TIFR GS 2026 Question Paper Computer Science

FOR TIFR GS EXAM PREPARATION

TIFR GS 2026
Question Paper ·
Computer Science
EXAM YEAR TYPE SUBJECT

TIFR GS 2026 Question Paper Computer Science

Notes · Sample Papers · Previous Year Papers · Mock Tests

m
m .co

m .co s e m
se g l a
a
1. Let S be a set, and (A1 , B1 ) and (A2 , B2 ) be two partitions of S. Hence for i ∈ {1, 2},
Ai and Bi are disjoint, and Ai ∪ Bi = S. Which of the following statements about
the sets A1 , B1 , A2 , and B2 MUST be true?
(a) (A1 ∪ A2 ) ∩ (B1 ∩ B2 ) = ∅. ✓
(b) (A1 ∪ A2 ) ∩ (B1 ∪ B2 ) = ∅.
m
m .co
(c) (A1 ∩ A2 ) ∪ (B1 ∩ B2 ) = S.

.co
(d) (A1 ∩ A2 ) ∪ (B1 ∩ B2 ) = ∅.
s e m
em
(e) None of the above.
s l a
a → S be any function. For X, Y ⊆ S, define
2. Let f :lS
g ag
a f (X) = { f (x) : x ∈ X }, f (Y ) = { y ∈ S : f (y) ∈ Y }.
−1

Which of the following statements holds for every function f and all subsets A, B?
(a) f (A ∩ B) = f (A) ∩ f (B) for all A, B ⊆ S.


(b) f −1 f (A) = A for all A ⊆ S.
(c) f (A \ B) = f (A) \ f (B) for all A, B ⊆ S.
m
.co
(d) If f is surjective, then f (S \ A) = S \ f (A) for all A ⊆ S.

m


(e) f f −1 (B) = B ∩ f (S) for all B ⊆ S. ✓
s e
l a
3. How many surjective functions f : {1, 2, 3, 4, 5} → {1, 2, 3, 4} are there?
(a) 45 ag
(b) 4!
(c) 45 − 35 + 25 − 15
(d) 45 − 4 × 35 + 6 × 25 − 4 × 15 ✓
5!
(e)
4!
4. “FizzBuzz” sequence starts with the integer 1, keeps incrementing by 1, but replacing
m
m any number divisible by 3 by ‘Fizz’, any number divisible by 5 by ‘Buzz’, and any
.co
.co em
number divisible by both 3 and 5 by ‘FizzBuzz’. The first few terms in the sequence

e m is of the form:
s
la 16, · · ·
las a g
1, 2, Fizz, 4, Buzz, Fizz, 7, 8, Fizz, Buzz, 11, Fizz, 13, 14, FizzBuzz,

ag The 8th number that appears in the above sequence is 14. Which of the following is
the 100th number?
(a) 151
(b) 173
(c) 178
(d) 179

m .
.co
(e) 187 ✓
e m
em l as
las ag
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 1 of 16

5. Consider the set of all distinct 6-letter sequences obtained by shuffling the letters
of the word ‘banana’. If you arrange this list in alphabetical order (with the first
sequence in this list being ’aaabnn’ and the last sequence in this list being ’nnbaaa’),
what is the 42nd sequence in this list?
(a) bannaa
(b) bnaaan
(c) bnaana
(d) naaanb ✓
(e) nabaan

6. Let S := {(x, y, z) ∈ R3 |x2 + y 2 + z 2 = 1} be the unit sphere in R3 . For 0 ≤ a ≤ b ≤
1, denote by A(a, b) the surface area of the subset

{(x, y, z) ∈ S|a ≤ z ≤ b} .

For t ∈ (0, 1), consider the function ft : [0, 1 − t] → R defined by

ft (a) := A(a, a + t).

We emphasize that for each t ∈ (0, 1), the domain of ft is the closed interval [0, 1 − t].
The shaded part in the figure below shows the subset above whose area is A(a, a + t),
as seen when viewing the sphere from a direction perpendicular to the Y Z-plane.

Z

1

t
a
−1 0 1 Y

−1

Consider the following statements.

