Wbjee

Let f : [a, b] → R be such that f is differentiable in (a, b), f is continuous at x = a and x = b and moreover f(a) = 0 = f(b). Then 

(A) there exists at least one point c in (a, b) such that f'(c) = f(c) 

(B) f'(x) = f(x) does not hold at any point in (a, b) 

(C) at every point of (a, b), f'(x) > f(x) 

(D) at every point of (a, b), f'(x) < f(x)

Let f : [a, b] → R be differentiable on [a, b] and k ∈ R. Let f(a) = 0 = f(b). Also let J(x) = f'(x) + kf(x). Then 

(A) J(x) > 0 for all x ∈ [a, b] 

(B) J(x) < 0 for all x ∈ [a, b] 

(C) J(x) = 0 has at least one root in (a, b) 

(D) J(x) = 0 through (a, b)

A proton of mass ‘m’ moving with a speed v (<< c, velocity of light in vacuum) completes a circular orbit in time ‘T’ in a uniform magnetic field. If the speed of the proton is increased to 2v, what will be time needed to complete the circular orbit?

(A) √2T

(B) T

(C) T/√2

(D) T/2

Let ρ be a relation defined on N, the set of natural numbers, as ρ = {(x, y) ∈ N × N : 2x + y = 41} Then 

(A) ρ is an equivalence relation 

(B) ρ is only reflexive relation 

(C) ρ is only symmetric relation 

(D) ρ is not transitive

The number of selection of n objects from 2n objects of which n are identical and the rest are different is 

(A) 2ⁿ 

(B) 2^{n – 1} 

(C) 2^{n – 1} 

(D) 2^{n –1} + 1