TIFR GS 2024 Question Paper Learning Information & Data Science

GS 2024 Common Section: Solutions Page 1 of 10

Common Section
1. If O is the center of the circle, what is the area of the shaded portion in square cm?

4cm

4cm

O

√
(a) 4π − 4 2
7
√
(b)
2
π − 4 2
7
√
(c)
2
π−4 3
√
(d) 83 π − 4 3 ✓
8
√
(e)
3
π−4 2

2. Let σ be a uniform random permutation of {1, . . . , 100}. What is the probability
that σ(1) < σ(2) < σ(3) (i.e., what is the probability that the
rst three elements
are in increasing order)?

In the options below, n! = 1 × 2 × . . . × n.

3
(a)
100!
3!
(b)
100!
6
(c)
100
1
(d) ✓
6
1
(e)
3
3. There is a 100cm long ruler that has 11 ants on positions 0cm, 10cm, 20cm, 30cm,
..., 100cm. The ant at the 0cm mark is facing towards the 100cm mark, and the ant
at the 100cm is facing towards the 0cm mark; all other ants is either facing towards
0cm or towards the 100cm mark.
All ants start moving in the direction they are facing at a speed of 1cm per second.
Whenever an ant collides with another ant, the two instantly reverse their direction
and continue moving in the other direction. All ants continue moving until they fall
o the ruler when they move past one of the two ends (at 0cm or 100cm marks).

How long will it take for all ants to fall o the ruler?

(a) At most 50 seconds, but cannot be determined without knowing the direc-
tions of all ants.

GS 2024 Common Section: Solutions Page 2 of 10

(b) 50 seconds.

(c) At least 50 seconds and at most 100 seconds, but cannot be determined
without knowing the directions of all ants.

(d) 100 seconds. ✓
(e) More than 100 seconds, but cannot be determined without knowing the
directions of all ants.

4. Let z1 , z2 , z3 , . . . , z2023 be a permutation of the numbers 1, 2, 3, . . . , 2023. Which of
Q2023
the following is true about the product i=1 (zi − i)?
Note: The parity of an integer n just denotes whether n is even or odd. Formally,
the parity of n is said to be odd if n is odd, and even if n is even.

(a) The above product is always even. ✓
(b) The above product is always odd.

(c) The parity of the above product always changes if we swap the values of
any two variables among z1 , z2 , . . . , z2023 .

(d) There always exist two variables among z1 , z2 , . . . , z2023 such that the par-
ity of the above product changes if we swap their values, but there may
also exist two variables among the z1 , z2 , . . . , z2023 such that swapping their
values does not change the parity of the above product.

(e) None of the above statements is true.

5. Let p(x) be a polynomial with real coe
cients which satis
es p(r) = p(−r) for every
real number r . Let n ≥ 5 be a positive integer. Suppose that p(i) = i for all 1 ≤ i ≤ n.
5
What is the maximum possible value of the absolute value of the coe
cient of x in
p(x)?
(a) 0 ✓
(b) 5

(c) 10

(d) n
(e) n+1

6. For each month in the year (i.e., January, February, March,...), let us assume the
probability that a person's birthday falls in that particular month is exactly 1/12,
and let us assume that this is independent for dierent persons.

What is the smallest value of the natural number n such that, among n independently
chosen persons the probability that there is a pair of them born in the same month
is at least 1/2?

(a) 3

(b) 4

(c) 5 ✓
(d) 6

GS 2024 Common Section: Solutions Page 3 of 10

(e) 7

7. Let S := {(a, b) | 0 ≤ a ≤ 1 , 0 ≤ b ≤ 1}, a unit square, in R2 . Let B :=
{(x, y) | x2 + y 2 ≤ 1}, a unit disk, in R2 . De
ne the set S + B as follows:

S + B := {(u, v) | ∃(a, b) ∈ S , (x, y) ∈ B such that u = a + x , v = b + y} .

What is the area of S + B ?

(a) π+4
(b) π + 5 ✓
(c) π+3
(d) π+2
(e) None of the above.

