One ticket is selected at random from 50 tickets numbered 00, 01, 02, ... , 49. Then the probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is zero, equals

•  1/14

• 1/7

• 5/14

• 5/14

Statement 1: ~ (p ↔ ~ q) is equivalent to p ↔ q
Statement 2 : ~ (p ↔ ~ q) is a tautology

• Statement–1 is true, Statement–2 is true, Statement–2 is a correct explanation for statement–1

• Statement–1 is true, Statement–2 is true; Statement–2 is not a correct explanation for statement–1.

• Statement–1 is true, statement–2 is false.

• Statement–1 is true, statement–2 is false.

Statement 1: The variance of first n even natural numbers is 
Statement 2: The sum of first n natural numbers is  and the sum of squares of first n natural numbers is 

• Statement–1 is true, Statement–2 is true, Statement–2 is a correct explanation for statement–1

• Statement–1 is true, Statement–2 is true; Statement–2 is not a correct explanation for statement–1.

• Statement–1 is true, statement–2 is false.

• Statement–1 is true, statement–2 is false.

Let a, b, c be such that 0 (a +c) ≠ . If ,then the value of 'n' is 

• 0

• any even integer

• any odd integer

• any odd integer

• π/2

• 1

• -1

• -1

For real x, let f(x) = x³+ 5x + 1, then

• f is one–one but not onto R

• f is onto R but not one–one

• f is one–one and onto R

• f is one–one and onto R

Let y be an implicit function of x defined by x^2x – 2x^xcoty – 1 = 0. Then y′ (1) equals 

• -1

• 1

• log 2

• log 2

Given P(x) = x⁴+ ax³ + cx + d such that x = 0 is the only real root of P′ (x) = 0. If P(–1) < P(1),then in the interval [–1, 1].

• P(–1) is the minimum and P(1) is the maximum of P

• P(–1) is not minimum but P(1) is the maximum of P

• P(–1) is the minimum but P(1) is not the maximum of P

• P(–1) is the minimum but P(1) is not the maximum of P

Let f(x) = (x + 1)2– 1, x ≥ – 1
Statement – 1: The set {x : f(x) = f–1(x)} = {0, –1}.
Statement – 2: f is a bijection.

• Statement–1 is true, Statement–2 is true,Statement–2 is a correct explanation for statement–1 

• Statement–1 is true, Statement–2 is true; Statement–2 is not a correct explanation for statement–1.

• Statement–1 is true, statement–2 is false.

• Statement–1 is true, statement–2 is false.

20.

Let f(x) = x|x| and g(x) = sinx

Statement 1 : gof is differentiable at x = 0 and its derivative is continuous atthat point
Statement 2: gof is twice differentiable at x = 0

• Statement–1 is true, Statement–2 is true, Statement–2 is a correct explanation for statement–1

• Statement–1 is true, Statement–2 is true;Statement–2 is not a correct explanation for statement–1.

• Statement–1 is true, statement–2 is false.

• Statement–1 is true, statement–2 is false.