Question
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- II I X P ⊇ π 2 , 3 π 2 ,   4 π ,   7 π I I Y Q an arithmetic progression I I I Z R NOT an arithmetic progression I V W S ⊇ π 6 , 7 π 6 , 13 π 6 T ⊇ π 3 , 2 π 3 ,   π U ⊇ π 6 , 3 π 4 Which of the following is the only correct combination?
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 = m π ,   m ∈ I ⇒ cos ⁡ x = m cos ⁡ x = 0 ,   1 ,   - 1 ⇒ x = n π 2 ⇒ X = n π 2 ,   n ∈ I ⇒ 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 m + 1 π 2 , m ∈ I ⇒ sin ⁡ x = 2 n + 1 4 ⇒ sin ⁡ x = - 1 4 , 1 4 , - 3 4 , 3 4 ⇒ Z = n π ± sin - 1 ⁡ 1 4 ,   n π ± sin - 1 ⁡ 3 4 , n ∈ I I I - Q ,   T I I I   f x = sin ⁡ π cos ⁡ x and Y = x : f ' x = 0 f ' x = cos ⁡ π cos ⁡ x . - π sin ⁡ x = 0 Now, cos ⁡ π cos ⁡ x = 0 ⇒ π cos ⁡ x = 2 m + 1 π 2 , m ∈ I ⇒ cos ⁡ x = 2 m + 1 2 ⇒ cos ⁡ x = - 1 2 , 1 2 ⇒ x = n π ± π 3 , n ∈ I or sin ⁡ x = 0 ⇒ x = n π Hence, Y = n π ,   n π ± π 3 , n ∈ I Y = π 3 , 2 π 3 ,   π , 4 π 3 , 5 π 3 ,   2 π … I I I → R I V g x = cos ⁡ 2 π sin ⁡ x and W = x : g ' x = 0 g ' x = - sin ⁡ 2 π sin ⁡ x 2 π cos ⁡ x = 0 Now, cos ⁡ x = 0 ⇒ x = 2 m + 1 π 2 , m ∈ I or sin ⁡ 2 π sin ⁡ x = 0 2 π sin ⁡ x = m π , m ∈ I ⇒ sin ⁡ x = m 2 ⇒ sin ⁡ x = - 1 ,   - 1 2 ,   0 , 1 2 ,   1 ⇒ x = n π 2 ,   n π ± π 6 , n ∈ I W = n π 2 ,   n π ± π 6 W = π 6 , π 2 , 5 π 6 , π ,   7 π 6 , 3 π 2 … . I V - P ,   R ,   S