The sum of the roots of the equation cos ⁡ 4 x + 6 = 7 cos ⁡ 2 x in the interval 0,314 is λ π , then the numerical value of λ is

Step-by-step solution

2 cos 2 ⁡ 2 x - 1 + 6 = 7 cos ⁡ 2 x On putting cos ⁡ 2 x = t , we get, 2 t 2 - 1 + 6 = 7 t 2 t 2 - 7 t + 5 = 0 2 t - 5 t - 1 = 0 t = 5 2 , 1 t = 5 2 (not possible) t = 1 ⇒ c o s 2 x = 1 ⇒ 2 x = 2 n π ⇒ x = n π The roots in 0,314 are π , 2 π , 3 π , . . . . . , 99 π 100 π > 314 Sum of roots = π + 2 π + 3 π + . . . . + 99 π = 4950 π ⇒ λ = 495 0