ISRO CS 2017

Question 1

Suppose A is a finite set with n elements. The number of elements and the rank of the largest equivalence relation on A are

• {n,1}
• {n, n}
• {n², 1}
• {1, n²}

Question 2

Consider the set of integers I. Let D denote “divides with an integer quotient” (e.g. 4D8 but 4D7). Then D is

• Reflexive, not symmetric, transitive
• Not reflexive, not antisymmetric, transitive
• Reflexive, antisymmetric, transitive
• Not reflexive, not antisymmetric, not transitive

Question 3

A bag contains 19 red balls and 19 black balls. Two balls are removed at a time repeatedly and discarded if they are of the same colour, but if they are different, black ball is discarded and red ball is returned to the bag. The probability that this process will terminate with one red ball is

• 1
• 1/21
• 0
• 0.5

Question 4

If x = -1 and x = 2 are extreme points of f(x) = α log |x| + β x² + x then

• α = -6, β = -1/2
• α = 2, β = -1/2
• α = 2, β = 1/2
• α = -6, β =1/2

Question 5

If T(x) denotes x is a trigonometric function, P(x) denotes x is a periodic function and C(x) denotes x is a continuous function then the statement “It is not the case that some trigonometric functions are not periodic” can be logically represented as

• ¬∃(x) [ T(x) ⋀ ¬P(x) ]
• ¬∃(x) [ T(x) ⋁ ¬P(x) ]
• ¬∃(x) [ ¬T(x) ⋀ ¬P(x) ]
• ¬∃(x) [ T(x) ⋀ P(x) ]

Question 6

Let f(x) = log|x| and g(x) = sin x . If A is the range of f(g(x)) and B is the range of g(f(x)) then A ∩ B is

• [-1, 0]
• [-1, 0)
• [-∞, 0]
• [-∞,1]

Question 7

The proposition (P⇒Q)⋀(Q⇒P) is a

• tautology
• contradiction
• contingency
• absurdity

Question 8

The number of elements in the power set of { {1, 2}, {2, 1, 1}, {2, 1, 1, 2} } is

• 3
• 8
• 4
• 2

Question 9

The function f: [0,3]→[1,29] defined by f(x) = 2x³ - 15x² + 36x + 1 is

• injective and surjective
• surjective but not injective
• injective but not surjective
• neither injective nor surjective

Question 10

If log(base a) 2 = x and log(base a) 8 = y, what is the value of log(base a) 16? 

• 2x + y 

• 2x - y 

• x + y 

• 4x