TIFR GS 2016 Question Paper Computer Science

Computer and Systems Sciences
CLARIFICATIONS

Page 13, Problem 2
The first sentence should read as follows.
Let X1 and X2 be two independent and identically distributed
real-valued random variables taking values in the unit in-
terval [0, 1].

Page 15, after Problem 6
Please ignore the blank space on the page. Problem 7
appears on page 16.

CSS 2016 Common Part Page 1 of 19

Part A: Common Part

1. Suppose the following statements about three persons in a room are true.

Chandni, Sooraj and Tara are in a room. Nobody else is in the room.
Chandni is looking at Sooraj. Sooraj is looking at Tara. Chandni is mar-
ried. Tara is not married. A married person in the room is looking at an
unmarried person.

Then, which of the following is necessarily true?

(a) Sooraj is married

(b) Sooraj is unmarried

(c) The situation described is impossible

(d) There is insufficent information to conclude if Sooraj is married or
unmarried X

(e) None of the above

2. Consider the graph shown below.

4 1 5

3 2

6

The following experiment is performed using this graph. First, an edge e = {i, j} of
the graph is chosen uniformly at random from the set of nine possibilities. Next, a
common neighbour k of i and j is chosen, again uniformly from the set of possibilities.
(Note that the set of possibilities is always non-empty.) Thus, {i, j, k} is a triangle
in the graph. What is the probability that the triangle finally picked is {1, 2, 3}?

(a) 61 X (b) 14 (c) 13 (d) 23 (e) 65

CSS 2016 Common Part Page 2 of 19

3. Consider the following set of 3n linear equations in 3n variables:

x1 − x2 = 0 x4 − x5 = 0 ··· x3n−2 − x3n−1 = 0
x2 − x3 = 0 x5 − x6 = 0 x3n−1 − x3n = 0
x1 − x3 = 0 x4 − x6 = 0 x3n−2 − x3n = 0
3n
Let S ⊆ R be the set of solutions to this set of equations. Then,

(a) S is empty

(b) S is a subspace of R3n of dimension 1

(c) S is a subspace of R3n of dimension n X

(d) S is a subspace of R3n of dimension n − 1

(e) S has exactly n elements

4. There are n balls b1 , . . . , bn and n boxes. Each ball is placed in a box chosen inde-
pendently and uniformly at random. We say that (bi , bj ) is a colliding pair if i < j,
and bi and bj are placed in the same box. What is the expected number of colliding
pairs?

n
(a) n−1


2
X (b) 0 (c) 1 (d) n/4 (e) 2

5. For a positive integer N ≥ 2, let
N
X 1
AN := ;
n=2
n
Z N
1
BN := dx.
x=1 x

Which of the following statements is true?

(a) As N → ∞, AN increases to infinity but BN converges to a finite number

(b) AN < BN and the difference decreases as N → ∞

(c) AN < BN < AN + 1 X

(d) BN < AN < BN + 1

(e) As N → ∞, BN increases to infinity but AN converges to a finite number

CSS 2016 Common Part Page 3 of 19

6. Which of the following statements about the eigen values of In , the n × n identity
matrix (over complex numbers), is true?

(a) The eigen values are 1, ω, ω 2 , . . . , ω n−1 , where ω is a primitive n-th root of unity

(b) The only eigen value is −1

(c) Both 0 and 1 are eigen values, but there are no other eigen values

(d) The eigen values are 1, 1/2, 1/3, . . . , 1/n

(e) The only eigen value is 1 X

7. Let S be the 4 × 4 square grid {(x, y) : x, y ∈ {0, 1, 2, 3}}. A monotone path in this
grid starts at (0, 0) and at each step either moves one unit up or one unit right. For
example, from the point (x, y) one can in one step either move to (x + 1, y) ∈ S or
(x, y + 1) ∈ S, but never leave S. Let the number of distinct monotone paths to reach
point (2, 2) starting from (0, 0) be z. How many distinct monotone paths are there
to reach point (3, 3) starting from (0, 0)?

