylow p-subgroups.
(D) G is abelian.

(6) Let (an )n≥1 be a sequence of positive real numbers. For each k ∈ N, define
Sk : R −→ R by
k
X
Sk (x) = an sin(nx).
n=1

Let D = {x ∈ R : limk−→∞ Sk (x) exists in R}, and f : D −→ R be defined by
f (x) = limk−→∞ Sk (x).
Pick the correct statement(s) from below.
(A) If an = √1n , then D = R.
2

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(B) If an = n12 and g : (0, 1] −→ R is defined by ag(x) = S (x), where N (x) is
N (x)
the greatest integer less than or equal to 1/x, then lim supx−→0+ g(x)
x
< ∞.
1
(C) If an = n3 , then f is Lipschitz continuous on D, i.e.,
( )
|f (x) − f (y)|
sup : x, y ∈ D, x 6= y < ∞.
|x − y|
(D) If an = n1 , then f is continuous on (0, 2π) ∩ D.
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(7) Let k be a field and let R = k[X]/(X n ) with n ≥ 2. Which of the following
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statements about R-modules are always true?
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(A) Every finitely generated R-module is a direct sum of cyclic modules.
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(B) Every R-module is free.
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(C) Every cyclic R-module is isomorphic to R or R/(X ) for some 1 ≤ m ≤ n.
(D) Every R-module is cyclic.

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(8) Let f (z) = ∞ n=0 an z be a power series with a radius of convergence exactly equal
P

to 1. Suppose f (z) can be analytically continued to a meromorphic function on
the open disk D = {z ∈ C : |z| < 2}, and its only singularity in this region is a
c−2 c−1
pole of order 2 at z = 1 with principal part (z−1) 2 + z−1 , where c−2 , c−1 6= 0.

Pick the correct statement(s) from below.
(A) limn−→∞ (an+1 − an ) = 0.
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(B) The series ∞ n=0 |an |(3/2) converges.
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(C) Let Γ be the contour |w| = 3/2 traversed counterclockwise. Then

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a g 2πi Γ
f (w) dw 6= 0.

(D) Let R be the radius of convergence of the Taylor expansion of f (z) centered
n
at zn = eiπ/n . Then limn−→∞ nRn = π.

(9) Let {an }n≥0 be a sequence of real numbers such that limn−→∞ an = 0. Pick the
correct statement(s) from below.
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(A) The limit limx−→1− ((1 − x) ∞ n=0 an x ) as x approaches 1 from below exists
P

and is equal to 0.
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(B) The limit limx−→1+ ((1 − x) ∞ n=0 an x ) as x approaches 1 from above exists
P

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and is equal to 0.
(C) The function f (x) = (1 − x) ∞
P n
e
n=0 an x is defined and continuous in the open m
e m interval (0, 1).
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las (D) If the function f (x) = (1 − x) ∞
P n
g
n=0 an x is defined and continuous in the
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ag open interval (0, 1), then |an | ≤ 1/n for all n ≥ 1.

(10) A sequence {Ck } of real 2 × 2 matrices is said to converge to a 2 × 2 matrix C,
written Ck −→ C, if the entries of Ck converge to the corresponding entries of
C. Now suppose that A is a real 2 × 2 matrix such that the sequence {Ak }k≥1
converges to the zero matrix: Ak −→ 0.
Pick the correct statement(s) from below.
(A) The eigenvalues of A are always real.
(B) The eigenvalues λj of A must satisfy |λj | < 1, j = 1, 2.
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(C) The eigenvalues of A P
must be distinct.
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(D) The geometric series ∞ −1
k=0 A converges to (I − A) .

Part B

(11) Let k = Z/pZ where p is a prime. Determine all the ring automorphisms of the
polynomial ring k[X].

(12) Let f (z) be a non-constant meromorphic function on C with only finitely many
poles. Assume that lim f (1/z) = 2026. Let m denote the number of poles of f
z−→0
and let t be the order of the zero of f (z) − 2026 at ∞. Determine the number of
solutions (counted with multiplicity) to f (z) = 2026 in terms of m and t.
Hint: First show that f is a rational function.

