## TIFR 2021

 Question 1
A box contains 5 red marbles, 8 green marbles, 11 blue marbles, and 15 yellow marbles. We draw marbles uniformly at random without replacement from the box. What is the minimum number of marbles to be drawn to ensure that out of the marbles drawn, at least 7 are of the same colour?
 A 7 B 8 C 23 D 24 E 39
 Question 2
The maximum area of a rectangle inscribed in the unit circle (i.e., all the vertices of the rectangle are on the circle) in a plane is:
 A 1 B 2 C 3 D 4 E 5
 Question 3
Let M be an n  m real matrix. Consider the following:
* Let k1 be the smallest number such that M can be factorized as AB, where A is an n x k1 and B is a k1 x m matrix.
*Let k2 be the smallest number such that M = Pk2 i=1 uivi, where each ui is an Nx1 matrix and each vi is an 1 x m matrix.
* Let k3 be the column-rank of M.
Which of the following statements is true?
 A k11 < k2 < k3 B k11 < k3 < k2 C k2 = k3 < k1 D k1 = k2 = k3 E No general relationship exists among k1; k2 and k3.
 Question 4
What is the probability that at least two out of four people have their birthdays in the same month, assuming their birthdays are uniformly distributed over the twelve months?
 A 25/48 B 5/5 C 5/12 D 41/96 E 55/96
 Question 5
A matching in a graph is a set of edges such that no two edges in the set share a common vertex. Let G be a graph on n vertices in which there is a subset M of m edges which is a matching. Consider a random process where each vertex in the graph is independently selected with probability 0 < p < 1 and let B be the set of vertices so obtained. What is the probability that there exists at least one edge from the matching M with both end points in the set B?
 A P2 B 1-(1-P2)m C P2m D (1-P2)m E 1-(1-p(1-p))m
 Question 6
Let d be the number of positive square integers (that is, it is a square of some integer) that are factors of 205 X 215. Which of the following is true about d?
 A 50 B 100 < d < 150 C 150 D 200 E 300 < d
 Question 7
 A The sequence fyng does not have a limit as n ! 1. B yn < 1 for all n = 2; 3; 4; : : : C limn!1 yn exists and is equal to 6=2. D limn!1 yn exists and is equal to 0. E The sequence {yn} first increases and then decreases as n takes values 2; 3; 4; : : :
 Question 8
Fix n > 6. Consider the set C of binary strings x1x2... xn of length n such that the bits satisfy the following set of equalities, all modulo 2: xi + xi+1 + xi+2 = 0 for all 1 < i < n - 2, xn-1 + xn + x1 = 0, and xn + x1 + x2 = 0. What is the size of the set C?
 A 1 for all n > 6 B 1 for all n >6 C 0 for all n> 6 D If n > 6 is divisible by 3 then |C|= 1. If n > 6 is not divisible by 3 then |C| = 4. E If n > 6 is divisible by 3 then |C| = 4. If n > 6 is not divisible by 3 then |C| = 1.
 Question 9
Lavanya and Ketak each flip a fair coin n times. What is the probability that Lavanya sees more heads than Ketak? In the following, the binomial coefficient (n/k) counts the number of k-element subsets of an n-element set.
 Question 10
Find the following sum.
 A 20/41 B 10/41 C 10/21 D 20/21 E 1
 Question 11
How many numbers in the range f0; 1; : : : ; 1365g have exactly four 1’s in their binary representation? (Hint: 136510 is 101010101012, that is, 1365 = 210 + 28 + 26 + 24 + 22 + 20🙂 In the following, the binomial coefficient (n/k) counts the number of k-element subsetsof an n-element set.
 A (6/4) B (10/4) C (10/4) + (8/3) + (6/2) + (5/1) D (11/4) + (9/3) + (7/2) + (5/1) E 1024
 Question 12
What are the last two digits of 72021?
 A 67 B 07 C 27 D 01 E 77
 Question 13
Five married couples attended a party. In the party, each person shook hands with those they did not know. Everyone knows his or her spouse. At the end of the party, Shyamal, one of the attendees, listed the number of hands that other attendees including his spouse shook. He got every number from 0 to 8 once in the list. How many persons shook hands with Shyamal at the party?
 A 2 B 4 C 6 D 8 E Insufficient information
 Question 14
Let P be a convex polygon with sides 5, 4, 4, 3. For example, the following:

Consider the shape in the plane that consists of all points within distance 1 from
some point in P. If ` is the perimeter of the shape, which of the following is always
correct?
 A L cannot be determined from the given information B 20 < L< 21 C 21 D 22 E 23
 Question 15
Let n, m and k be three positive integers such that n> m > k. Let S be a subset of {1, 2......, n}of size k. Consider sampling a function f uniformly at random from the set of all functions mapping {1,....n} to {1....m}. What is the probability that f is not injective on the set S, i.e., there exist i; j 2 S such that f(i) = f(j)? In the following, the binomial coefficient (n/k) counts the number of k-element subsets of an n-element set.