(a) 2z + 6 (b) 3z + 6 (c) 2z + 8 X (d) 3z + 8 (e) 3z + 4

8. Let A and B be finite sets such that A ⊆ B. Then, what is the value of the expression:
X
(−1)|C\A| ,
C:A⊆C⊆B

where C \ A = {x ∈ C : x 6∈ A}?

(a) Always 0

(b) Always 1

(c) 0 if A = B and 1 otherwise.

(d) 1 if A = B and 0 otherwise X

(e) Depends on the size of the universe

CSS 2016 Common Part Page 4 of 19

9. Suppose a rectangular farm has area 100 square metres. The lengths of its sides are
not known. It is known, however, that all the edges are at least 2 metres in length.
Which of the following statements about the rectangle’s perimeter p (in metres) is
FALSE?

(a) p can take all values between 45 and 50

(b) p can be 52 for some configuration

(c) p can take all values between 55 and 60

(d) p can be 70 for some configuration

(e) p can be 39 for some configuration X

10. Consider the sequence hsn : n ≥ 0i defined as follows: s0 = 0, s1 = 1, s2 = 1, and
sn = sn−1 + sn−2 + sn−3 , for n ≥ 3. Which of the following statements is FALSE?

(a) s4k is even, for any k ≥ 0

(b) s4k+1 is odd, for any k ≥ 0

(c) s4k+2 is odd, for any k ≥ 0

(d) sn is a multiple of 3, for only finitely many values of n X

(e) s4k+3 is even, for any k ≥ 0

11. In one of the islands that his travels took him to, Gulliver noticed that the probability
that a (uniformly) randomly chosen inhabitant has height at least 2 metres is 0.2.
Also, 0.2 is the probability that a (uniformly) randomly chosen inhabitant has height
at most 1.5 metres. What can we conclude about the average height h in metres of
the inhabitants of the island?

(i) 1.5 ≤ h ≤ 2 (ii) h ≥ 1.3 (iii) h ≤ 2.2

Which of the above statements is necessarily true?

(a) (ii) only X

(b) (iii) only

(c) (i), (ii), and (iii)

(d) (ii) and (iii) only

(e) None of the above

CSS 2016 Common Part Page 5 of 19

12. There are two rocks A and B, located close to each other, in a lily pond. There is a
frog that jumps randomly between the two rocks at time t = 0, 1, 2, . . .. The location
of the frog is determined as follows. Initially, at time t = 0, the frog is at A. From
then on, the frog’s location is determined as follows. If the frog is at A at time t,
then at time t + 1, with probability 2/3 it jumps to B and with probability 1/3, it
jumps on the spot and stays at A. If the frog is at B at time t, then at time t + 1,
with probability 1/2 it jumps to A and with probability 1/2 it jumps on the spot
and stays at B. What is the probability that the frog is at B at time 3 (just after its
third jump)?

(a) 21 (b) 31
54
X (c) 14
27
61
(d) 108 (e) 23 .

13. Let n ≥ 2 be any integer. Which of the following statements is not necessarily true?

n n−1
+ n−1




(a) i
= i i−1
, where 1 ≤ i ≤ n − 1

(b) n! divides the product of any n consecutive integers
Pn n

(c) i=0 i = 2n

(d) n divides ni , for all i ∈ {1, 2, . . . , n − 1} X



(e) If n is an odd prime, then n divides 2n−1 − 1

14. A diagonal in a polygon is a straight line segment that connects two non-adjacent
vertices, and is contained in the interior of the polygon (except for its end points).
Two such diagonals are said to cross if they have a point in common in the interior
of the polygon. In one such polygon with n vertices, a certain number (say k) of
non-crossing diagonals were drawn to cut up the inside of the polygon into regions,
each of which was a quadrilateral. How many diagonals were drawn, that is, what is
k?

(a) Cannot be determined from the information given

(b) n2 − 2 X

(c) n4 − 1

(d) n − 4

(e) n2 − 9.5n + 22

CSS 2016 Common Part Page 6 of 19

15. In a tournament with 7 teams, each team plays one match with every other team.
For each match, the team earns two points if it wins, one point if it ties, and no points
if it loses. At the end of all matches, the teams are ordered in the descending order
of their total points (the order among the teams with the same total are determined
by a whimsical tournament referee). The first three teams in this ordering are then
chosen to play in the next round. What is the minimum total number of points a
team must earn in order to be guaranteed a place in the next round?

