TIFR GS 2026 Question Paper Mathematics

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TIFR GS 2026
Question Paper ·
Mathematics
EXAM YEAR TYPE SUBJECT

TIFR GS 2026 Question Paper Mathematics

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Notation and Conventions
‚ N denotes the set of natural numbers t0, 1, . . . u, Z the set of in-
tegers, Q the set of rational numbers, R the set of real numbers,
and C the set of complex numbers. These sets are assumed to
carry the usual algebraic and metric structures.
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‚ For x ą 0, logpxq denotes the natural logarithm of x.
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‚ For a metric space pX, dq, with x P X and r ą 0, Bpx, rq denotes
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s
the open ball ty P X | dpx, yq ă ru.
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‚ Rn denotes the Euclidean space of dimension n. Subsets of

g l a Rn are viewed as metric spaces using the standard Euclidean
a distance on Rn . For x P Rn , }x} denotes the standard Euclidean
norm of x, i.e., the distance from x to 0.
‚ All rings are associative and assumed to contain a multiplica-
tive identity. A ring homomorphisms is assumed to map the
multiplicative identity to the multiplicative identity.
‚ For a ring R, Mn pRq denotes the ring of n ˆ n matrices with
entries in R. The identity matrix in Mn pRq will be denoted by
Id or by Idn . For a matrix A P Mn pRq, AJ will stand for its
m
transpose, tracepAq for its trace, and detpAq for its determinant.

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‚ For a ring R, GLn pRq Ă Mn pRq denotes the multiplicative group
of invertible n ˆ n matrices with entries in R.
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‚ Mn pRq will also be viewed as a real vector space, and Mn pCq
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as a complex vector space, equipped with the usual Euclidean
topology. Subsets of Mn pRq are given the subspace topology.
‚ For a ring R, Rrx1 , . . . , xn s denotes the polynomial ring in n
variables x1 , . . . , xn over R.
‚ For a compact metric space X, CpX, Rq denotes the real vector
space of continuous real valued functions on X. We define a
metric d on CpX, Rq by dpf, gq “ sup |f pxq ´ gpxq|.
xPX
‚ If A is a set, #A stands for the cardinality of A. We denote
#A “ 8 if A is infinite.
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‚ If B is a subset of a set A, we write AzB for the set ta P A |
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a R Bu.
‚ Let G be a group, and let S Ď G. The subgroup of G generated
e m
e m by S is defined to be the smallest subgroup of G that contains
las
las S.
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ag

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GS 2026, Mathematics: Stage I Questions

Part A - MCQ.
(1) For α ą 0, and for all positive integers n, let
an “ logp1 ` n´α q.
Then,
(a) ř8
ř
n“1 an is convergent for all α ą 0.
X (b) ř8
n“1 an is convergent for all α ą 1.
(c) ř8
n“1 an is convergent for all 0 ă α ď 1.
8
(d) n“1 an is divergent for all α ą 1.

(2) What is the value of
1
n ˆ ˙n
ź k
lim ?
nÑ8
k“1
k ` n

(a) 1{e
(b) 1{2
(c) 1{ logp4q
X (d) 1{4

(3) For x P r0, 1s, let
8
ÿ xn
f pxq “ .
n“1
n2
Then
(a) f is continuous but nowhere differentiable.
(b) f is differentiable on p0, 1q and its derivative extends con-
tinuously to r0, 1s.
X (c) f is differentiable on p0, 1q but the derivative diverges as
x Ñ 1´ .
(d) f can be extended to a real analytic function on p0, 1 ` q
for some  ą 0.

(4) For each positive integer n and each x P r0, 1s, define
fn pxq “ sinpnq exp ´ n4 px ´ n1 q2 .
` ˘

Which of the following is true about the sequence of functions
pfn q on r0, 1s?
(a) pfn q converges to 0 uniformly on r0, 1s.
X (b) pfn q converges to 0 pointwise, but not uniformly, on r0, 1s.
(c) pfn q does not converge pointwise on r0, 1s.
(d) pfn q converges uniformly to a nonzero function on r0, 1s.

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(5) Let f : r0, 8q Ñ R be a continuous function. Consider the
statements:
(i) If lim f pxq “ 8, then f is not uniformly continuous.
xÑ8
(ii) If f is bounded, then f is uniformly continuous.
Which of the following is correct?
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(a) Both (i) and (ii) are true.
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(b) (i) is true, (ii) is false.
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(c) (i) is false, (ii) is true.
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X (d) Both (i) and (ii) are false.