8. A palindrome is a string that reads the same in reverse (e.g. ABBA or KAYAK or
MALAYALAM).

How many strings of length 5 using the letters from {A, B, C, D, E} have no palin-
dromic substring of length at least 2?

(a) 243

(b) 405

(c) 540 ✓
(d) 675

(e) 1280
R∞ 1
9. Compute 16 x
· √√1 dx.
x−1

(a) 0
π
(b)
3
π
(c)
2
2π
(d) 3 ✓
(e) 2π

10. Let M be a 3 × 3 matrix over the real numbers such that M T M = I. Consider the
following statements.

3
(i) There exists a non-zero vector x ∈ R such that M x = 0.
3
(ii) There exist two non-zero vectors x, y ∈ R such that the angle between them
and that between M x and M y are dierent.

(iii) There exists a non-zero vector x such that its length is dierent from that of
M x.

Which of the above statements is/are true?

(a) Only (i).

GS 2024 Common Section: Solutions Page 4 of 10

(b) Only (ii).

(c) Only (iii).

(d) All three statements.

(e) None of the three statements. ✓

11. Consider the following sequence of polynomials with real coe
cients.

P0 (x) = 1
P1 (x) = 2x
Pn+1 (x) = 2xPn (x) − Pn−1 (x), for all natural numbers n ≥ 1.

What is the dimension of the linear span of the set

{P0 (x), P1 (x), P2 (x), P3 (x), P4 (x)}

in the vector space of polynomials in variable x with real coe
cients?

(a) 1
(b) 2
(c) 3
(d) 4
(e) 5 ✓

12. A subset S of the rational numbers is said to be nice if for every in
nite sequence
of x1 , x2 , . . . of elements from S , there is always two indices i < j such that xi ≤ xj .
Consider the following statements.

(i) The set of natural numbers IN is nice.

(ii) The set of integers Z is nice.

(iii) The set of positive rational numbers is nice.

Which of the above statements is/are true?

(a) Only (i). ✓
(b) Only (i) and (ii).

(c) Only (i) and (iii).

(d) All three statements are true.

(e) None of the three statements is true.

13. Let n ≥ 100 be a positive integer. X1 , X2 , . . . , Xn be independent random
Let
2
variables, each taking values in the set {0, 1} such that Pr[Xi = 1] = for each
Pn 3
1 ≤ i ≤ n. De
ne S := i=1 Xi , and let p(x) be a polynomial such that for each
j ′
non-negative integer j , the coe
cient of x in p(x) is Pr[S = j]. What is p (1)?

Note: p′ (x) denotes the polynomial obtained by dierentiating p with respect to x.
(a) 0

GS 2024 Common Section: Solutions Page 5 of 10

(b) n
(c) 2n
3
✓
4n2 +2n
(d)
9
4n2 −4n
(e)
9

14. Let A and B be two n × n invertible matrices with real entries such that every row in
A sums to 1 and every row in B sums to 2. Consider the following three statements:

(i) Every row in the matrix AB sums to 2.
−1
(ii) Every row in the matrix A B sums to 2.
−1 −1 1
(iii) Every row in the matrix A B sums to .
2

Which of the above statements is/are true?

(a) None of the statements (i), (ii), (iii) is true.

(b) All the three statements (i), (ii), and (iii) are true. ✓
(c) Statement (i) is true but not necessarily statements (ii) or (iii).

(d) Statements (i) and (ii) are true but not necessarily statement (iii).

(e) Statements (i) and (iii) are true but not necessarily statement (ii).

15. Suppose Michelle gives Asna and Badri two dierent numbers from N = {1, 2, 3, . . .}.
It is commonly known to both Asna and Badri that they each know only their own
number and that it is dierent from the other one. The following conversation ensues.

Michelle: I privately gave each of you a dierent natural number. Which of you has the
larger of the two numbers?
Asna: I don't know.
Badri: I don't know either.
Asna: Oh, then I know who has the larger number.
Badri: In that case, I know both numbers. What numbers were Asna and Badri
respectively given?

(a) Asna was given 2, Badri was given 3. ✓
(b) Asna was given 3, Badri was given 2.