(a) 13 (b) 12 (c) 11 (d) 10 X (e) 9

CSS 2016 Computer Science Page 7 of 19

Part B: Computer Science

1. A Boolean formula is said to be a tautology if it evaluates to TRUE for all assignments
to its variables. Which one of the following is NOT a tautology?

(a) ((p ∨ q) ∧ (r ∨ s)) ⇒ ((p ∧ r) ∨ q ∨ s)

(b) ((p ∨ q) ∧ (r ∨ s)) ⇒ (q ∨ s) X

(c) ((p ∨ q) ∧ (r ∨ s)) ⇒ (r ∨ q ∨ s)

(d) ((p ∨ q) ∧ (r ∨ s)) ⇒ (p ∨ q ∨ s)

(e) ((p ∨ q) ∧ (r ∨ s)) ⇒ (p ∨ q)

2. Which language class has the following properties?

It is closed under union and intersection but not complement.

(a) Regular language

(b) Context-free languages

(c) Recursive languages

(d) Recursively enumerable languages X

(e) Languages that are not recursively enumerable

3. Assume P 6= N P . Which of the following is not TRUE?

(a) 2-SAT is in NP

(b) 2-SAT is in coNP

(c) 3-SAT is polyonomial-time reducible to 2-SAT X

(d) 4-SAT is polynomial-time reducible to 3-SAT

(e) 2-SAT is in P

CSS 2016 Computer Science Page 8 of 19

4. In the following, A stands for a set of apples, and S(x, y) stands for “x is sweeter
than y”. Let

Ψ ≡ ∃x : x ∈ A
Φ ≡ ∀x ∈ A : ∃y ∈ A : S(x, y).

Which of the following statements implies that there are infinitely many apples (i.e.,
A is an infinite set)?

(a) Ψ ∧ Φ ∧ [∀x ∈ A : ¬S(x, x)]

(b) Ψ ∧ Φ ∧ [∀x ∈ A : S(x, x)]

(c) Ψ ∧ Φ ∧ [∀x, y ∈ A : S(x, x) ∧ S(x, y) → S(y, y)]

(d) Ψ ∧ Φ ∧ [∀x ∈ A : ¬S(x, x)] ∧ [∀x, y, z ∈ A : S(x, y) ∧ S(y, z) → S(y, x)]

(e) Ψ ∧ Φ ∧ [∀x ∈ A : ¬S(x, x)] ∧ [∀x, y, z ∈ A : S(x, y) ∧ S(y, z) → S(x, z)] X

5. Consider the following recursive function mc91.

int mc91(int n)
{
print n
if (n > 100) {
return n - 10;
}
else {
return mc91(mc91(n+11));
}
}

Let

Out = {n : there is an x ∈ {0, 1, . . . , 100} such that n is one of the integers
printed by mc91(x)}.

Then, which of the following is Out?

(a) {n : −∞ < n ≤ 100}

(b) {n : 0 ≤ n ≤ 101}

(c) {n : 0 ≤ n ≤ 110}

(d) {n : 0 ≤ n ≤ 111} X

(e) {n : 0 ≤ n < +∞}

CSS 2016 Computer Science Page 9 of 19

6. A subset X of Rn is convex if for all x, y ∈ X and all λ ∈ (0, 1), we have λx+(1−λ)y ∈
X. If X is a convex set, which of the following statements is necessarily TRUE?

(a) For every x ∈ X, there exist y, z ∈ X −{x} and λ ∈ (0, 1) so that x = λy+(1−λ)z

(b) If x, y ∈ X and λ ≥ 0, then λx + (1 − λ)y ∈ X

(c) If x1 , . . . , xn ∈ X (n ≥ 1), then (x1 + · · · + xn )/n ∈ X X

(d) If x ∈ X, then λx ∈ X for all scalars λ

(e) If x, y ∈ X, then x − y ∈ X

7. Let n = m!. Which of the following is TRUE?

(a) m = Θ(log n/ log log n) X

(b) m = Ω(log n/ log log n) but not m = O(log n/ log log n)

(c) m = Θ(log2 n)

(d) m = Ω(log2 n) but not m = O(log2 n)

(e) m = Θ(log1.5 n)

8. Consider the following language
 
 
PRIMES = 111 · · · 11 : p is prime .
| {z } 
p times

Then, which of the following is TRUE?

