## 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?
 A 2.5 B 3 C 3.5 D 4 E None of the above
Engineering-Mathematics       Probability-and-statistics
 Question 2
Let M be a real nXn matrix such that for every non-zero vector xεRn, we have xT Mx > 0. Then
 A Such an M cannot exist. B Such M s exist and their rank is always n. C Such M s exist, but their eigenvalues are always real. D No eigenvalue of any such M can be real. E None of the above.
Engineering-Mathematics       Linear-Algebra
 Question 3 A a B b C c D d E e
Engineering-Mathematics       Polynomials-and-vectors
 Question 4  A a B b C c D d E e
Engineering-Mathematics       Probability-and-statistics
 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 A2 sums to 2.
(2) Every column in the matrix A3 sums to 3.
(3) Every column in the matrix A-1 sums to 1.
Which of the following is TRUE?
 A none of the statements (1), (2), (3) is correct B statement (1) is correct but not statements (2) or (3) C statement (2) is correct but not statements (1) or (3) D statement (3) is correct but not statements (1) or (2) E all the 3 statements (1), (2), and (3) are correct
Engineering-Mathematics       Linear-Algebra
 Question 6
What is the maximum number of regions that the plane R2 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).
 A 25 B 50 C 55 D 56 E 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
 A 17 / 2401 B 48 / 2401 C 105 / 2401 D 175 / 2401 E 294 / 2401
Engineering-Mathematics       Probability-and-statistics
 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)?
 A f' is zero at exactly two points, f'' need not be zero anywhere B f' is zero at exactly two points, f'' is zero at exactly one point C f' is zero at at least two points, f'' is zero at exactly one point D f' is zero at at least two points, f'' is zero at at least one point E f' is zero at at least two points, f'' is zero at at least two points
Engineering-Mathematics       Calculus
 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?
 A 45 B 135 C 136 D 198 E 450
Aptitude       Numerical
 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)?
 A Sunday B Monday C Wednesday D Friday E None of the others
Aptitude       Numerical
 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?
 A P(A) > P(B) B P(A) = 1/27 C P(A) > P(A|B) D P(A) < P(A|B) E None of the above
Engineering-Mathematics       Probability-and-statistics
There are 11 questions to complete.

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