TIFR GS 2025 Question Paper Mathematics PhD at CAM

FOR TIFR GS EXAM PREPARATION

TIFR GS 2025
Question Paper ·
Mathematics
EXAM YEAR TYPE SUBJECT DETAILS

TIFR GS 2025 Question Paper Mathematics PhD at CAM

Notes · Sample Papers · Previous Year Papers · Mock Tests

m
m .co

m .co s e m
se g l a
a

m
m
Notations and Conventions
co m .co
m . s e
• N denotes the set of natural numbers {0, 1, . . .}, Z the set of integers, Q the set of rational
s e l a
a ag
numbers, R the set of real numbers, and C the set of complex numbers. These sets are

g l
assumed to carry the usual algebraic and metric structures.
a
• For sets A and B, A \ B = {x ∈ A : x ∈
/ B}.
√
• i= −1.

• Rn , n ⩾ 1, is equipped with the Euclidean topology.

• For a subset A of a topological space, A denotes the closure of A.

• For a real normed linear space X,
m
.co
· the dual of X is X ∗ = {φ : X → R : φ is linear and continuous}, and

s em
· ⟨T x, y⟩ := (T x)(y) for T : X → X ∗ and x, y ∈ X.

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

• For k ∈ N, C k ([0, 1]) denotes the set of real-valued functions having k continuous deriva-
tives on (0, 1), and f (k) ∈ C([0, 1]).

• For a set S ⊆ R of positive Lebesgue measure and 1 ⩽ p < ∞,
Z
p
L (S) = {f : S → R : f is measurable and |f (x)|p dx < ∞}.
S

m
m
• For an n × n matrix A, det(A) denotes the determinant of A.
.co
m .co s e m
s e g la
g la a
a

.co s e m
s em l a
g la 1
ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 1 of 7

1. For 0 < a < 1, let λ = a1 + ai and
Z
2
zn = einx e−λx dx.
R

What is lim zn ?
n→∞

(a) a
(b) ∞
(c) 0
(d) 1

2. Let D = {(x, y) ∈ R2 : x2 + y 2 < 1}. For p ∈ R, define

sin2 (x2 + y 2 )
Z
Cp = dxdy.
(x2 + y 2 )p
D

(a) Cp < ∞ for all p > 0.
(b) Cp < ∞ for all p > 1.
(c) Cp < ∞ for all p < 3.
(d) Cp < ∞ for all p < 4.

3. Let p(x) = x3 − x2 − x + 1. Consider the following statements.

(I) If T : R3 → R3 is a linear transformation and p(T ) = 0, then T is invertible.
(II) If T : R3 → R3 is a linear transformation and p(T ) = 0, then T is diagonalizable.

(a) (I) and (II) are both true.
(b) (I) and (II) are both false.
(c) (I) is true and (II) is false.
(d) (I) is false and (II) is true.

4. What is the cardinality of {A ⊆ R : R \ A = R \ A}?

(a) 0
(b) 1
(c) 2
(d) infinite

2
For more Question Papers, Sample Papers, Notes & Syllabus visit Page 2 of 7

m
m .co

m .co s e m
se 5. Let y ∈ C ([0, 1]) be a non identically zero solution to gl
a
a
2

2
dy 2
(t) + et sin t y(t) = 0,


2
t ∈ (0, 1).
dt
(a) y can have infinitely many zeroes.
(b) Either y has no zeroes or finitely many zeroes.
(c) The third derivate of y does not exist at 1/2.
m
m .co
.co
(d) The equation has a unique non identically zero solution.
e m
l as
em function which satisfies |f (z)| ⩾ |z| for all z ∈ C, and assume moreover
6. Let f be ansentire
a ag
2

that fg(i)l = −i. Then,
a
(a) f (0) ̸= 0
(b) f (1) = −3
(c) f (2) cannot be uniquely determined from the given information.
(d) f (3) = 9i

m
.co
7. Let H = {x + iy : x ∈ R, y > 0} and f : H → H be a bijective holomorphic function such
that lim|z|→∞ |f (z)| = ∞. Choose the correct option.

s em
(a) f is a linear function.
(b) f is a rational function that g
a
islnot a polynomial.
a
(c) f ({it : t > 0}) is a subset of a circle of finite radius.
(d) f ({it : t > 0}) is a subset of a horizontal line.

8. Let f : R → R be defined by f (y) = y sin(1/y) for y ̸= 0, and f (0) = 0. Then, the initial
value problem
dy


m
 (t) = f (y(t)), t ∈ R,


.co
dt
m
c. o has m

y(0) = 0,
e


e m las
las ag
g
(a) at least one strictly monotonic solution.
a (b) a unique solution; which is y ≡ 0.
(c) infinitely many solutions.
(d) at least one oscillatory solution (i.e., the solution is not identically zero and has
infinitely many zeroes).

9. Choose the correct option.
.co
For any real Banach space X and any linear operator T : X → X ∗ , if ⟨T x, y⟩ = ⟨T y, x⟩
e m
for all x, y ∈ X, then
em l as
las 3
ag
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 3 of 7

(a) T is invertible.
(b) T is surjective.
(c) T is bounded.
(d) T = 0.

10. Choose the correct option.

(a) If f : R → R is a bijective differentiable map then its inverse is also a bijective
differentiable map.
(b) If f : R → R is a bijective differentiable map then for all x ∈ R there exists an
ϵ > 0 and a differentiable map g : (x − ϵ, x + ϵ) → R such that f ◦ g(t) = t for all
t ∈ (x − ϵ, x + ϵ).
(c) If f : R → R is a bijective differentiable map then f is either monotonically increasing
or monotonically decreasing.
(d) If f : R → R is a bijective differentiable map then f ′ (x) > 0 for all x ∈ R or
f ′ (x) < 0 for all x ∈ R.