(i) There exists a t ∈ (0, 1) for which ft is strictly increasing on its domain.
(ii) There exists a t ∈ (0, 1) for which ft is strictly decreasing on its domain.
(iii) There exists a t ∈ (0, 1) for which ft is constant on its domain.

Which of the above statements is/are true?
(a) (i) only

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 2 of 16

m
m .co

m .co s e m
se g l a
a
(b) (ii) only
(c) (iii) only ✓
(d) (i), (ii), and (iii) are all true.
(e) None of (i), (ii), and (iii) is true.

m
.co
7. Consider the unit interval I = [0, 1]. A point P is chosen randomly from I according
m
.co
to a probability density function on I in which the density at any point x ∈ I is
e m
s
proportional to the distance of x from the endpoint of I nearest to x. What is
e m l a
s
expected distance of the randomly chosen point P from the endpoint of I nearest
to P ?
gl
(a) a1/2
a ag
(b) 1/3 ✓
(c) 1/4
(d) 1/5
(e) 1/6

8. Two points are chosen independently and uniformly at random on the circumference
m
.co
of a circle of radius 1. What is the expected length of the chord joining the two
chosen points?
e m
(a) 6/π
l as
ag
(b) 5/π
(c) 4/π ✓
(d) 3/π
(e) 2/π

9. Two positive real numbers x and y are chosen independently and uniformly at random
in [0, 2]. What is the probability that xy < 1 and xy < 2?
(a) 3 ln4 2
m
.co
(b) 3 ln42+1
m
c. o (c) 3 ln 2
8
s e m
e m (d) 3 ln 2+1
✓
la
las 8
(e) None of the above ag
ag 10. Let N be the set of all positive integers and a, b ∈ N be such that their greatest
common divisor is 1. Let

S(a, b) := {a · x + b · y | x, y ∈ N}.

Let T (a, b) := N \ S(a, b). Consider the following statements.

(i) T (a, b) is always a finite set.
m .
.co
(ii) T (a, b) need not be a finite set.
e m
em l as
las ag
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 3 of 16

(iii) The largest number in T (a, b) is at least a · b.
(iv) The largest number in T (a, b) is at most a · b.

Which of the above is/are true?
(a) Only (i)
(b) Only (ii)
(c) Only (ii) and (iii)
(d) Only (i) and (iv) ✓
(e) Only (ii) and (iv)

11. Let Mn (R) be the set of n × n matrices with real entries. Let A ∈ Mn (R) be a
diagonal matrix with n distinct entries on the diagonal. Consider the linear map

Φ : Mn (R) → Mn (R), Φ(X) = AX − XA.

What is the dimension of the image of Φ?
(a) 0
(b) n
(c) n(n − 1)/2
(d) n2 − n ✓
(e) n2

12. Define n = 2020 · 2525 . What is the number of factors of n2 that are smaller than n?
(a) 500
(b) 1490
(c) 5710 ✓
(d) 4255
(e) 2800

13. Uma loves numbers that are multiples of 3, while Gauri does not. Anytime Uma
sees an integer n that is not a multiple of 3, she is willing to pay |1 to round it down
to the nearest multiple of 3 and anytime Gauri sees an integer n that is a multiple
of 3, she is willing to pay |1 to divide the number by 3. The process stops when the
number becomes 0. Given an integer n, let u(n) and g(n) be the money spent by
Uma and Gauri during the process respectively. Consider the following statements
for 0 ≤ n < 32025 :

(i) u(n) + g(n) > 2025.
(ii) u(n) > 2025.
(iii) u(n) > g(n) + 1.
(iv) g(n) > 2025.

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 4 of 16

m
m .co

m .co s e m
se g l a
a
Which of the above are possible?
(a) None of (i), (ii), (iii), and (iv).
(b) Only (i) but not (ii), (iii), and (iv). ✓
(c) Only (i) and (ii) but not (iii) and (iv).

m
.co
(d) Only (i), (ii), and (iii) but not (iv).

o m
(e) All of (i), (ii), (iii), and (iv).
m
. c s e
14. A 4x4 sudoku
following s emhave to be obeyed.
is a 4 × 4 grid partitioned into four 2 × 2 squares or “boxes”. The
g l a
l a rules
a
•agRows: Each row is a permutation of 1, 2, 3, 4.
• Columns: Each column is a permutation of 1, 2, 3, 4.
• Boxes: Each box is a permutation of 1, 2, 3, 4.

