Let `cos^(-1)(x)+cos^(-1)(2x)+cos^(-1)(3x)` be `pi.` If `x` satisfies the equation `a x^3+b x^2+c x-c_1=0,` then the value of `(b-a-c)` is_________

Answer

3

Explanation

`cos^(-1) (x) + cos^(-1) (2x) + cos^(-1) cos^(-1) (3x) = pi`
or `cos^(-1) (2x) + cos^(-1) (3x)= pi - cos^(-1) (x) = cos^(-1) (-x)`
or `cos^(-1) [(2x) (3x) - sqrt(1 -4x^(2)) sqrt(1 -9x^(2))] = cos^(-1) (-x)`
or `6x^(2) - sqrt(1-4x^(2)) sqrt(1 -9x^(2)) = -x`
or `(6x^(2) + x)^(2) = (1-4x^(2)) (1-9x^(2))`
or `x^(2) + 12x^(3) = 1 - 13x^(2)`
or `12x^(3) + 14x^(2) -1 = 0`
`rArr a = 12, b = 14, c =0`
`rArr a + b + c = 12 + 14 -1 = 25`