(a) PRIMES is regular

(b) PRIMES is undecidable

(c) PRIMES is decidable in polynomial time X

(d) PRIMES context free but not regular

(e) PRIMES is NP-complete and P 6= NP

CSS 2016 Computer Science Page 10 of 19

9. Which of the following graphs DOES NOT have an Eulerian circuit? (Recall that an
Eulerian circuit in an undirected graph is a walk in the graph that starts at a vertex
and returns to the vertex after travelling on each edge exactly once.)

(a) K9,9 X

(b) K8,8

(c) K12,12

(d) K9

(e) The graph G on vertex set {1, 2, . . . , 9} with edge set

E(G) = {{i, j} : 1 ≤ i < j ≤ 5 or 5 ≤ i < j ≤ 9}.

10. A vertex cover in an undirected graph G is a subset C ⊆ V (G) such that every edge
of G has an endpoint in C. An independent set in G is a subset I ⊆ V (G) such that
no edge has both its endpoints in I. Which of the following is TRUE of every graph
G and every vertex cover C of G?

(a) There exists an independent set of size |C|

(b) V (G) − C is an independent set X

(c) |C| ≥ |E(G)|/2

(d) |C| ≥ |V (G)|/2

(e) C intersects every independent set

11. Let n ≥ 4 be an integer. Regard the set Rn as a vector space over R. Consider the
following undirected graph H.

V (H) = {S ⊆ Rn : S is a basis for Rn };
E(H) = {{S, T } : |S \ T | = 1 and |T \ S| = 1},

where S \ T = {x ∈ S : x 6∈ T }. Which of the following statements is FALSE?

(a) H has an infinite number of vertices

(b) The diameter of H is infinite X

(c) H is connected

(d) H contains an infinite clique

(e) H contains an infinite independent set

CSS 2016 Computer Science Page 11 of 19

12. A computer program computes a function f : {0, 1}∗ × {0, 1}∗ → {0, 1}∗ . Suppose
f (a, b) has length |b|2 , where |a| and |b| are the lengths of a and b. Suppose, using
this program, the following computation is performed.

x = "01"
for i = 1,...,n do
x = f("01", x)

Suppose at the end, the length of the string x is t. Which of the following is TRUE
(assume n ≥ 10)?

(a) t ≤ 2n

(b) n < t ≤ n2

(c) n2 < t ≤ nlog2 n
n
(d) nlog2 n < t ≤ 2(2 ) X
n
(e) 2(2 ) < t

13. An undirected graph G = (V, E) is said to be k-colourable if there exists a mapping
c : V → {1, 2, . . . , k} such that for every edge {u, v} ∈ E we have c(u) 6= c(v). Which
of the following statements is FALSE?

(a) G is |V |-colourable

(b) G is 2-colourable iff there are no odd cycles in G

(c) G is (∆ + 1)-colourable where ∆ is the maximum degree in G

(d) There is a polynomial time algorithm to check if G is 2-colourable

(e) If G has no triangle then it is 3-colourable X

14. Consider a family F of subsets of {1, 2, . . . , n} such that for any two distinct sets
A and B in F we have: A ⊂ B or B ⊂ A or A ∩ B = ∅. Which of the following
statements is TRUE? (Hint: what does the Venn diagram of this family look like?)

