Let f x = sin ⁡ π cos ⁡ x and g x = cos ⁡ 2 π sin ⁡ x be two functions defined for x > 0 . Define the following sets whose elements are written in the increasing order: X = x : f x = 0 , Y = x : f ' x = 0 Z = x : g x = 0 , W = x : g ' x = 0 List- I contains the sets X , Y , Z and W . List- I I contains some information regarding these sets. List- I List- I I I X P ⊇

Step-by-step solution

I f x = sin ⁡ π cos ⁡ x   and X = x : f x = 0 f x = 0 sin ⁡ π cos ⁡ x = 0 π cos ⁡ x = n π cos ⁡ x = n cos ⁡ x = 0 ,   1 ,   - 1 ⇒ x = n π 2 X = n π 2 ,   n ∈ N ⇒ X = π 2 ,   π , 3 π 2 ,   2 π … … I - P ,   Q I I   g x = cos ⁡ 2 π sin ⁡ x and Z = x : g x = 0 cos ⁡ 2 π sin ⁡ x = 0 2 π sin ⁡ x = 2 n + 1 π 2 si