An example of a sudoku grid is the following.

1 2 3 4

o m2
c
3 4 1
2 1m.4 3
e
4as 3 2 1
l
ag
What is the total number of distinct sudoku grids?
(a) 288 ✓
(b) 384
(c) 576
(d) 1152
(e) None of the above
o m
c
m Let B denote the Euclidean unit ball and C = [−1, 1] the cubemin. R . Which
c. o15. statement n
2
s e n
n n

e m is true?
l a
las (a) Vol(B ) n
grows with n.
ag
g
2

a n
(b) Vol(C )/Vol(B ) tends to 0.
n 2

(c) Vol(B2n ) tends to 0 as n increases. ✓
(d) B2n and Cn have the same volume for all n.
(e) Vol(B2n ) approaches ∞.

m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 5 of 16

1. What is the decimal representation of the hexadecimal number (A1.4C)16 ?
13
(a) 11 256
13
(b) 17 256
71
(c) 161 256
13
(d) 161 256
(e) None of the above. ✓

2. Given an undirected graph G = (V, E), the edge chromatic number of G, denoted by
χ′ (G), is the mimimum number of colours required to colour the edges of the graph
G such that no two edges incident on the same vertex have the same colour. It is not
hard to see that χ′ (G) ≥ ∆(G), where ∆(G) is the maximum degree of the graph.
For which of the following graphs χ′ (G) > ∆(G)?
(a) C6 , the cycle of length 6
(b) C5 , the cycle of length 5 ✓
(c) K4 , the complete graph on 4 vertices
(d) K2,2 , the complete bipartite graph with 2 vertices on each side
(e) The balanced binary tree of depth 3

3. A coder claims to have implemented the depth first search algorithm correctly and
a tester has to verify this claim. For this, the tester runs the implementation on an
input graph G and the result is a complete binary tree of depth 4, like in the picture
below.

1

2 3

4 5 6 7

8 9 10 11 12 13 14 15

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

The presence of which of the following edges in G would prove that the implementa-
tion is incorrect?
(a) (1, 5)
(b) (2, 10)
(c) (3, 15)

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 6 of 16

m
m .co

m .co s e m
se g l a
a
(d) (4, 20) ✓
(e) None of the above
4. Let G be a weighted graph where the weight of an edge e is w(e). Let S be a subset
of the edges of G that form a minimum-weight spanning tree. Which of the following
does not depend on the choice of S?
m
.co
P
m
c. o w(e)
(i) w(e)
e∈S: w(e) is prime
e m
(ii)
e
P
m l as
(iii) l as
e∈S: w(e) is irrational
P
ag
ag
w(e)
e∈S: w(e)<2025

(a) (i) only
(b) (ii) only
(c) (iii) only
(d) More than one, but not all of (i), (ii), and (iii)
(e) All of (i), (ii), and (iii) ✓

o m
5. For a language L, consider the language perm(L) defined as follows.
c
. u is a permutation of w}
m
perm(L) = {w | ∃u ∈ L such that
e
l a
Now consider the following statements.s
ag is also regular.
(i) If L is regular, then perm(L)
(ii) If L is context-free, then perm(L) is also context-free.
(iii) If L is decidable, then perm(L) is decidable.
Which of the above are true?
(a) Only (i) and (iii)
(b) Only (ii) and (iii)
m
.co
(c) Only (iii) ✓
m(d) All of (i), (ii), and (iii) are true
c. o (e) None of (i), (ii), or (iii) is true s e m
s em 6. Consider an alphabet Σ. Let Σ denote the set of non-empty
g l a words over Σ. Let
la a
+

ag + ω
U, V ⊆ Σ . Let U denote the set of infinite words formed by concatenating words
from U infinitely many times, that is,
U ω = {w = u1 u2 u3 · · · | ui ∈ U for all i ⩾ 0}.
Also let lim(U ) be the set of infinite words that have infinitely many prefixes in U ,
that is,
lim(U ) = {w = a1 a2 a3 · · · |aj ∈ Σ for all j ⩾ 1 and a1 · · · ai ∈ U for infinitely many i ⩾ 0}.