11. Let f : R → R be a continuous function satisfying the following condition: for each x0
and for each r > 0,
xZ0 +r
1
f (x0 ) = f (y)dy.
r
x0 −r

Choose the correct statement.

(a) There exists a non constant function f ∈ L2 (R) satisfying the above property.
(b) There exists at most one function in L2 (R) that satisfies the above property.
(c) The space of functions f ∈ L2 (R) and satisfying the above property is one dimen-
sional.
(d) The space of functions f ∈ L2 (R) and satisfying the above property is infinite
dimensional.

12. Consider the 11 × 11 matrix:
 
−2 1 − 212 1
32
··· − 1012
 
 
 12 2 − 1212 1
··· − 2012 
 11 132 
 
 
 1 − 2212 −2 1
··· − 3012 
A =  212 232 .
 

 .. .. .. .. .. 

 . ...
 . . . .  
 
1 1
1012
··· ··· ··· − 110 2 −2

Consider the following statements.

4
For more Question Papers, Sample Papers, Notes & Syllabus visit Page 4 of 7

m
m .co

m .co s e m
se g l a
(I) The eigenvalues of A are all non-negative.
(II) A is invertible.
a
(a) (I) and (II) are both true.
(b) (I) and (II) are both false.
(c) (I) is true and (II) is false.

m
.co
(d) (I) is false and (II) is true.

o m m
13. For sets A, B m
. c s e
s e
n
⊆ R , define
l a
g l a A + B = {a + b : a ∈ A and b ∈ B}. ag
a
(a) If A is compact and B is closed, then A + B is compact.
(b) If A is closed and B is closed, then A + B is closed.
(c) If A is open and B is compact, then A + B is open.
(d) If A is open and B is compact, then A + B is compact.

m
.co
em
14. Choose the correct option.

as ∞

If g : R → R is a positive continuouslfunction such that e g(x)dx < ∞ for some α > 0,
Z

g
αx

then
a 0

Z∞
g(x)
(a) dx < ∞ for all n ∈ N.
xn
0
(b) lim g(x) exists.
x→∞
Z∞
1
m
.co
(c) dx < ∞ for some β > 0.
m g β (x)
c. o (d) there exist C > 0 and β > 0 such that m
0

m s e
s e g la
la
∞

g(x)dx ⩽ Ce , a
Z
g
−βt

a t

for all t > 0.

15. Let fn : R → R be sequence of differentiable functions such that
Z 1
fn (x) dx = 0 and |fn′ (x)| ⩽ x−1/2 for all x ∈ (0, 1].
.co
0

e m
Consider the following statements.
em l as
las 5
ag
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 5 of 7

(I) {fn } is uniformly bounded on [0, 1].
(II) {fn } has a subsequence that converges uniformly on [0, 1].

(a) (I) and (II) are both true.
(b) (I) and (II) are both false.
(c) (I) is true and (II) is false.
(d) (I) is false and (II) is true.

16. Let n ⩾ 2, and A be an n × n real symmetric and positive definite matrix satisfying
trace(A) = det(A).
Choose the correct statement that holds for every such A.
n
(a) trace(A) < n n−1 .
(b) trace(A) ⩽ n.
n
(c) det(A) ⩾ n n−1 .
(d) all of the eigenvalues of A are ⩽ 12 .

17. A box contains 2024 balls in which 200 are blue and the rest are red. Balls are chosen
from the box one by one at random, and discarded.
Consider the following statements.

(I) The probability of picking a red ball at the k-th pick is the same for all k =
1, . . . , 2024.
(II) The probability of picking a blue ball at the 200-th pick is the same as the probability
of picking a red ball at 1824-th pick.
(III) The probability of picking all the blue balls in the first 200 draws is the same as the
probability of picking all the red balls in the first 1824 draws.

(a) (I) and (II) are true, (III) is false.
(b) (I) and (III) are true, (II) is false.
(c) (II) and (III) are true, (I) is false.
(d) (I), (II), and (III) are all true.


R1
18. Let X denote the inner product space C([0, 1]), ⟨·, ·⟩ where ⟨f, g⟩ = f (x)g(x) dx.
0
R1
Define T : X → X by (T f )(x) = f (t) dt for f ∈ X and x ∈ [0, 1]. Choose the correct
x
statement.

(a) T is surjective.

6
For more Question Papers, Sample Papers, Notes & Syllabus visit Page 6 of 7

m
m .co

m .co s em
se g a that U ◦ T is the identity
lsuch
a
(b) There exists a bounded linear operator U : X → X
map.
(c) T (X) is dense in X.
(d) T (X) is a closed subset of X.

√
19. Let T : L2 (0, 1) 7→ L2 (0, 1) defined by (T f )(x) = f ( x). We denote the operator norm
of T by ∥T ∥.
m
m .co

.co m
√
(a) ∥T ∥ = 2.
s e
s em l a
ag
(b) ∥T ∥ = 2.
(c) ∥T l∥a= .
a g √1

1
2

(d) ∥T ∥ = .2

20. Let f : R → R defined by
( 1
x− 2 , if 0 < x < 1, and
f (x) =
0, otherwise.
m
.co
Let rn be an enumeration of rational numbers and correspondingly define
m
∞
s e
la2 f (x − r ), for x ∈ R.
X
−n
h(x) =
g
n

a n=1

Choose the correct statement.

(a) h ∈ L1 (R) and h ∈ L2 (R).
(b) h ∈ L1 (R) and h ∈
/ L2 (R).
/ L1 (R) and h ∈ L2 (R).
(c) h ∈
/ L1 (R) and h ∈
(d) h ∈ / L2 (R).
m
m .co
m .co s e m
s e g la
g la a
a

.co s e m
s em l a
g la 7
ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 7 of 7