Page 35 - ISC-12

12Jai Baba Ki ISC Mathematics – Class XII by Gupta–Bansal

52. Let S be the area of the region enclosed by y = e −x 2 , y = 0, x = 0, and x = 1. Then
1 1
(A) S ≥ . (B) S ≥ 1 − .
e e

1 1 1 1 1
(C) S ≤ 1 + √ . (D) S ≤ √ + √ 1 − √ .
4 e 2 e 2
[JEE 2012]

53. A ship is fitted with three engines E 1 , E 2 and E 3 . The engines function independently of each other
1 1 1
with respective probabilities , and . For the ship to be operational at least two of its engines must
2 4 4
function. Let X denote the event that the ship is operational and let X 1 , X 2 and X 3 denote respectively
the events that the engines E 1 , E 2 and E 3 are functioning. Which of the following is (are) true?

3
c
(A) P[X |X] = .
1
16
7
(B) P[Exactly two engines of the ship are functioning|X] = .
8
5 7
(C) P[X|X 2 ] = . (D) P[X|X 1 ] = .
16 16
[JEE 2012]

0
54. If y(x) satisfies the differential equation y − y tan x = 2x sec x and y(0) = 0, then
π π π π
2 2
(A) y = √ . (B) y 0 = .
4 8 2 4 18
π π π 4π 2π
2 2
(C) y = . (D) y 0 = + √ .
3 9 3 3 3 3
[JEE 2012]

55. For every integer n, let a n and b n be real numbers. Let function f : R → R be given by
(
a n + sin πx for x ∈ [2n, 2n + 1]
f(x) =
b n + cos πx for x ∈ (2n − 1, 2n)
for all integers n. If f is continuous, then which of the following hold(s) for all n?

(A) a n−1 − b n−1 = 0. (B) a n − b n = 1. (C) a n − b n+1 = 1. (D) a n−1 − b n = −1.
[JEE 2012]

56. If
Z x
2
t
f(x) = e (t − 2)(t − 3) dt, for all x ∈ (0, ∞),
0
then

(A) f has a local maximum at x = 2.
(B) f is decreasing on (2, 3).
00
(C) there exists some c ∈ (0, ∞) such that f (c) = 0.
(D) f has a local minimum at x = 3.

[JEE 2012]