(a) |F| ≤ 2n and there exists a family F such that |F| = 2n X

(b) |F| ≤ n2 and there exists a family F such that |F| = n2

(c) |F| ≤ 2n2 and there exists a family F such that |F| = 2n2

(d) |F| ≤ 2n−1 and there exists a family F such that |F| = 2n−1

(e) None of the above

CSS 2016 Computer Science Page 12 of 19

15. Let G be an undirected graph. For a pair (x, y) of distinct vertices of G, let
mincut(x, y) be the least number of edges that should be deleted from G so that
the resulting graph has no x-y path. Let a, b, c be three vertices in G such that
mincut(a, b) ≤ mincut(b, c) ≤ mincut(c, a). Consider the following possibilities:

(i) mincut(a, b) < mincut(b, c) < mincut(c, a)

(ii) mincut(a, b) = mincut(b, c) < mincut(c, a)

(iii) mincut(a, b) < mincut(b, c) = mincut(c, a)

(iv) mincut(a, b) = mincut(b, c) = mincut(c, a)

Which of the following is TRUE?

(a) All of (i), (ii), (iii), (iv) are possible

(b) (i), (ii), (iii) are possible but not (iv)

(c) (i) and (iv) are possible but neither (ii) nor (iii)

(d) (ii) and (iv) are possible but neither (i) nor (iii) X

(e) (iii) and (iv) are possible but neither (i) nor (ii)

CSS 2015 Systems Science Page 13 of 19

Part C: Systems Science
R∞
1. Suppose f (x) = cx−α for some c > 0 and α > 0 such that 1
f (x)dx = 1. Then,
which of the following is possible?
R∞
(a) 1 xf (x)dx = ∞ X
R ∞ (x)
(b) 1 f1+x dx = ∞
R∞
(c) 1 (ln x)f (x)dx = ∞
R ∞ f (x)
(d) 1 1+ln x
dx = ∞
(e) None of the above

2. Let X1 and X2 be two independent continuous real-valued random ( variables taking
1 if X1 = Y
values in the unit interval [0, 1]. Let Y = max{X1 , X2 } and Z = .
2 otherwise
Which of the following is true?
(a) Pr[Z = 1] = Pr[Z = 1 | Y ≥ 0.3] X
(b) Pr[Z = 1] = Pr[Z = 1 | Y = 0.3, X1 = 0.2]
(c) Pr[Z = 1] = Pr[Z = 1 | Y = 0.3, X1 = 0.3, X2 = 0.2]
(d) Pr[Z = 1] > Pr[Z = 2] = 21
(e) Pr[Z = 1] < Pr[Z = 2]

3. Let (X, Y ) be a pair of independent random variables. Suppose X takes values in
{1, . . . , 6} with equal probability, and Y takes values in {2, 3} with Pr[Y = 2] = p.
Let Z = (X mod Y ) + 1.
Which of the statements is true?
(a) Pr[Z = 1] = 25 for some value of p X
(b) Pr[Z = 1] = 12 for no value of p
(c) Pr[Z = 1] = 12 for p = 21
(d) Pr[Z = 1] = p(1 − p)
(e) None of the above

CSS 2015 Systems Science Page 14 of 19

4. Consider a system which in response to input x(t) outputs

y(t) = x(t2 ).

Which of the following describes the system?

(a) linear, time-invariant, causal

(b) linear, time-invariant, non-causal

(c) linear, time-variant X

(d) non-linear, time-invariant

(e) non-linear, time-variant

5. Consider the opamp circuit in the figure. Approximately what is Vo ?

R 4R
V1

2R
V2 −
Vo
+

V1


(a) − 2
+ V2
V1
+ V22


(b) − 4
(c) − (V1 + 2V2 )
(d) − (4V1 + 2V2 ) X
(e) None of the above

6. What is the Laplace transform F (s) of the signal f (t), t ≥ 0 defined below? In
t ∈ [0, 1),

1, t ∈ [0, 21 )

f (t) =
0, t ∈ [ 12 , 1)

and in t ≥ 1, f (t) = f (t − n), n ≤ t < n + 1, n = 1, 2, 3, . . .
−s/2
(a) 1−es
−s/2
(b) 1+es
1
(c) s(1−e −s )

CSS 2015 Systems Science Page 15 of 19
(d) s 1−e1−s/2
( )
(e) s 1+e1−s/2 X
( )

Please ignore the blank space below.
Problem 7 appears on page 16.

