The exhaustive set of value of a for which `a-cot^(-1)3x=2tan^(-1)3x+cos^1xsqrt(3)+sin^(-1)xsqrt(3)m a yh a v e` solution, is `[-pi/4,pi/4]` (b) `[pi/2,(3pi)/2]` (c)`[(2pi)/3,(4pi)/3]` (d) `[-(3pi)/6,(7pi)/6]`

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)]`