Page 15 - ISC-12

12Jai Baba Ki ISC Mathematics – Class XII by Gupta–Bansal
r r
1 + sin x 1 − sin x
60. The area of the region between the curves y = and y = bounded by the lines
cos x cos x
π
x = 0 and x = is
4
√
Z 2−1 t
(A) √ dt.
2
0 (1 + t ) 1 − t 2
√
Z 2−1 4t
(B) √ dt.
2
0 (1 + t ) 1 − t 2
√
2+1 4t
Z
(C) √ dt.
2
0 (1 + t ) 1 − t 2
√
2+1 t
Z
(D) √ dt.
2
0 (1 + t ) 1 − t 2
[JEE 2008]
61. An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the
experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and
B are independent, is

(A) 2, 4 or 8. (B) 3, 6 or 9.

(C) 4 or 8. (D) 5 or 10.

[JEE 2008]

ˆ
62. Let two non-collinear unit vectors ˆa and b form an acute angle. A point P moves so that at any time t
−→
ˆ
the position vector OP (where O is the origin) is given by ˆa cos t + b sin t. When P is farthest from
−→ −→
origin O, let M be the length of OP and ˆu be the unit vector along OP. Then,
ˆ
ˆ
ˆ a + b ˆ a − b
ˆ 1/2
ˆ 1/2
(A) ˆu = and M = (1 + ˆa · b) . (B) ˆu = and M = (1 + ˆa · b) .
ˆ
ˆ
|ˆa + b| |ˆa − b|
ˆ
ˆ
ˆ a + b ˆ a − b
ˆ 1/2
ˆ 1/2
(C) ˆu = and M = (1 + 2ˆa · b) . (D) ˆu = and M = (1 + 2ˆa · b) .
ˆ
ˆ
|ˆa + b| |ˆa − b|
[JEE 2008]
63. Let
e e
Z x Z −x
I = dx, J = dx.
e 4x + e 2x + 1 e −4x + e −2x + 1
Then, for an arbitrary constant C , the value of J − I equals
4x 2x 2x x
1 e − e + 1 1 e + e + 1
(A) log + C. (B) log + C.
x
2 e 4x + e 2x + 1 2 e 2x − e + 1
2x x 4x 2x
1 e − e + 1 1 e + e + 1
(C) log + C. (D) log + C.
x
2 e 2x + e + 1 2 e 4x − e 2x + 1
[JEE 2008]