m .
.co
Now consider the following statements.
e m
em l as
las ag
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 7 of 16

(i) (U ∪ V )ω = U ω ∪ V ω .
(ii) lim(U ∪ V ) = lim(U ) ∪ lim(V ).

Which of the above are true?
(a) (i) is true but (ii) is false
(b) (i) is false but (ii) is true ✓
(c) Both (i) and (ii) are true
(d) Both (i) and (ii) are false
(e) Both (i) and (ii) are true if and only if Σ has at least five letters

7. There are n different bottles, each containing a distinct type of wine. We know that
exactly one of them is poisoned. Additionally, we have access to test kits that can
take a drop of wine from a set of bottles and detect whether the poisoned bottle is
in the set. This detection takes one day. What is the minimum number of test kits
needed to find out the poisoned bottle within a day?
(a) Θ(n)
√
(b) Θ( n)
(c) Θ(log n) ✓
(d) Θ(log log n)
(e) None of the above

8. Let G = (V, E) be an undirected simple graph on n vertices that has two distinct
perfect matchings M1 and M2 . Let E1 be the set of those edges in M1 that are
not contained in M2 . Similarly, let E2 be the set of those edges in M2 that are not
contained in M1 . Consider the edge subgraph H = (V, E1 ∪ E2 ) of G. What is the
maximum possible number of vertices in H whose degree in H is an odd number?
(Assume that n ≥ 100.)
(a) 0 ✓
(b) 2
(c) 50
(d) n/2
(e) n

9. Consider a real m × n matrix A whose entries are drawn from the set {0, 1}. Given
subsets S and T of {1, 2, . . . , m} and {1, 2, . . . , n} respectively, we denote by AS,T
the |S| × |T | submatrix of A obtained by picking from A the rows whose indices are
in S and the columns whose indices are in T . Such a submatrix is called a rectangle.
A rectangle is said to be on if all its entries are 1. A rectangle AS,T is said to include
the (i, j) entry of A if i ∈ S and j ∈ T . Rectangles AS,T and AS ′ ,T ′ are said to be
disjoint is the sets S × T and S ′ × T ′ are disjoint.

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 8 of 16

m
m .co

m .co s e m
se g l a
a
Suppose that there exist 100 disjoint rectangles of A, all of which are “on”, such that
only those entries of A are 1 that are included in one of these rectangles. Let r(A)
denote the rank of A. Which of the following statements is true for all such A?
(Assume that m, n are both at least 1000.)
(a) r(A) = 100
m
.co
(b) r(A) ≤ 100, and r(A) = 1 is possible ✓
m
.co
(c) 50 ≤ r(A) ≤ 100, and r(A) = 50 is possible
s e m
s em
(d) 100 ≤ r(A) ≤ 200, and r(A) > 100 is possible
l a
(e) Allla
g
of the above statements are false ag
a the path graph G = (V , E ) on n vertices as shown below. So V =
10. Consider n n n n
{1, . . . , n} and En = {(i, i + 1) : 1 ≤ i ≤ n − 1}.

1 2 n

A matching M in Gn is a subset of En that contains at most one edge incident to

m
each vertex. Note that the empty set is also a matching. So G1 has one matching,
o
. c
M = ∅, while G2 has two matchings, M = ∅ and M = {(1, 2)}.
How many distinct matchings are there m
s e in G ? 10

(a) 89 ✓
l a
(b) 100 ag
(c) 512
(d) 1024
(e) None of the above

11. Consider two polynomials p(x), q(x) ∈ Z[x] of degree at most n, with each coefficient
in {−1, 0, 1}. For which smallest positive integer M is it guaranteed that if p(M ) =
m
.co
q(M ) then p = q?
m(a) 2
c. o (b) s e m
e m 3✓
la
las (c) n
ag
ag (d) n!
(e) there is no such M that works