(c) Asna was given 3, Badri was given 4.

(d) Asna was given 4, Badri was given 3.

(e) None of the above.

GS 2024 LIDS Section: Solutions Page 6 of 10

LIDS Section
2
1. Let f (x) = √12π exp(− x2 ) denote the probability density of a standard Gaussian.
R∞
Then
−∞
f (x)2 dx equals
q
2
(a) .
π
q
1
(b) 4π
. ✓
√
(c) 2 π .
√
(d) 4 π .
√
(e) 8 π .

2. An urn contains 15 white and 10 black balls. Balls are drawn one by one from the
urn without replacement. What is the probability that the 5
th draw results in a white
ball and the 15
th draw results in a black ball?

2
(a)
15
(b) 14 ✓
1
(c)
5
2
(d)
5
(e) None of the above

3. Let (x1 , x2 ) be a random sample from the
 uniform probability
 distribution on a unit
E[x21 ] E[x1 x2 ]
square of side 1 centered at 0. Let M = . Let λ1 ≥ λ2 be the two
E[x2 x1 ] E[x22 ]
(possibly equal) eigenvalues of M . Then λ1 − λ2 equals the following.
1
(a)
4
.
1
(b) .
3
1
(c) .
2
1
(d) .
12
(e) 0. ✓

4. An n × n matrix A is called positive semi-de
nite (p.s.d.) if xT Ax ≥ 0 for all x ∈ Rn .
Consider the following statements.

1. A p.s.d. matrix is always full-rank.

2. All eigenvalues of a p.s.d. matrix are non-negative.

3. All eigenvalues of a p.s.d. matrix are distinct.

Which of the following is TRUE?

(a) Statement 1 only.

(b) Statement 2 only ✓
(c) Statement 3 only.

GS 2024 LIDS Section: Solutions Page 7 of 10

(d) Both Statements 1 and 3.

(e) Both Statements 2 and 3.

5. Consider a signal f (t) which peaks in the time domain at a particular time point t0
in a Gaussian manner. Formally,

f (t) = exp(−(t − t0 )2 ).
Which of the following is the best description of the frequency domain power spectrum
2
(i.e., the graph of |F (ω)| against the angular frequency ω , where F denotes the
continuous time Fourier transform of f ) of f ?

Note: The following information may be useful. For any complex number b and any
complex number a with positive real part,

Z∞ r
π
exp(−a(x − b)2 ) dx = ,
a
−∞

where the square root of a is chosen to be the one that has positive real part.

(a) The power spectrum of f does not exist.

(b) The power spectrum of f is shaped like a square pulse.

(c) The power spectrum of f is shaped like the probability density function of
some probability distribution whose tails follow an inverse power law.

(d) The power spectrum of f is shaped like a delta function.

(e) The power spectrum of f is shaped like the probability density
function of some Gaussian distribution. ✓

6. Let Xi , i = 1, 2, . . . , be independent and uniformly distributed random variables
1
over [0, 1] with mean . Let
2
 
1
T = min i : Xi ≥ .
2
What is the E[XT −1 ]?

[Hint: Note that T is a random variable and XT −1 is not a uniformly distributed
random variable over [0, 1].]
1
(a)
3
1
(b)
4
3
(c)
4
5
(d)
6
(e) None of the above ✓

7. Consider the function
f (x) = x2 − loge x
for values of x which lie in the interval [0.5, 1]. In this domain, suppose the function
attains the maximum ∗
value at x . Which of the following is true?

GS 2024 LIDS Section: Solutions Page 8 of 10

√
(a) 0.5 < x∗ < 1/ 2
∗
√
(b) x = 1/ 2
√ ∗
(c) 1/ 2 < x < 1

(d) x∗ ∈ {0.5, 1} ✓
(e) The maximum value is attained at more than one value of x

8. Let   x 2 
2
P = (x, y) : x ≤ 1, y ≤ 1, +y ≤1 .
2
Compute
max x + y.
(x,y)∈P
√
(a) 1 + 35
√
3
(b) 1 + 2
✓
2
(c) 1 + √
5
√
(d) 5
(e) None of the above

9. Recall that the Fourier transform of a signal f (t), t ∈ R is given by
Z ∞
F (ω) = f (t)e−jωt dt.
−∞

Suppose that for a signal f (t), it is known that F (ω) = 0 for |ω| > 10. Consider the
following signals:

1
 t=0
sin(t)
(i) 0 < |t| ≤ 10π
 t
0 otherwise

(
sin(t)
t
π ≤ t ≤ 10π
(ii)
0 otherwise
(
1 t=0
(iii) sin(t)
t
t ̸= 0

Which of the above could possibly be f (t)?

