TIFR GS 2026 Question Paper with Answer Key Mathematics PhD at CAM

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TIFR GS 2026
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Mathematics
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TIFR GS 2026 Question Paper Mathematics PhD at CAM

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GS 2026 – Mathematics PhD laat CAM
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Notations and Conventions
• All sets considered here are assumed to be nonempty.

• N denotes the set of natural numbers {1, 2, . . .}, Z the set of integers, 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 sets A and B, A \ B = {x ∈ A : x ∈
/ B}.
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• For z ∈ C, Rem
.
z denotes the real part of z and Im z denotes the imaginary part of z. se
s e g l a
g lathe unit disc {z ∈ C : |z| < 1}.
• D denotes a
n
• R ,na ⩾ 1, is equipped with the Euclidean topology.
• For a subset A of a topological space, A denotes the closure of A.

• C([0, 1]) denotes the set of continuous real-valued functions on the interval [0, 1].

• C 1 ([0, 1]) denotes the set of real-valued functions on [0, 1] that are differentiable on [0, 1]
with f ′ ∈ C([0, 1]). Differentiability at 0 and 1 means that one-sided derivatives exist at
0 and 1, respectively.
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• For a measurable set S ⊆ R and 1 ⩽ p ⩽ ∞, Lp (S) denotes the standard Lebesgue space
with the standard Lp norm ∥ · ∥p .
s em
a
Hlequipped with the inner product ⟨·, ·⟩,
• For a subset S of a Hilbert space g
a
S ⊥ = {h ∈ H : ⟨h, s⟩ = 0 for all s ∈ S}.

Part A
1. Consider the following subsets of R2 .

• X = {(t, sin(1/t) : t ∈ (0, 1)}
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m • Z = {(t, exp (1/t)) : t ∈ (0, 1)}
• Y = {(t, cos(1/t) : t ∈ (0, 1)}
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s e Choose the correct option.
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a (a) Only one of the sets X, Y , Z is connected.
(b) Exactly two of the sets X, Y , Z are connected.
(c) All of the three sets X, Y , Z are connected.
(d) None of the sets X, Y , Z are connected.

2. Theorem: Let (X, d) be a metric space. Suppose E and F are disjoint subsets of X, and
that E is unbounded. If , then there exists a δ > 0 such that d(e, f ) ≥ δ
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for all e ∈ E and f ∈ F .
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Choose the statement below that when used in the blank above makes the theorem true.
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(a) E is open and F is open.
(b) E is open and F is compact.
(c) E is closed and F is closed.
(d) E is closed and F is compact.

3. Let X = (0, 1) and fn , f : X → R be continuous functions. Consider the following
statements.

(i) fn converges uniformly to f on X.
(ii) For any convergent sequence (xn ) in X, that is lim xn ∈ X, we have that lim fn (xn ) =
f (lim xn ).

Choose the correct option.

(a) (i) =⇒ (ii) and (ii) =⇒ (i).
(b) (i) =⇒ (ii) but (ii) =⇒
̸ (i).
(c) (i) =⇒
̸ (ii) but (ii) =⇒ (i).
(d) (i) =⇒
̸ (ii) and (ii) =⇒
̸ (i).

4. For the two dimensional Lebesgue measure on the set

X = {(t2 , t3 ) ∈ R2 : t ∈ R},

consider the following statements.

(i) If Y ⊆ X, then Y is measurable.
(ii) R2 \ X is measurable.
(iii) X has infinite measure.

Choose the correct option.

(a) (i), (ii), and (iii) are true.
(b) (i) and (ii) are true, but (iii) is false.
(c) (ii) and (iii) are true, but (i) is false.
(d) (i), (ii), and (iii) are false.

5. Let f : R → R be continuous such that f (0) = 1 and f (x) = 0 for |x| > 1. Define
fn (x) = f (x + n) for n ∈ N and x ∈ R. Consider the following statements.