12. Consider the recurrence
j n k l n m
T (n) = T +T − T (1), T (1) = 1.
2 2
Here n is a positive integer. What is the asymptotic growth of T (n)?
m .
.co
(a) Θ(1) ✓
e m
em l as
las ag
ag For more Question Papers, Sample Papers, Notes & Syllabus visit Page 9 of 16

(b) Θ(log n)
√
(c) Θ( n)
(d) Θ(n)
(e) Θ(n log n)

13. Consider the following non-deterministic automaton.

q1a 0 q2a 0 q3a 0 q4a
0,1
ε
q0

ε 1 1 1
q1b q2b q3b q4b

Let p be the probability that a uniformly random string from {0, 1}10 is accepted by
the above automaton. Which of the following is true about p?
(a) 0 < p ≤ 0.05
(b) 0.05 < p ≤ 0.10
(c) 0.10 < p ≤ 0.15
(d) 0.15 < p ≤ 0.20
(e) p > 0.20 ✓

14. Consider the following pseudocode that, on input a non-negative integer n, computes
the remainder of the n-th Fibonacci number when divided by 100.

1: function Fib(n)
2: if n = 0 then
3: return 0
4: else if n = 1 then
5: return 1
6: else
7: a ←Fib(n − 1)
8: b ←Fib(n − 2)
9: return (a + b) mod 100
10: end if
11: end function

In the above pseudocode, a and b are variables local to the function Fib.
Define T : N → N and S : N → N as follows: T (n) and S(n) is the time and
space, respectively, taken by the above pseudocode to compute Fib(n). Assume that
recursion is implemented using the standard stack model.

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 10 of 16

m
m .co

m .co s e m
se g l a
a
If

PolyBounded = {f : N → N : ∃c > 0 such that f (n) = O(nc )}
Exp = {f : N → N : f (n) = 2Ω(n) }

m
then which of the following statements is true about T (n) and S(n)?

m
(a) T (n) ∈ PolyBounded and S(n) ∈ PolyBounded.
.co
.co
(b) T (n) ∈ PolyBounded and S(n) ∈ Exp
s e m
s em l a
ag
(c) T (n) ∈ Exp and S(n) ∈ PolyBounded ✓

g la∈ Exp and S(n) ∈ Exp
(d) T (n)
(e) aT (n) ∈ Exp and S(n) ∈
/ Exp and S(n) ∈
/ PolyBounded

15. Let f (x) be a non-zero univariate polynomial of degree d > 1 with integer coefficients.
Over any ring or a field R that contains the ring of integers, we say that f is irreducible
over R if f cannot be written as the product of two polynomials g(x) and h(x) over
R such that each of g, h has degree at least one and has coefficients in the ring/field
R.
Which of the following statements is true?
m
.co
(a) f is irreducible over the ring of integers if and only if f is irreducible
m
over the field of rational numbers. ✓
s e
(b) f is irreducible over the ring ofaintegers if and only if f is irreducible over the
field of real numbers.
a gl
(c) f is irreducible over the field of rational numbers if and only if f is irreducible
over the field of real numbers.
(d) f is irreducible over the field of rational numbers if and only if f is irreducible
over the field of complex numbers.
(e) f is irreducible over the field of real numbers if and only if f is irreducible over
the field of complex numbers.
m
m .co
m .co s e m
s e g la
g la a
a

m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 11 of 16

1. Let B be the three dimensional Euclidean ball of radius 1 and let Q be the three
dimensional cube [−1, 1]3 . Let B + Q be set of all vectors x + y where x ∈ B and
y ∈ Q. Which of the following is FALSE?
(a) B + Q is a convex set
(b) The volume of B + Q is more than the sum of the volumes of B and Q
(c) B + Q is not a cube
(d) The volume of B + Q is 4π/3 + 24 ✓
(e) B + Q is not a Euclidean ball

