If the sum of solutions of the system of equations 2 sin 2 θ - cos 2 θ = 0 and 2 cos 2 θ + 3 sin θ = 0 in the interval 0 , 2 π is k π , then k is equal to _______.

Step-by-step solution

Given, 2 sin 2 θ - cos 2 θ = 0 ⇒ 2 sin 2 θ - 1 - 2 sin 2 θ = 0 ⇒ sin 2 θ = 1 2 2 ⇒ θ = π 6 , 5 π 6 , 7 π 6 , 11 π 6 Also given, 2 cos 2 θ + 3 sin θ = 0 ⇒ 2 sin 2 θ - 3 sin θ - 2 = 0 ⇒   sin θ = - 1 2 ⇒ θ = 7 π 6 , 11 π 6 So, the common solution is θ = 7 π 6 , 11 π 6 Sum = 7 π + 11 π 6 = 3 π = k π ⇒ K = 3