(i) (fn ) converges in L∞ (R).
(ii) (fn ) converges in Lp (R) for some 1 ≤ p < ∞.
(ii) (fn ) converges in Lp (R) for all 1 ≤ p < ∞.

Choose the correct option.

(a) (i), (ii), and (iii) are true.
(b) (i) and (ii) are true, but (iii) is false.
(c) (i) and (iii) are true, but (ii) is false.

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(d) (i), (ii), and (iii) are false.
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6. Let M = {f ∈ L2 (−1, 1) : −1
f (x) dx = 0}, and g(x) = x2 . What is

inf {∥g − f ∥2 : f ∈ M }?
√
2
(a) 3
.
1
(b) .
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2
(c) .
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(d) 1.
e
as
em1]), ∥ · ∥
→ C([0, 1]), ∥ · ∥
be defined by T (f ) = f . Considerglthe
7. Let T : C s([0,
lastatements. a
1 ′
∞ ∞

a g
following

(i) T is a closed operator.


(ii) C 1 ([0, 1]), ∥ · ∥∞ is a Banach space.

Choose the correct option.

(a) (i) is true but (ii) is false.
(b) (i) and (ii) are both true.
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(c) (i) and (ii) are both false.
(d) (i) is false but (ii) is true.
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8. Consider the holomorphic functions
 
sin(z) 1
f (z) = and g(w) = f
z3 w

for z, w ∈ C \ {0}. Choose the correct statement.

(a) f (z) has a pole of order 3 at z = 0 and g(w) has a pole of order 3 at w = 0.
(b) f has a pole of order 3 at z = 0 and g(w) has an essential singularity at w = 0.
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m (c) f has a pole of order 2 at z = 0 and g(w) has a pole of order 3 at w = 0.
c. o (d) f has a pole of order 2 at z = 0 and g(w) has an essential singularity
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s em 9. Consider the following domains in C. g l a
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ag (i) {z ∈ C : Re z > 0 and Im z > 0}.
(ii) A bounded domain.
(iii) C \ {0}.
(iv) C \ U where U is an open subset of C with U ̸= C.

From which of the above domains does there exist an injective holomorphic function into
the unit disc D?
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(a) (i), (ii), and (iii).
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(b) (i), (ii), and (iv).
(c) (i), (iii), and (iv).
(d) (i) and (ii).

10. Let X be a Hausdorff topological space such that for all closed sets A ⊆ X and continuous
functions f : A → [0, 1] there exists a continuous function f˜: X → [0, 1] such that f˜|A = f .
Which of the following statements is necessarily true?

(a) If C1 and C2 are disjoint closed sets, then there are disjoint open sets O1 and O2
such that C1 ⊆ O1 and C2 ⊆ O2 .
(b) X is compact.
(c) X is connected.
(d) All singleton subsets of X are open.

Part B
1. Let X and Y be metric spaces, and f : X → Y be a bijective continuous function. Choose
the correct statement.

(a) If X is compact, then f is a homeomorphism.
(b) If Y is compact, then X is compact.
(c) If Y is connected, then X is connected.
(d) If X = R and Y ⊆ R2 , then Y is a closed set.

2. Let E be a closed convex subset of a Hilbert space H. For each x ∈ H, there exists a
unique y0 ∈ E minimizing {∥x − y∥ : y ∈ E}. Which of the following does this y0 satisfy?

(a) ⟨x − y0 , z − y0 ⟩ = 0, ∀z ∈ E.
(b) ⟨x − y0 , z − y0 ⟩ ≤ 0, ∀z ∈ E.
(c) ⟨x − y0 , z − y0 ⟩ ≥ 0, ∀z ∈ E.
(d) y0 ∈ x + E ⊥ .

3. Consider the following functional equation for f in C([0, 1]):
Z 1 p
2f (x) − cos(f (y)) 1 + (x − y)2 dy + sin(ex ) = 0, for x ∈ [0, 1].
0

Choose the correct statement.

