List I List II
P. Let y(x) = cos(3 cos –1 x), x ∈ [–1, 1], x ≠ ± 
. Then 1. 1
equals
Q. Let A 1 , A 2 ,......, A n (n > 2) be the vertices of a regular polygon of n 2. 2
sides with its centre at the origin. Let 
be the position vector of
the point A k , k = 1, 2,...., n. If 
, then
the minimum value of n is
R. If the normal from the point P(h, 1) on the ellipse 
is 3. 8
perpendicular to the line x + y = 8, then the value of h is
S. Number of positive solutions satisfying the equation 4. 9
is 
P Q R S
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Sol. (P) y = 4x 3 – 3x where cos θ = x
= 12x 2 – 3
= (x 2 – 1) . 24x + x(12x 2 – 3)
= 36x 3 – 27x = 9(4x 3 – 3x) = 9y
Hence 
= 9
(Q) 
= 
Let = 
. . . . . = 
= λ (as centre is origin)
More over angle between 2 consecutive 
is 
Hence given equation reduces to
(n – 1) λ 2 sin 
= (n – 1) λ 2 cos 
⇒ tan 
= 1 ⇒ 
⇒ n = 8
(R) Equation of normal 
= 3 
slope = 
= 1 (as it is perpendicular to z + y = 1) ⇒ h = 2
(S) tan –1 
+ tan –1 
+ tan –1 
⇒ 
⇒ 
⇒ 3x 3 + x 2 = 8x 2 + 6x ⇒ 3x 3 – 7x 2 – 6x = 0
⇒ 3x 2 – 7x + 6 = 0 (as x ≠ 0)
⇒ (x – 3) (3x + 2) = 0 ⇒ x = – 
, 3 
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