## TIFR PHD CS & SS 2017

 Question 1
A suitcase weighs one kilogram plus half of its weight. How much does the suitcase weigh?
 A 1.333... kilograms B 1.5 kilograms C 1.666... kilograms D 2 kilograms E cannot be determined from the given data
 Question 2
 A (a, b)≤ ǁbǁ B (a, b)≤ ǁaǁ C (a, b) = ǁaǁǁbǁ D (a, b)≥ ǁbǁ E (a, b)≥ ǁaǁ
 Question 3
On planet TIFR, the acceleration of an object due to gravity is half that on planet earth. An object on planet earth dropped from a height h takes time t to reach the ground. On planet TIFR, how much time would an object dropped from height h take to reach the ground?
 A t/√2 B √2t C 2t D h/t E h/2t
 Question 4
Which of the following functions asymptotically grows the fastest as n goes to inﬁnity?
 A (log log n)! B (log log n)log n C (log log n)log log log n D (log n)log log n E 2√(log log n)
 Question 5
How many distinct ways are there to split 50 identical coins among three people so that each person gets at least 5 coins?
 A 335 B 350 − 250 C (35 2) D (50 15)*335 E (37 2)
 Question 6
How many distinct words can be formed by permuting the letters of the word ABRACADABRA ?
 A 11! / (5! 2! 2!) B 11! / (5! 4!) C 11! 5! 2! 2! D 11! E 11! 5! 4!
 Question 7
Consider the sequence S0, S1, S2,... deﬁned as follows: S0 = 0, S1 = 1, and Sn = 2Sn−1 + Sn−2 for n ≥ 2. Which of the following statements is FALSE?
 A for every n ≥ 1, S2n is even B for every n ≥ 1, S2n+1 is odd C for every n ≥ 1, S3n is a multiple of 3 D for every n ≥ 1, S4n is a multiple of 6 E none of the above
 Question 8
In a tutorial on geometrical constructions, the teacher asks a student to construct a right-angled triangle ABC where the hypotenuse BC is 8 inches and the length of the perpendicular dropped from A onto the hypotenuse is h inches, and oﬀers various choices for the value of h. For which value of h can such a triangle NOT exist?
 A 3.90 inches B 2√2 inches C 2√3 inches D 4.1 inches E none of the above
 Question 9
 A 3α B α2 C 6α(1 − α) D 3α2(1 − α) E 6α(1 − α)+ α2
 Question 10
For a set A, deﬁne (A) to be the set of all subsets of A. For example, if A = { 1, 2 }, then P(A) = {Φ,{1, 2},{1},{2}}. Let f : A -->P(A) be a function and A is not empty. Which of the following must be TRUE?
 A f cannot be one-to-one (injective) B f cannot be onto (surjective) C f is both one-to-one and onto (bijective) D there is no such f possible E if such a function f exists, then A is inﬁnite
 Question 11
Let fog denote function composition such that (fog)(x) = f (g(x)). Let f :A --> B such that for all g : B --> A and h : B-->A we have fog = foh ==> g=h. Which of the following must be TRUE?
 A f is onto (surjective) B f is one-to-one (injective) C f is both one-to-one and onto (bijective) D the range of f is ﬁnite E the domain of f is ﬁnite
There are 11 questions to complete.

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