Page 30 - ISC-12

Multiple Correct Choices Type Jai Baba Ki7
3
28. In R , let L be a straight line passing through the origin. Suppose that all the points on L are at a
constant distance from the two planes P 1 : x + 2y − z + 1 = 0 and P 2 : 2x − y + z − 1 = 0. Let M be
the locus of the feet of the perpendiculars drawn from the points on L to the plane P 1 . Which of the
following points lie(s) on M?

5 2 1 1 1 5 1 1 2
(A) 0, − , − . (B) − , − , . (C) − , 0, . (D) − , 0, .
6 3 6 3 6 6 6 3 3

[JEE 2015]

29. Let y(x) be a solution of the differential equation
0
x
x
(1 + e )y + ye = 1.
If y(0) = 2, then which of the following statements is (are) true?
(A) y(−4) = 0. (B) y(−2) = 0.

(C) y(x) has a critical point in the interval (−1, 0).
(D) y(x) has no critical point in the interval (−1, 0).

[JEE 2015]

30. Consider the family of all circles whose centres lie on the straight line y = x. If this family of circles
0
00
is represented by the differential equation Py + Qy + 1 = 0, where P, Q are functions of x, y and
2
dy d y
00
0
y 0 here y = , y = , then which of the following statements is (are) true?
dx dx 2
(A) P = y + x. (B) P = y − x.
0
0 2
0
0 2
(C) P + Q = 1 − x + y + y + (y ) . (D) P − Q = x + y − y − (y ) .
[JEE 2015]
0
0
31. Let g : R → R be a differentiable function with g(0) = 0, g (0) = 0 and g (1) 6= 0. Let
x

 g(x) if x 6= 0
f(x) = |x|
0 if x = 0

and h(x) = e |x| for all x ∈ R. Let (f ◦ h)(x) denote f(h(x)) and (h ◦ f)(x) denote h(f(x)). Then
which of the following is (are) true?

(A) f is differentiable at x = 0. (B) h is differentiable at x = 0.
(C) f ◦ h is differentiable at x = 0. (D) h ◦ f is differentiable at x = 0.

[JEE 2015]

π π π

32. Let f(x) = sin sin sin x for all x ∈ R and g(x) = sin x for all x ∈ R. Let (f ◦ g)(x)
6 2 2
denote f(g(x)) and (g ◦ f)(x) denote g(f(x)). Then which of the following is (are) true?

1 1 1 1
(A) Range of f is − , . (B) Range of f ◦ g is − , .
2 2 2 2