 A 1- k!/kk B 1- m!/mk C 1-k!(m/k)/mk D 1-k!(n/k)/nk E 1-k!(n/k)/mk
 Question 16
Consider the following statements about propositional formulas.
(i) (p ^ q) ! r and (p ! r) ^ (q ! r) are not logically equivalent.
(ii) (:a ! b) ^ (:b _ (:a _ :b)) and :(a \$ b) are not logically equivalent.
 A Both (i) and (ii) are true. B (i) is true and (ii) is false. C (i) is false and (ii) is true. D Both (i) and (ii) are false. E Depending on the values of p and q, (i) can be either true or false, while (ii) is always false.
 Question 17
Let L be a singly-linked list and X and Y be additional pointer variables such that X points to the first element of L and Y points to the last element of L. Which of the following operations cannot be done in time that is bounded above by a constant?
 A Delete the first element of L. B Delete the last element of L. C Add an element after the last element of L. D Add an element before the first element of L. E Interchange the first two elements of L.
 Question 18
What is the prefix expression corresponding to the expression: ((9 + 8)  7 + (6  (5 + 4))  3) + 2? You may assume that  has precedence over +.
 A * + + 9 8 7 ** 6 + + 5 4 3 2 B * + ++ 9 8 7   6 + + 5 4 3 2 C +* + +9 8 7 **6 + 5 4 3 2 D ++* + 9 8 7 **6 + 5 4 3 2 E +*+* 9 8 7 ++6 * 5 4 3 2
 Question 19
Consider the following two languages. PRIME = f1n j n is a prime number g; FACTOR = f1n01a01b j n has a factor in the range [a; b]g: What can you say about the languages PRIME and FACTOR?
 A PRIME is in P, but FACTOR is not in P. B Neither PRIME nor FACTOR are in P. C Both PRIME and FACTOR are in P. D PRIME is not in P, but FACTOR is in P. E None of the above since we can answer this question only if we resolve the status of the NP vs. P question.
 Question 20
For a language L over the alphabet fa; bg, let L denote the complement of L and let L* denote the Kleene-closure of L. Consider the following sentences. (i) L and L are both context-free. (ii) L is not context-free but L is context-free. (iii) L is context-free but L is regular. Which of the above sentence(s) is/are true if L = {anbn | n > 0}?
 A Both (i) and (iii) B Only (i) C Only (iii) D Only (ii) E None of the above
 Question 21
Consider the following pseudocode: procedure HowManyDash(n) if n = 0 then print ‘-’ else if n = 1 then print ‘-’ else HowManyDash(n - 1) HowManyDash(n - 2) end if end procedure How many ‘-’ does HowManyDash(10) print?
 A 9 B 10 C 55 D 89 E 1024
 Question 22
Which of the following regular expressions defines a language that is different from the other choices?
 A b*(a + b)*a(a + b*ab*(a + b)* B a*(a + b)*ab*(a + b)*a(a + b)* C (a + b)*ab*(a + b)*a(a + b)*b* D (a + b)*a(a + b)*b*a(a + b)*a* E (a + b)*b*a(a + b)*b*(a + b)*
 Question 23
Let A and B be two matrices of size nn and with real-valued entries. Consider the following statements. 1. If AB = B, then A must be the identity matrix. 2. If A is an idempotent (i.e. A2 = A) nonsingular matrix, then A must be the identity matrix. 3. If A-1 = A, then A must be the identity matrix. Which of the above statements MUST be true of A?
 A 1, 2, and 3 B Only 2 and 3 C Only 1 and 2 D Only 1 and 3 E Only 2
 Question 24
Let L be a context-free language generated by the context-free grammar G = (V;E;R; S) where V is the finite set of variables, E the finite set of terminals (disjoint from V ), R the finite set of rules and S e V the start variable. Consider the context-free grammar G0 obtained by adding S ! SS to the set of rules in G. What must be true for the language L0 generated by G"?
 A L'= LL B L' = L C L' = L* D L' = {xx|x e L} E None of the above
 Question 25
Let G be a connected bipartite simple graph (i.e., no parallel edges) with distinct edge weights. Which of the following statements on MST (minimum spanning tree) need not be TRUE?
 A G has a unique MST. B Every MST in G contains the lightest edge. C Every MST in G contains the second lightest edge. D Every MST in G contains the third lightest edge E No MST in G contains the heaviest edge.
 Question 26
Suppose we toss a fair coin repeatedly until the first time by which at least two heads and at least two tails have appeared in the sequence of tosses made. What is the expected number of coin tosses that we would have to make?
 A 8 B 4 C 5.5 D 7.5 E 4.5
 Question 27
Let G be an undirected graph. For any two vertices u; v in G, let cut(u, v) be the minimum number of edges that should be deleted from G so that there is no path between u and v in the resulting graph. Let a; b; c; d be 4 vertices in G. Which of the following statements is impossible?
 A cut(a; b) = 3, cut(a; c) = 2, and cut(a; d) = 1 B cut(a, b) = 3, cut(b; c) = 1, and cut(b, d) = 1 C cut(a; b) = 3, cut(a; c) = 2, and cut(b; c) = 2 D cut(a; c) = 2, cut(b; c) = 2, and cut(c; d) = 2 E cut(b; d) = 2, cut(b; c) = 2, and cut(c; d) = 1.
