Page 33 - ISC-12

10Jai Baba Ki ISC Mathematics – Class XII by Gupta–Bansal

42. Let f : [a, b] → [1, ∞) be a continuous function and let g : R → R be defined as

0 if x < a





 Z x


g(x) = f(t) dt if a ≤ x < b .
a



 Z b


f(t) dt if x > b


a
Then
(A) g(x) is continuous but not differentiable at a.
(B) g(x) is differentiable on R.
(C) g(x) is continuous but not differentiable at b.
(D) g(x) is continuous and differentiable at either a or b but not both.
[JEE 2014]
π π

3
43. Let f : − , → R be given by f(x) = [log (sec x + tan x)] . Then
2 2
(A) f(x) is an odd function. (B) f(x) is a one-one function.
(C) f(x) is an onto function. (D) f(x) is an even function.

[JEE 2014]
√ π
44. Let ~x, ~y and ~z be three vectors each of magnitude 2 and the angle between each pair of them is . If
3
~
~a is a non-zero vector perpendicular to ~x and ~y × ~z and b is a non-zero vector perpendicular to ~y and
~z × ~x, then

~
~
~
~
(A) b = (b · ~z)(~z − ~x). (B) ~a = (~a · ~y)(~y − ~z). (C) ~a·b = −(~a·~y)(b·~z). (D) ~a = (~a · ~y)(~z − ~y).
[JEE 2014]
45. Let M be a 2 × 2 symmetric matrix with integer entries. Then M is invertible if

(A) the first column of M is the transpose of the second row of M.
(B) the second row of M is the transpose of the first column of M.
(C) M is a diagonal matrix with non-zero entries in the main diagonal.

(D) the product of entries in the main diagonal of M is not the square of an integer.
[JEE 2014]

46. For 3 × 3 matrices M and N, which of the following statement(s) is (are) NOT correct?

T
(A) N MN is symmetric or skew symmetric, according as M is symmetric or skew symmetric.
(B) MN − NM is skew symmetric for all symmetric matrices M and N.
(C) MN is symmetric for all symmetric matrices M and N.

(D) (adj M)(adj N) = adj (MN) for all invertible matrices M and N.

[JEE 2014]