2. Consider a uniform random point in the cube [−1, 1]n . As n increases, which state-
ment is true?
(a) The average squared distance from the origin grows linearly with n ✓
(b) The average squared distance remains bounded
(c) The expected distance to the origin tends to 0
(d) The expectation of ∥x∥2 decreases as n increases
(e) The expectation of ∥x∥3 decreases as n increases

3. Let X and Y be independent standard normal random variables, i.e., independent
Gaussian random variables with mean 0 and variance 1. Define the 2 × 2 random
matrix  
X Y
M= .
Y X
Let λ1 and λ2 be the two eigenvalues of M . Which of the following statements about
(λ1 , λ2 ) is FALSE?
(a) E[λ1 + λ2 ] = 0
(b) Cov(λ1 , λ2 ) = 0
(c) The eigenvalues are λ1 = X + Y and λ2 = X − Y
(d) λ1 and λ2 are independent random variables
(e) None of the above ✓

4. Consider a 2 × 2 symmetric matrix A with eigenvalues 5 and 2 corresponding to or-
√ √
T √ √
T
thogonal eigenvectors ⃗v1 = 2/ 5 1/ 5 and ⃗v2 = −1/ 5 2/ 5 respectively.
That is,
A = 5⃗v1⃗v1T + 2⃗v2⃗v2T .
Consider the function f (⃗x) = ⃗xT A⃗x, where ⃗x ∈ R2 , and the set
n
T o
S = x1 x2 ∈ R2 : x21 + x22 ≤ 1 and 2x1 + x2 = 0 .

What is the maximum value of f (⃗x) for ⃗x ∈ S?

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 12 of 16

m
m .co

m .co s e m
se g l a
(a) 5 a
(b) 2 ✓
(c) 0
(d) 3
(e) None of the above

5. Consider a sequence of independent coin tosses, where each toss shows heads with
m
m .co
probability p and tails with probability 1 − p. A “run” is a maximal sequence of

.co m
consecutive identical outcomes, e.g., in HHTTTHTHH, the runs are HH, TTT, H,
s e
em the number of runs in the first n tosses. For large n, what is the
T, and HH.
s g l a
g
expected a
Let R denote
l
n
number of runs per toss, that is, lim E[R ]? 1 a
(a) ap(1 − p)
n→∞ n n

(b) (p(1 − p))2
(c) 2p(1 − p) ✓
(d) None of the above, but the limit exists
(e) The limit does not exist

m
6. Consider a particle starting at 0 on the integer number line. At each time step, it

.co
moves according to the following rule: with probability 1/2, it moves right by 1 unit;

em
and with probability 1/2, it moves left by 1 unit. The particle stops moving when it
s
la reaches +3 before it reaches -2?
first reaches either position +3 or position -2.
What is the probability that thegparticle
(a) 1/3
a
(b) 2/5 ✓
(c) 1/2
(d) 3/5
(e) 2/3

m
.co
7. Suppose X is a uniform random variable which takes values in the interval [0, 1].
m We write this as X ∼ U[0, 1]. Independent of the realization of X, with probability

m .co p = 0.1, an adversary is able to intercept X. On intercepting X, the adversary may
s e m
e la
replace it with a random value drawn according to a distribution of its choosing which

las may depend on the realization of X. Let Y denote the outcome after the possible
ag
ag intervention by the adversary. i.e., Y = X with probability 1 − p = 0.9 and, Y is set
by the adversary (possibly dependent on X) with probability p = 0.1. Consider the
following distributions for Y :

(i) Y ∼ U[0, 1]
(ii) Y ∼ U[0.1, 1.1]
(iii) Y ∼ U[0, 1.1]

m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 13 of 16

Which of the above is a distribution of Y the adversary CANNOT induce?
(a) Only (i)
(b) Only (ii) ✓
(c) Only (iii)
(d) Only (ii) and (iii)
(e) None of the above answers is correct

8. A fair coin is tossed m + n times, where m is strictly greater than n. Then the
probability of getting m consecutive heads is
(a) 21m
(b) 21n
(c) m+n
2m
(d) (n + 1)/2m+1
(e) None of the above ✓