(a) There exists no solution in C([0, 1]).
(b) There exists a unique solution f ∈ C([0, 1]).
(c) There exist exactly two solutions in C([0, 1]).
(d) There exist infinitely many solutions in C([0, 1]).

4. Consider the following functions on R \ {0}.

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(i) f (t) = t + 2t2 sin(1/t).
(ii) g(t) = t + 2t3 sin(1/t).
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(iii) h(t) = t3 cos(t).

Which among the above functions has an inverse in (−ϵ, ϵ) \ {0} for some ϵ > 0?

(a) Only (iii).
(b) Only (ii) and (iii).
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(c) Only (ii).
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(d) All the functions (i), (ii), and (iii).

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5. Let f andsfe be real-valued Lebesgue measurable functions on [0, 1]. Suppose that (f
g l a
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g l
converges ato f pointwise. Consider the following statements. a
) n

a
(i) For all ϵ > 0 there exists a compact set K ⊆ [0, 1] such that (f ) converges uniformlyn
to f on K and the Lebesgue measure of K is greater than 1 − ϵ.
(ii) There exists a dense set U ⊆ [0, 1] such that (fn ) converges uniformly to f on U .

Which is of the above statements are necessarily true?

(a) (i) and (ii) are both true.

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(b) (i) is true but (ii) is false.

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(c) (ii) is true but (i) is false.

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(d) (i) and (ii) are both false.
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6. Choose the correct option. ag
(a) There exists a function f ∈ C(R) such that
Z 1
lim (f (x + ϵ) − f (x))dx ̸= 0.
ϵ→0 0

(b) There exists a function f ∈ L∞ (R) such that
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Z 1

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lim (f (x + ϵ) − f (x))dx ̸= 0.
ϵ→0
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0

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ag
ag lim
Z 1
(f (x + ϵ) − f (x))dx ̸= 0.
ϵ→0 0

(d) For all functions f ∈ L1 (R),
Z 1
lim (f (x + ϵ) − f (x))dx = 0.
ϵ→0 0

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7. Let fn (x) = sin(nπx). Choose the correct statement.
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(a) (fn ) converges in L2 (0, 2).
(b) (fn ) does not have any convergent subsequence in L2 (0, 2).


(c) There exists E ⊆ (0, 2) such that E has Lebesgue measure 2 and fn |E converges
pointwise.


(d) There exists an interval F ⊂ (0, 2) such that fn |F converges in L2 (F ).

P2025
8. Let X denote R2025 equipped with the norm ∥(x1 , . . . , x2025 )∥ = i=1 |xi | and T : X → X
be defined by 
u, if ∥u∥ ≤ 1, and
T (u) =
u/∥u∥, if ∥u∥ > 1.
Consider the following statements.
(i) ∥T (u) − T (v)∥ ≤ ∥u − v∥ for all u, v ∈ X.
(ii) ∥T (u) − T (v)∥ ≤ 2∥u − v∥ for all u, v ∈ X.
(iii) ∥T (u) − T (v)∥ ≤ 3∥u − v∥ for all u, v ∈ X.
Choose the correct option.
(a) (i), (ii) and (iii) are true.
(b) Only (ii) and (iii) are true.
(c) Only (iii) is true.
(d) (i), (ii) and (iii) are false.

9. Let 

F = f : D → D : f is holomorphic and f 12 = 0
and
α = sup {Im f (0) : f ∈ F} .
Choose the correct option.
(a) α = 1.
(b) α = 31 .
(c) α = 21 .
(d) α = 32 .

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10. Let f : D → C be holomorphic such that |f (z)| ≥ |z| for all z ∈ D. Consider the
following statements.
(i) There exists z0 ∈ D such that |f (z0 )| < 1.
(ii) There exists z0 ∈ D such that |f (z0 )| ≥ 1.
(iii) The function f is bounded.
Choose the correct option.
(a) Only (i) is true.
(b) Only (ii) is true.
(c) Only (i) and (ii) are true.
(d) Only (ii) and (iii) are true.

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