TIFR PHD CS & SS 2020
Question 1 
Two balls are drawn uniformly at random without replacement from a set of five balls numbered 1, 2, 3, 4, 5. What is the expected value of the larger number on the balls drawn?
2.5  
3  
3.5  
4  
None of the above 
Question 2 
Let M be a real nXn matrix such that for every nonzero vector xεR^{n}, we have x^{T} Mx > 0. Then
Such an M cannot exist.  
Such M s exist and their rank is always n.  
Such M s exist, but their eigenvalues are always real.  
No eigenvalue of any such M can be real.  
None of the above. 
Question 5 
Let A be an nXn invertible matrix with real entries whose column sums are all equal to 1. Consider the following statements:
(1) Every column in the matrix A^{2} sums to 2.
(2) Every column in the matrix A^{3} sums to 3.
(3) Every column in the matrix A^{1} sums to 1.
Which of the following is TRUE?
(1) Every column in the matrix A^{2} sums to 2.
(2) Every column in the matrix A^{3} sums to 3.
(3) Every column in the matrix A^{1} sums to 1.
Which of the following is TRUE?
none of the statements (1), (2), (3) is correct  
statement (1) is correct but not statements (2) or (3)  
statement (2) is correct but not statements (1) or (3)  
statement (3) is correct but not statements (1) or (2)  
all the 3 statements (1), (2), and (3) are correct

Question 6 
What is the maximum number of regions that the plane R^{2} can be partitioned into using 10 lines?
Hint: Let A(n) be the maximum number of partitions that can be made by n lines. Observe that A(0) = 1, A(2) = 2, A(2) = 4 etc. Come up with a recurrence equation for A(n).
Hint: Let A(n) be the maximum number of partitions that can be made by n lines. Observe that A(0) = 1, A(2) = 2, A(2) = 4 etc. Come up with a recurrence equation for A(n).
25  
50  
55  
56  
1024 
Question 7 
A lottery chooses four random winners. What is the probability that at least three of them are born on the same day of the week? Assume that the pool of candidates is so large that each winner is equally likely to be born on any of the seven days of the week independent of the other winners
17 / 2401
 
48 / 2401
 
105 / 2401  
175 / 2401  
294 / 2401 
Question 8 
Consider a function f : [0, 1] > [0, 1] which is twice differentiable in (0, 1). Suppose it has exactly one global maximum and exactly one global minimum inside (0, 1). What can you say about the behaviour of the first derivative f' and and second derivative f'' on (0, 1) (give the most precise answer)?
f' is zero at exactly two points, f'' need not be zero anywhere  
f' is zero at exactly two points, f'' is zero at exactly one point  
f' is zero at at least two points, f'' is zero at exactly one point  
f' is zero at at least two points, f'' is zero at at least one point  
f' is zero at at least two points, f'' is zero at at least two points 
Question 9 
A contiguous part, i.e., a set of adjacent sheets, is missing from Tharoor’s GRE preparation book. The number on the first missing page is 183, and it is known that the number on the last missing page has the same three digits, but in a different order. Note that every sheet has two pages, one at the front and one at the back. How many pages are missing from Tharoor’s book?
45  
135  
136  
198  
450 
Question 10 
In a certain year there were exactly four Fridays and exactly four Mondays in January. On what day of the week did the 20th of January fall that year (recall that January has 31 days)?
Sunday  
Monday  
Wednesday  
Friday  
None of the others

Question 11 
Suppose we toss m = 3 labelled balls into n = 3 numbered bins. Let A be the event that the first bin is empty while B be the event that the second bin is empty. P(A) and P(B) denote their respective probabilities. Which of the following is true?
P(A) > P(B)  
P(A) = 1/27  
P(A) > P(AB)  
P(A) < P(AB)  
None of the above 
There are 11 questions to complete.