9. Let c1 , c2 , . . . , cn and z be complex numbers such that
1 1 1
+ + ... + = 0.
z − c1 z − c2 z − cn
Assume that the numbers c1 , c2 , . . . , cn are represented in the complex plane by the
vertices of a convex n-gon C. Then
(a) z always lies strictly outside C
(b) z always lies inside or on the boundary of C ✓
(c) z may lie inside or outside C and both are possible
Pn
(d) if z lies inside C then z must be the centroid, i.e., z = n1 k=1 ck
(e) None of the above

10. Let X = (X1 , X2 , X3 ) be a random vector with covariance matrix Σ. Suppose aT X
and bT X are uncorrelated for any orthogonal vectors a and b. Then
(a) rank(Σ) = 1
(b) rank(Σ) = 2
(c) Σ is a scalar multiple of the identity matrix ✓
(d) rank(Σ) = 3 but Σ is not necessarily diagonal
(e) None of the above is always true

11. Let the point X = (X1 , X2 , X3 ) be chosen uniformly at random from the surface of
the 3-dimensional unit ball, i.e., from the set {(x, y, z) ∈ R3 |x2 + y 2 + z 2 = 1}. Then
(a) E(X14 ) = 12
(b) E(X14 ) = 13
(c) E(X14 ) = 15 ✓

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 14 of 16

m
m .co

m .co s e m
se g l a
(d) E(X14 ) = 19 a
(e) None of the above

12. Let X, Y, Z be random variables defined on the same probability space, taking values
in a finite set S. Assume that |S| > 10 and also that for all s1 , s2 , s3 ∈ S, Pr[X =
s1 , Y = S2 , Z = s3 ] is strictly positive.
Under the above assumptions, consider the following statements.
m
m .co
(i) If X is independent of Y , and Y is independent of Z, then X is independent of
Z.
c o m
(ii) If X and Y. are independent, then X and Y are independent conditioned on Z. se
em
(iii) If X sand Y are independent conditioned on Z, then X and Y are independent.
g l a
g la a
a the above setup, choose the correct statement from the following.
Given
(a) Only statement (i) is always true
(b) Only statement (ii) is always true
(c) Only statement (iii) is always true
(d) Statements (i), (ii), and (iii) are all always true
(e) None of the statements (i), (ii), and (iii) is always true ✓
m
.co
13. Let X be a random variable satisfying 0 ≤ X ≤ 1, and let p = E[X].
Consider the following statements:
s em
2 2

g la surely.
P1: E[X ] = p iff X is constant almost
P2: E[X ] = p iff X takes onlyathe values 0 and 1 almost surely.
2

P3: For every such X, one always has E[X 2 ] ∈ [p2 , p].

Which of the following is true?
(a) Only P1 and P2 are true
(b) Only P2 and P3 are true
(c) Only P1 and P3 are true
m
.co
(d) All of P1, P2, and P3 are true ✓
m
c. o (e) None of the statements P1, P2, or P3 is true s e m
s em 14. Which of the following is true in 4 dimensions? g l a
g la a of the origin centred
(a) The origin centred Euclidean ball of volume 1 is a subset
a axis-aligned cube of volume 1
(b) The origin centred Euclidean ball of radius 1 is a subset of a origin
centered cube of volume 16 21 ✓
(c) The origin centred Euclidean ball of radius 1 contains an origin centered cube of
side length 45
(d) The volume of a Euclidean ball of radius 1 is 34 π

m .
.co s e m
s em l a
g la ag
a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 15 of 16

(e) The volume of a Euclidean ball of radius 2 is 64
3
π

15. Let X be a random variable uniformly distributed in a regular hexagon of side length
1. Let ∆ be the difference between the highest eigenvalue of the matrix E[XX T ] and
the lowest eigenvalue of the same matrix. Which of the following is TRUE?
(a) ∆ = 0 ✓
(b) ∆ = 1
(c) ∆ = 2
(d) ∆ = 3
(e) ∆ = 4

For more Question Papers, Sample Papers, Notes & Syllabus visit Page 16 of 16