Page 37 - ISC-12

14Jai Baba Ki ISC Mathematics – Class XII by Gupta–Bansal
11
62. Let E and F be two independent events. The probability that exactly one of them occurs is and
25
2
the probability of none of them occurring is . If P(T) denotes the probability of occurrence of the
25
event T, then
4 3
(A) P(E) = , P(F) = .
5 5
1 2
(B) P(E) = , P(F) = .
5 5
2 1
(C) P(E) = , P(F) = .
5 5
3 4
(D) P(E) = , P(F) = .
5 5
[JEE 2011]

63. Let f be a real-valued function defined on the interval (0, ∞) by
x √
Z
f(x) = ln x + 1 + sin t dt.
0
Then which of the following statement(s) is (are) true?

00
(A) f (x) exists for all x ∈ (0, ∞).
0
0
(B) f (x) exists for all x ∈ (0, ∞) and f is continuous on (0, ∞), but not differentiable on (0, ∞).
0
(C) there exists α > 1 such that |f (x)| < |f(x)| for all x ∈ (α, ∞).
0
(D) there exists β > 0 such that |f(x)| + |f (x)| ≤ β for all x ∈ (0, ∞).
[JEE 2010]

x
64. Area of the region bounded by the curve y = e and lines x = 0 and y = e is
(A) e − 1.
Z e
(B) ln (e + 1 − y) dy.
1
Z 1
x
(C) e − e dx.
0
e
Z
(D) ln y dy.
1
[JEE 2009]

65. If
Z π sin nx
I n = dx, n = 0, 1, 2, . . . ,
x
(1 + π ) sin x
−π
then
10
P
(A) I n = I n+2 . (B) I 2m+1 = 10π.
m=1
10
P
(C) I 2m = 0. (D) I n = I n+1 .
m=1
[JEE 2009]