Page 38 - ISC-12

Multiple Correct Choices Type Jai Baba Ki15

66. For the function
1
f(x) = x cos , x ≥ 1,
x

(A) at least one x in the interval [1, ∞), f(x + 2) − f(x) < 2.
0
(B) lim f (x) = 1.
x→∞
(C) for all x in the interval [1, ∞), f(x + 2) − f(x) > 2.
0
(D) f (x) is strictly decreasing in the interval [1, ∞).
[JEE 2009]

67. Let f(x) be a non-constant twice differentiable function defined on (−∞, ∞) such that

1
f(x) = f(1 − x) and f 0 = 0.
4
Then

1
00
(A) f (x) vanishes at least twice on [0, 1]. (B) f 0 = 0.
2
Z 1/2 1 Z 1/2 Z 1
(C) f x + sin x dx = 0. (D) f(t) e sin πt dt = f(1 − t) e sin πt dt.
−1/2 2 0 1/2
[JEE 2008]