g la(6) Consider the set
a A “ tA P M pRq | A “ 0u. 10
10

For each A P M10 pRq, define the linear transformation TA :
M10 pRq Ñ M10 pRq by TA pXq “ AX. Then which of the follow-
ing is true?
X (a) For all A P A, we have detpTA q “ 0 and tracepTA q “ 0.
(b) There exists A P A for which detpTA q ‰ 0 whereas tracepTA q “
0.
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(c) There exists A P A for which detpTA q “ 0 whereas tracepTA q ‰
0.
em
(d) There exists A P A for which detpTA q ‰ 0 and tracepTA q ‰
s
0.
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a metric space and U be a proper dense
(7) Let pX, dq be a complete
subset of X. Let f : U Ñ R be a function. Consider the
following two statements:
(i) If f is continuous, then it can be extended to a continuous
function g : X Ñ R.
(ii) If f is uniformly continuous, then it can be uniquely ex-
tended to a continuous function g : X Ñ R.
Then which of the following is correct?
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(a) Both (i) and (ii) are true.
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(b) Both (i) and (ii) are false.

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(c) (i) is true but (ii) is false.
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s e X (d) (ii) is true but (i) is false.
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(8) Let pX, dq be a metric space with the additional
a dpx, yq ď max tdpx, zq, dpy, zqu for all x, y, z P X.
Consider the following statements:
(i) For all x, y, z P X, at least two of the numbers dpx, yq, dpy, zq, dpx, zq
are equal.
(ii) For each positive real number r, and x, y P X, the sets
Bpx, rq and Bpy, rq are either equal or disjoint.
Which of the following is true?

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X (a) (i) and (ii) are true.
(b) (i) and (ii) are false.
(c) (i) is true and (ii) is false.
(d) (ii) is true and (i) is false.

(9) Let T : Rn Ñ Rn be a linear map, such that T ‰ 0, and
Rn “ prangepT qq ‘ pnull spacepT qq. Consider the following
statements:
(i) T is diagonalisable.
(ii) There does not exist any integer k ą 0 such that T k “ 0.
Which of the following must be true?
X (a) (i) is false and (ii) is true.
(b) (i) is true and (ii) is false.
(c) (i) and (ii) are true.
(d) (i) and (ii) are false.

(10) For any differentiable function g : R Ñ R, let
Ag “ tx P r0, 1s | gpxq “ 0u,
Bg “ tx P r0, 1s | g 1 pxq “ 0u.
Let S “ tg : R Ñ R | g is differentiable and Ag X Bg “ Hu.
Which of the following statements is true?
(a) For all g P S, the set Bg is finite.
X (b) For all g P S, the set Ag is finite.
(c) There exists g P S such that both Ag and Bg are infinite.
(d) None of the other statements is true.

(11) Let g : r1, 8q Ñ R be a function such that
1
gp1q “ 1 and g 1 pxq “ , for all x ě 1.
x2 ` g 2 pxq
Then which of the following statements is true?
(a) lim gpxq exists, and belongs to p2, 8q.
xÑ8
X (b) lim gpxq exists, and belongs to p´8, 2s.
xÑ8
(c) None of the other statements is true.
(d) lim gpxq “ 8.
xÑ8

(12) Let A, B be linear endomorphisms of a finite dimensional vector
space over a field. Consider the following statements:
(i) Every eigenvalue of AB is an eigenvalue of BA.
(ii) Every eigenvector of AB is an eigenvector of BA.
Which of the following is correct?
(a) Both (i) and (ii) are true.
X (b) (i) is true, (ii) is false.
(c) (i) is false, (ii) is true.

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(d) Both (i) and (ii) are false.

(13) For a real number a, let tau denote the greatest integer that is
less than or equal to a, and let tau “ a ´ tau. Let n ě 4 be an
integer. Then the expression
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equals
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ag
ag (c) ntpn´´1.2qeu.
(b)

(d) None of the other options.

(14) Consider the space
" ˇ ˆ ˙*
ˇ J 2 ´2
X “ A P M2 pRq ˇˇ AA “ .
´2 3
Then the number of connected components of X equals
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(a) 0.
(b) 1.
X (c) 2.
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(d) 4.
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(15) Let V be the realavector space of all polynomials in Rrxs of
degree less than or equal to 10. Consider the linear operator
T : V Ñ V defined by
pT pqpxq “ pp2x ` 1q, for all p P V.
What is the trace of T ?
(a) 11.
X (b) 2047.
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(c) 121.
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(d) None of the other options.
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Crx, ys
.
pf, Bf ,
Bx By
Bf
q
Then the dimension of V over C is:
(a) pq.
X (b) pq ´ p ´ q ` 1.
(c) p ` q.
(d) Infinite.

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(17) Let Γ Ă R2 be the graph of y “ sinpxq and let ` be a line in R2
such that #pΓ X `q “ n P N. Which of the following statements
is necessarily true?
(a) n is even.
(b) n is odd.
(c) n ď 10.
X (d) None of the remaining three options.

