If 2 sin 3 x + sin 2 x cos x + 4 sin x - 4 = 0 has exactly 3 solutions in the interval 0 , n π 2 , n ∈ N , then the roots of the equation x 2 + n x + ( n - 3 ) = 0 belong to :

Step-by-step solution

Given: 2 sin 3 x + sin 2 x cos x + 4 sin x - 4 = 0 ⇒ 2 sin 3 x + 2 sin x · cos 2 x + 4 sin x - 4 = 0 ⇒ 2 sin 3 x + 2 sin x · 1 - sin 2 x + 4 sin x - 4 = 0 ⇒ 2 sin 3 x + 2 sin x - 2 sin 3 x + 4 sin x - 4 = 0 ⇒ 6 sin x - 4 = 0 ⇒ sin x = 2 3 Now, for exactly three solution we get, ⇒ n = 5 (in the given interval) So, x 2 + n x + n - 3 = 0 ⇒ x 2 + 5 x + 2 = 0 ⇒ x = - 5 ± 17 2 So, the required interval is ( - ∞ , 0 ) .