Page 32 - ISC-12

Multiple Correct Choices Type Jai Baba Ki9
π π
4
6
8
2
37. Let f(x) = 7 tan x + 7 tan x − 3 tan x − 3 tan x for all x ∈ − , . Then the correct
2 2
expression(s) is (are)
π/4 1 π/4
Z Z
(A) x f(x) dx = . (B) f(x) dx = 0.
0 12 0
Z π/4 1 Z π/4
(C) x f(x) dx = . (D) f(x) dx = 1.
0 6 0
[JEE 2015]

1
192x 3 Z 1
0
38. Let f (x) = for all x ∈ R with f = 0. If m ≤ f(x) dx ≤ M, then the possible
4
2 + sin πx 2 1/2
values of m and M are
1 1
(A) m = 13, M = 24. (B) m = , M = . (C) m = −11, M = 0. (D) m = 1, M = 12.
4 2
[JEE 2015]

2
4
2
39. Let M and N be two 3 × 3 matrices such that MN = NM. Further, if M 6= N and M = N , then
2
2
(A) determinant of (M + MN ) is 0.
2
2
(B) there is a 3 × 3 non-zero matrix U such that (M + MN )U is the zero matrix.
2
2
(C) determinant of (M + MN ) ≥ 1.
2
2
(D) for a 3 × 3 matrix U, if (M + MN )U equals the zero matrix then U is the zero matrix.
[JEE 2014]
40. For every pair of continuous functions f, g : [0, 1] → R such that
max{f(x) : x ∈ [0, 1]} = max{g(x) : x ∈ [0, 1]},
the correct statement(s) is (are):

2
2
(A) (f(c)) + 3f(c) = (g(c)) + 3g(c) for some c ∈ [0, 1].
2
2
(B) (f(c)) + f(c) = (g(c)) + 3g(c) for some c ∈ [0, 1].
2
2
(C) (f(c)) + 3f(c) = (g(c)) + g(c) for some c ∈ [0, 1].
2
2
(D) (f(c)) = (g(c)) for some c ∈ [0, 1].
[JEE 2014]
41. Let f : (0, ∞) → R be given by
1
 
x 1 
Z −  t+
f(x) = e t dt.
1/x t
Then

(A) f(x) is monotonically increasing on [1, ∞).
(B) f(x) is monotonically decreasing on (0, 1).

1
(C) f(x) + f = 0, for all x ∈ (0, ∞).
x
x
(D) f(2 ) is an odd function of x on R.
[JEE 2014]