(18) Suppose that f : p´1, 1q Ñ R is infinitely differentiable, and
|f pxq ´ 1 ´ x ´ x2 | ď |x|3
for all x P p´ 13 , 31 q. Then,
(a) f p0q “ f 1 p0q “ f 2 p0q “ 1.
(b) f p0q “ f 1 p0q “ 1 and f 3 p0q “ 6.
X (c) f p0q “ f 1 p0q “ 1 and f 2 p0q “ 2.
(d) None of the other statements is true.

(19) Consider the following two assertions about a group G.
(I) G is abelian.
(II) The map ϕ : G Ñ G defined by ϕpgq “ g 2 is a group
homomorphism.
Which of the following is true?
X (a) (I) implies (II), and (II) implies (I).
(b) (I) implies (II), but (II) does not imply (I).
(c) (I) does not imply (II), but (II) implies (I).
(d) (I) does not imply (II), and (II) does not imply (I).

(20) Suppose that fn : r0, 1s Ñ r0, 1s is continuous and increasing
for each n ě 1, and the sequence pfn pxqqně1 of real numbers
is decreasing for all x P r0, 1s. Let f : r0, 1s Ñ r0, 1s be the
function defined as f pxq “ lim fn pxq. Then
nÑ8
(a) f is continuous.
X (b) f is right-continuous but not necessarily left-continuous.
(c) f is left-continuous but not necessarily right-continuous.
(d) None of the other assertions is true.

Part B - T/F.
T (1) If f : R Ñ R is a continuously differentiable function and S Ă R
is a bounded subset, then there exists a constant CS ą 0 such
that |f pxq ´ f pyq| ď CS |x ´ y| for all x, y P S.

T (2) There exists a continuous surjection from Q to t0, 1, . . . , 2026u.

F (3) The subset
X “ tf P Cpr0, 1s, Rq : |f pxq ´ f pyq| ď |x ´ y|1{2 for all x, y P r0, 1su

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of Cpr0, 1s, Rq is compact.

T (4) There is no continuous bijection f : R Ñ r0, 1s.

F (5) Let A, B be linear endomorphisms of a finite dimensional vector
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space over a field. If A and B are diagonalizable, then AB is

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also diagonalizable.
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. P GL pCq and A is diagonalizable, then A is diagonaliz- s e
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2
T (6) If A
eable. l
5

l as ag
agF (7) A function f : R Ñ R is said to be periodic if there exists a
positive real number c such that f px ` cq “ f pxq for all x P R.
Then the set of all periodic functions is a linear subspace of the
real vector space of all functions from R to R.

T (8) If ppxq P Crxs is a polynomial which is not equal to x, then the
polynomial ppxq ´ x divides the polynomial ppppxqq ´ x.

T (9) Let A be a square matrix with non-negative real entries such
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that the sum of entries in each row of A is 1. Then each eigen-
value of A has absolute value less than or equal to 1.
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F (10) Let X be a metric space having exactly 4 points. Then, for
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some positive integer n, there exists a distance preserving map
f : X Ñ Rn , where Rn is equipped with its Euclidean metric.

T (11) Let f : R Ñ R be a continuous function. For each positive
integer n, let fn : R Ñ R be defined by fn pxq “ f pxn q. If the
sequence of functions pfn q converges uniformly on R, then f
must be a constant function.

T (12) Let X, Y be finite metric spaces. Then X and Y are homeomor-
phic if and only if the real vector spaces CpX, Rq and CpY, Rq
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are isomorphic.

m .co F (13) Suppose a finite group G is the union of its Sylow subgroups.
s e m
s e Then the order of G is a power of some prime number.
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g la F (14) For a 2 ˆ 2 real matrix A, if the limit a
a
N
ÿ
n´1 A2n´1
sinpAq “ lim p´1q
N Ñ8
n“1
p2n ´ 1q!

exists in M2 pRq, then all eigenvalues of the limit matrix sinpAq
have absolute value less than or equal to 1.

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F (15) Let A be a finite abelian group of order 2025. Then for every
subgroup H ď A we have A – H ˆ A{H as groups.

F (16) Consider the map L : Rrts Ñ Rrts given by Lpf q “ 3f `
2f 1 , where f 1 denotes the derivative (with respect to t) of the
polynomial f . Then L is injective, but not surjective.

T (17) Let σ1 , . . . , σ80 be elements of the permutation group S236 such
that σi3 is the identity permutation for each i. Then there exist
1 ď i ă j ď 80 such that σi is conjugate to σj .

T (18) For a permutation σ P S10 , let Fixpσq Ď t1, 2, . . . , 10u denote
the set of fixed points of σ. Then the average
1 ÿ
#Fixpσq
10! σPS
10

is equal to 1.

F (19) The set
X “ tpb, cq P R2 | the polynomial x2 ` bx ` c has distinct roots in Cu
is connected.

T (20) The integer 16n ` 26n ` 36n ` . . . ` 20256n is divisible by 7 for
all integers n ě 10.

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