Let α = max x ∈ R 8 2 sin 3 x · 4 4 cos 3 x and β = min x ∈ R 8 2 sin 3 x · 4 4 cos 3 x . If 8 x 2 + b x + c = 0 is a quadratic equation whose roots are α 1 / 5 and β 1 / 5 , then the value of c - b is equal to :

Step-by-step solution

α = max 8 2 sin 3 x · 4 4 cos 3 x = max 2 6 sin 3 x · 2 8 cos 3 x = max 2 6 sin 3 x + 8 cos 3 x and β = min 8 2 sin 3 x · 4 4 cos 3 x = min 2 6 sin 3 x + 8 cos 3 x Now range of 6 sin 3 x + 8 cos 3 x = - 6 2 + 8 2 , + 6 2 + 8 2 = - 10 , 10 α = 2 10   &   β = 2 - 10 So, α 1 / 5 = 2 2 = 4 ⇒ β 1 / 5 = 2 - 2 = 1 4 The quadratic 8 x 2 + b x + c = 0 , sum of roots = - b 8 and product of roots = c 8 ∴   c - b = 8 × [ (product of roots ) + ( su