KVPY Government Jobs Exams

The roots of (x – 41)^49 + (x – 49)^41 + (x – 2009)^2009 = 0 are -
(A) all necessarily real
(B) non-real except one positive real root
(C) non-real except three positive real roots
(D) non-real except for three real roots of which exactly one is positive

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KVPY Other Question

Let f : R → R be a differentiable function such that f (a) = 0 = f (b) and f ′(a) f ′(b) > 0 for some a < b. Then
the minimum number of roots of f ′(x = 0 in the interval (a, b) is-
(A) 3 (B) 2 (C) 1 (D) 0 Consider the regions A = {(x, y) | x^2 + y^2 ≤ 100} and = {(x, y) | sin (x + y) > 0} in the plane. Then the area
of the region A ∩ B is-
(A) 10 π (B) 100 (C) 100 π (D) 50 π