## TIFR PHD CS & SS 2017

Question 1 |

A suitcase weighs one kilogram plus half of its weight. How much does the suitcase weigh?

1.333... kilograms | |

1.5 kilograms | |

1.666... kilograms | |

2 kilograms | |

cannot be determined from the given data |

Question 2 |

(a, b)≤ ǁbǁ | |

(a, b)≤ ǁaǁ | |

(a, b) = ǁaǁǁbǁ | |

(a, b)≥ ǁbǁ | |

(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?

t/√2 | |

√2t | |

2t | |

h/t | |

h/2t |

Question 4 |

Which of the following functions asymptotically grows the fastest as n goes to inﬁnity?

(log log n)! | |

(log log n) ^{log n} | |

(log log n) ^{log log log n} | |

(log n) ^{log log n} | |

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?

3 ^{35} | |

3 ^{50} − 2^{50} | |

(35 2) | |

(50 15)*3 ^{35} | |

(37 2) |

Question 6 |

How many distinct words can be formed by permuting the letters of the word ABRACADABRA ?

11! / (5! 2! 2!) | |

11! / (5! 4!) | |

11! 5! 2! 2! | |

11! | |

11! 5! 4! |

Question 7 |

Consider the sequence S0, S1, S2,... deﬁned as follows: S0 = 0, S1 = 1, and Sn = 2S

_{n−1}+ S_{n−2}for n ≥ 2. Which of the following statements is FALSE?for every n ≥ 1, S _{2n} is even | |

for every n ≥ 1, S _{2n+1} is odd | |

for every n ≥ 1, S _{3n} is a multiple of 3 | |

for every n ≥ 1, S _{4n} is a multiple of 6 | |

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?

3.90 inches | |

2√2 inches | |

2√3 inches | |

4.1 inches | |

none of the above |

Question 9 |

3α | |

α ^{2} | |

6α(1 − α) | |

3α ^{2(1 − α)} | |

6α(1 − α)+ α ^{2} |

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?

f cannot be one-to-one (injective) | |

f cannot be onto (surjective) | |

f is both one-to-one and onto (bijective) | |

there is no such f possible | |

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?

f is onto (surjective) | |

f is one-to-one (injective) | |

f is both one-to-one and onto (bijective) | |

the range of f is ﬁnite | |

the domain of f is ﬁnite |