CSS 2015 Systems Science Page 16 of 19

7. Suppose X and Y are independent Gaussian random variables, whose pdfs are rep-
resented below.
0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0
−2 0 2 4 −2 0 2 4

Which of the following describes the pdf of the X + Y ?
0.8

0.6

(a) 0.4
0.2

0
−2 0 2 4

0.6

0.4

(b)
0.2

0
−2 0 2 4

0.6

0.4

(c) X
0.2

0
−2 0 2 4

0.3

(d) 0.2
0.1

−2 0 2 4

0.3

(e) 0.2
0.1

−2 0 2 4

CSS 2015 Systems Science Page 17 of 19

8. In terms of their frequency responses, which of the following is the odd one out?
C C

(a)
L/2

C C

(b)
L L

C

L L
(c)
C C

C

L

(d) 2C 2C X

L

(e) All four circuits are equivalent

CSS 2015 Systems Science Page 18 of 19

9. Suppose Y = X + Z, where X and Z are independent zero-mean random variables
each with variance 1. Let X̂(Y ) = aY be the optimal linear least-squares estimate
of X from Y , i.e., a is chosen such that E[(X − aY )2 ] is minimized. What is the
resulting minimum E[(X − X̂(Y ))2 ]?

(a) 1 (b) 32 (c) 12 X (d) 31 (e) 14

10. Let U1 , U2 , U3 be independent random variables that are each uniformly distributed
between zero and one. What is the probability that the second highest value amongst
the three lies between 1/3 and 2/3?

(a) 29 1
(b) 27 (c) 13
27
X (d) 31 7
(e) 18

11. Suppose that a random variable X has a probability density function (pdf) given by

f (x) = c exp(−2x)

for x ≥ 1, and f (x) = 0, for x < 1, where c is an appropriate constant so that f (x)
is a valid pdf.
What is the expected value of X given that X ≥ 5?
(a) 5 12 X (b) 7 (c) 10 (d) 8 12 (e) 6

12. Recall that the Shannon entropy of a random variables X taking values in a finite
set S is given by X
H[X] = − Pr[X = x] log2 Pr[X = x].
x∈S

(We set 0 log2 0 = 0.) For a pair of random variables (X, Y ) taking values in the
finite set S × T , we write
X
H[X | Y = y] = − Pr[X = x | Y = y] log2 Pr[X = x | Y = y]
x∈S
X
and H[X | Y ] = − Pr[Y = y]H[X | Y = y].
y∈T

Now, consider an 1024 × 1024 chess board. Suppose 1024 rooks are placed one after
another randomly at distinct locations on a 1024 × 1024 chess board so that no rook
attacks another: that is, the i-th rook (i = 1, 2, . . . , 1024) is placed at a location
chosen uniformly from among the available possibilities so that it does not attack
any of the previously placed rooks. Let Ri be the row number of the i-th rook and
Ci its column number. What is H[R513 , C513 | R1 , R2 , . . . , R512 ]?
(a) log2 513 (b) 9 (c) 10 (d) 19 X (e) 81

CSS 2015 Systems Science Page 19 of 19

13. Suppose m and n are positive integers, m 6= n, and A is an m × n matrix with real
entires. Consider the following statements.

(i) rank(AAT ) = rank(AT A)

(ii) det(AT A) = det(AAT )

(iii) Trace(AAT ) = Trace(AT A)

Which of the above statements is true for all such A?
(a) Only (i)

(b) Only (ii)

(c) Only (iii)

(d) (i) and (iii) X

(e) None of them

14. Consider matrices A ∈ Rn×m , B ∈ Rm×m , and C ∈ Rm×n . Let r = rank(ABC).
Which of the following must be true?
(a) min(m, n) ≤ r ≤ max(m, n)
(b) r ≤ min(m, n) X
(c) min(rank(A), rank(B), rank(C)) ≤ r ≤ max(m, n)
(d) min(m, n) ≤ r ≤ max(rank(A), rank(B), rank(C))
(e) None of the above

15. What is
 √  
√3 2 x
 
max x y
x,y 2 2 y

subject to

x2 + y 2 = 1?
√
(a) 1 (b) 2 (c) 2 (d) 3 (e) 4 X