 Question 28
Let A be a 36 matrix with real-valued entries. Matrix A has rank 3. We construct a graph with 6 vertices where each vertex represents a distinct column in A, and there is an edge between two vertices if the two columns represented by the vertices are linearly independent. Which of the following statements MUST be true of the graph constructed?
 A Each vertex has degree at most 2. B The graph is connected C There is a clique of size 3. D The graph has a cycle of length 4. E The graph is 3-colourable.
 Question 29
Consider the following greedy algorithm for colouring an n-vertex undirected graph G with colours c1; c2; : : :: consider the vertices of G in any sequence and assign the chosen vertex the first colour that has not already been assigned to any of its neighbours. Let m(n; r) be the minimum number of edges in a graph that causes this greedy algorithm to use r colours. Which of the following is correct?
 A m(n; r) = 0(r) B m(n; r) = 0(r[log2 r] C m(n; r) = (r/2) D m(n; r) = nr E m(n; r) = n(r/2)
 Question 30
Consider a system with input x(t) and output y(t) such that y(t) = t x(t): Consider the following statements: 1. The system is linear. 2. The system is time-invariant. 3. The system is causal. Then 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 statements 1 and 3 are correct. E All three statements 1, 2, and 3 are correct.
 Question 31
Given a fixed perimeter of 1, among the following shapes, which one has the largest area?
 A Square B A regular pentagon C A regular hexagon D A regular septagon E A regular octagon
 Question 32
 A Only Statement 1 is correct. B Only Statements 1 and 2 are correct. C Only Statements 1 and 3 are correct. D All of Statements 1, 2, and 3 are correct. E None of the three Statements 1,2, and 3 are correct.
 Question 33
 A Only (1) is correct B Only (1) and (2) are correct C All (1), (2) and (3) are correct. D Only (2) and (3) are correct. E None of the above
 Question 34
 A f < a B f > a C f < a-1 D f > a-1 E None of the above
 Question 35
Consider a fair coin (i.e., both heads and tails have equal probability of appearing). Suppose we toss the coin repeatedly until both sides have been seen. What is the expected number of times we would have seen heads?
 A 1 B 5/4 C 3/2 D 2 E None of the above
 Question 36
 A The function f(y) is non-positive for all y > 1. B The function f(y) first increases and then decreases with y for y > 1. C The function f(y) first decreases and then increases with y for y > 1 D The function f(y) oscillates infinitely often between negative and positive value for y > 1. E The derivative of function f(y) does not exist at y = 1.
 Question 37
The maximum area of a parallelogram inscribed in the ellipse (i.e. all the vertices of the parallelogram are on the ellipse) x2 + 4y2 = 1 is:
 A 2 B 4 C 1 D 5 E 3
 Question 38
A stick of length 1 is broken at a point chosen uniformly at random. Which of the following is false?
 A Twice the length of the smaller piece is greater than the length of the larger piece with positive probability B One half of the length of the larger piece is greater than the length of smaller piece with positive probability C The product of the length of the smaller piece and the larger piece is less than 1/4 in expectation D The ratio of the length of larger piece to the smaller piece is greater than 100 with positive probability. E The product of the length of the smaller piece and the larger piece is greater than 1/4 with positive probability
 Question 39
 A ||u - P1|| B ||u - P2|| C ||u - P2-P1|| D ||u - P2-P1|| E 0
 Question 40
Suppose that X1 and X2 denote the output of rolls of two independent dices that can each take integer values f1; 2; 3; 4; 5; 6g with probability 1=6 for each outcome. Further, U denotes a continuous random variable that is independent of X1 and X2 and is uniformly distributed in the interval [0; 1]. Suppose that the sum of the three random variables, that is, X1 + X2 + U, equals 6.63. Conditioned on this sum what is the probability that X1 equals 2?
 A 2.21 B 3 C 1/6 D 1/5 E 1/3
 Question 41
An ant does a random walk in a two dimensional plane starting at the origin at time 0. At every integer time greater than 0, it moves one centimeter away from its earlier position in a random direction independent of its past. After 4 steps, what is the expected square of the distance (measured in centimeters) from its starting point?
 A 4 B 1 C 2 D pi E 0
 Question 42
Consider a unit Euclidean ball in 4 dimensions, and let Vn be its volume and Sn its surface area. Then Sn=Vn is equal to:
 A 1 B 4 C 5 D 2 E 3
 Question 43
A tourist starts by taking one of the n available paths, denoted by 1; 2; ... ; n. An hour into the journey, the path i subdivides into further 1 + i subpaths, only one of which leads to the destination. The tourist has no map and makes random choices of the path and the subpaths. What is the probability of reaching the destination if n = 3?
 A 10/36 B 11/36 C 12/36 D 13/36 E 14/36
 Question 44
 A H(X) < 3 B H(X)e(3; 5] C H(X) e (5; 10] D H(X) > 10 but finite E H(X) is unbounded
There are 44 questions to complete.