(13) Let A be a set (possibly uncountable). For each finite subset F of A, let SF
be a non-empty set of functions from F to {0, 1}. Assume that the collection
{SF | F ⊆ A, F finite} is consistent, i.e., {s|F | s ∈ SG } = SF for every pair
F ⊆ G of finite subsets of A. Show that there exists a function f : A −→ {0, 1}
such that f |F ∈ SF for every finite subset F of A.

(14) (A) (5 marks) Let M2 (R) ∼ = R4 be the space of real 2 × 2 matrices with the Eu-
clidean topology. Show that the set of matrices with two distinct eigenvalues
(in C) is open in M2 (R) .
(B) (5 marks) Let n be a positive integer. Let V be an n-dimensional vector space
over a finite field with q elements. Calculate the number of k-dimensional
subspaces of V for 1 ≤ k ≤ n.

(15) Let V be a finite-dimensional vector space over a field and let GL(V ) denote the
group of invertible linear transformations V −→ V . Let V = ⊕ni=1 Vi , where Vi are
subspaces of V . Let G ⊂ GL(V ) be a subgroup such that for all g ∈ G, g(Vi ) ⊂ Vi
for all 1 ≤ i ≤ n. Suppose further that the centraliser of G in GL(V ) is contained
in the center of GL(V ). Show that n = 1 and V1 = V .
Hint: Consider the projections πi : V −→ Vi and show that every element of G
commutes with each πi .

(16) Let F be a finite field with q elements.
(A) (7 marks) Show that every element of F can be written as a sum of two
squares. That is, for every a ∈ F , show that there exist b, c ∈ F such that
a = b2 + c 2 .
(B) (3 marks) For a ∈ F , let N (a) denote the cardinality of the set {(b, c) ∈ F 2 |
b2 + c2 = a}. Show that N (a) = q 2 .
P
a∈F
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∗ 2 a C([0, 1]) denote the set of all
(17 ) Let K : [0, 1] −→ R be a continuous function. Let
real-valued continuous functions on [0, 1].
Let f ∈ C([0, 1]). For x ∈ [0, 1], define
Z 1
T (f )(x) := K(x, y)f (y) dy.
0
Also, set ||f k∞ := sup |f (x)|.
x∈[0,1]
(A) Show that T (f ) ∈ C([0, 1]).
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(B) Show that there exists a constant M > 0, depending only on K, such that

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. c kT (f )k ≤ M kf k .
{f ∈ C([0, 1]) : kf k ≤ 1}. Show that the family {T (f ) : f ∈ F} se
∞ ∞

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(C) Let F =
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∞
issuniformly
(D)laDeduce that any sequence (f ) in F has a subsequence (f ) such that T (f g
a)
a g converges uniformly on [0, 1].
n nk nk

(18∗ ) Let K be a field and consider the ring of formal power series
(∞ )
n
X
K[[X]] = an X : an ∈ K .
n=0
n
(A) Show that an element f (X) = ∞ n=0 an X ∈ K[[X]] is invertible if and only
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if a0 6= 0.
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(B) Show that every nonzero element f ∈ K[[X]] can be written uniquely in the
form
e
f (X) = X m u(X), m
where m ≥ 0 and u(X) is a unit.
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(C) Deduce that every ideal of K[[X]] is of the form (X n ) for some n ≥ 0. In
particular, K[[X]] is a principal ideal domain.
(D) Determine all prime ideals of K[[X]].

(19∗ ) (A) (7 marks) Let G be a finite, non-abelian group satisfying the following prop-
erty: if H1 , H2 are distinct proper maximal subgroups of G, then H1 ∩ H2 is
trivial. Show that G cannot be simple, i.e., show that G has a non-trivial,
proper normal subgroup.
Recall that a proper subgroup H of G is called maximal if no other proper
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(B) (3 marks) Let G be a finite, non-abelian group such that every proper sub-

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group of G is abelian. Show that G cannot be simple.
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(20 ) Let f : R −→ R be a continuously differentiable function.
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exists a constant c > 0 such that for all x, y ∈ R , the following inequality holds:
kf (x) − f (y)k ≥ ckx − yk.
Prove that f is a global diffeomorphism from Rn onto Rn .

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