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
       Aptitude       Numerical
Question 2
A
(a, b)≤ ǁbǁ
B
(a, b)≤ ǁaǁ
C
(a, b) = ǁaǁǁbǁ
D
(a, b)≥ ǁbǁ
E
(a, b)≥ ǁaǁ
       Engineering-Mathematics       Vectors
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
       Aptitude       Numerical
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)
       Algorithms       Asymptotic-Complexity
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)
       Engineering-Mathematics       Combinatorics
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!
       Engineering-Mathematics       Combinatorics
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
       Algorithms       Recurrences
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
       Aptitude       Numerical
Question 9
A
B
α2
C
6α(1 − α)
D
2(1 − α)
E
6α(1 − α)+ α2
       Engineering-Mathematics       Probability-and-statistics
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
       Engineering-Mathematics       Relations-and-Functions
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
       Engineering-Mathematics       Relations-and-Functions
There are 11 questions to complete.

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