(a) Only (i)

(b) Only (ii)

(c) Only (iii) ✓
(d) Only (i) and (iii)

(e) Only (i) and (ii)

GS 2024 LIDS Section: Solutions Page 9 of 10

10. Suppose A ∈ C5×4 such that Rank(A) = 3 (i.e., A is a 5 × 4 matrix with entries
4×5
which are complex numbers and of rank 3), and B ∈ C such that Rank(B) = 3.
Of the options below, choose the tightest bound on the rank of AB which holds for
all such A and B . Note that to receive credit, you need to choose the tightest bound
and not just a correct bound.

(a) 0 ≤ Rank(AB) ≤ 3
(b) 1 ≤ Rank(AB) ≤ 4
(c) 1 ≤ Rank(AB) ≤ 3
(d) 2 ≤ Rank(AB) ≤ 3 ✓
(e) None of the above

11. Let α = (1, 2, 3, 4)T and w = ( 21 , 12 , 12 , 21 )T be two vectors in R4 . Find the vector z
which, among the set of all orthogonal vectors to w , minimizes the Euclidean distance
to the vector α.

(a) (−1.5, −0.5, 0.5, 1.5)T ✓
(b) (−2, −0.5, 0.5, 2)T
(c) (−1.5, 0.5, 2.5, −1.5)T
(d) (1.5, 2.5, −2.5, −1.5)T
(e) (0.5, 0, 1.5, −2)T

12. Let f : [0, 1] × [0, 1] → R be a continuous and dierentiable function such that
R1 R1 R 1 R 1−x
x=0 y=0
f (x, y)dxdy = 1. Let
x=0 y=0
f (x, y)dxdy = A. Consider the following
statements

1. A ≤ 1/2.
2. A ≥ 1/2.
3. A ≥ 1/4.
4. A ≤ 1/4.

Which of the following is TRUE.

(a) Only statement 1 is correct.

(b) Only statement 2 is correct.

(c) Only statement 3 is correct.

(d) Only statement 4 is correct.

(e) None of the above. ✓

13. Let X = X0 , X1 , X2 , . . . be independent identically distributed random variables,
1
Pn 4
such that P[X = −1] = P[X = 1] = .. Then E[( i=1 Xi ) ] is equal to which of the
2
following?
n
+ n2 .



(a)
1
1 + n + n2 .


(b)

GS 2024 LIDS Section: Solutions Page 10 of 10

1 + n + n2 + n4 .



(c)
n n



(d)
2
+ 4
.

(e) None of the above. ✓

14. Let Wi = (Xi , Yi ) be a random variable which takes values in Z2 for i = 0, 1, . . . , n.
Suppose that all Wi − Wi−1 are independent and take values (0, 1) and (1, 0) with
probability 1/2 each. Suppose W0 is deterministic and equal to (0, 0).
What is the expected squared distance of Wn from the point (n/2, n/2)?

(a) 0.
(b) n/2. ✓
(c) n.
(d) 2n.
(e) None of the above.

15. Consider the function
f (x, y) = (x2 − 4)2 + y 2 .

Consider the following statements.

(1) (−2, 0) is a global minimum.
(2) (+2, 0) is a global minimum.
(3) ∇f (0, 0) = 0 but (0, 0) is neither a local minimum nor a local maximum.

Which of the following statements is correct.

(a) Only (1) is true.

(b) Only (2) is true.

(c) Only (1) and (2) are true.

(d) Only (2) and (3) are true.

(e) All of (1), (2), (3), are true. ✓