The exhaustive set of values of a for which `a - cot^(-1) 3x = 2 tan^(-1) 3x + cos^(-1) x sqrt3 + sin^(-1) x sqrt3` may have solution, is

Options

A. `[-(pi)/(4), (pi)/(4)]`
B. `((pi)/(2), (3pi)/(2))`
C. `[(2pi)/(3), (4pi)/(3)]`
D. `[-(3pi)/(6), (7pi)/(6)]`

Answer

C

Explanation

`a - cot^(-1) 3x = 2 tan^(-1) 3x + cos^(-1) x sqrt3 + sin^(-1) x sqrt3`
Clearly, given equation is meaningful when `-1 le x sqrt3 le 1`
Given equation becomes
`a= tan^(-1) 3x + (tan^(-1) 3x+ cot^(-1) 3x) + (cos^(-1) x sqrt3 + sin^(-1) x sqrt3)`
`= tan^(-1) 3x + pi//2 + pi//2`
`pi + tan^(-1) 3x`
Now `-1 le x sqrt3 le 1`
`rArr -sqrt3 le 3x le sqrt3`
`-(pi)/(3) le tan^(-1) 3x le (pi)/(3)`
`rArr (2pi)/(3) le pi + tan^(-1) 3x le (4pi)/(3)`
`rArr a in [(2pi)/(3), (4pi)/(3)]`