P. Let y(x) = cos(3cos^-1x), x∈ [-1, 1], x ± √3/2. Then 1/y(x){(x^2 - 1)(d^2y(x)/dx^2) + x(dy(x)/dx)} equals

Question

Match List I to List II.

List I	List II
P. Let y(x) = cos(3cos-1x), x∈ [-1, 1], x ± √3/2. Then 1/y(x){(x2 - 1)(d2y(x)/dx2) + x(dy(x)/dx)} equals	1. 1
Q. Let A1, AT, …, An (n > 2) be the vertices of a regular polygon of n sides with its centre at the origin. Let  vector ak be the position vector of the point Ak, then the minimum value of n is	2. 2
R. If the normal from the point P(h, 1) on the ellipse x2/6 + x2/3 =1 is perpendicular to the line x + y = 8, then the value of h is	3. 8
S. Number of positive solutions satisfying the equation tan-1(1/(2x + 1) + tan-1(1/(4x + 1)) = tan-1(2/x2) is	4. 9

 

	P	Q	R	S
(A)	4	3	2	1
(B)	2	4	3	1
(C)	4	3	1	2
(D)	2	4	1	3

Answer 1

Correct option (A)

Explanation:

For (P) in List I:

y(x) = cos(3cos^-1x)

Therefore,

6x³ + 2x² = 16x² + 12x

⇒ 2x (3x² + x - 8x - 6) = 0

⇒ x (3x² - 7x - 6) = 0

⇒ x (3x² - 9x + 2x - 6) = 0

⇒ x [3x (x - 3) + 2(x - 3)] = 0

⇒ x (x - 3) (3x + 2) = 0

Therefore,

x = 0, 3, - 2/3

Hence, number of +ve solutions = 1

Therefore,

(S) → (1)