If `sin^(-1)(x-1)+cos^(-1)(x-3)+tan^(-1)(x/(2-x^2))=cos^(-1)k+pi,` then the value of `k` is

Options

A. 1
B. `-(1)/(sqrt2)`
C. `(1)/(sqrt2)`
D. none of these

Answer

C

Explanation

`sin^(-1) (x -1)`
`rArr -1 le x -1 le 1`
`rArr 0 le x le 2`
`cos^(-1) (x -3)`
`rArr -1 le x -3 le 1`
`rArr 2 le x le 4`
`:. X = 2`
So, `sin^(-1) (2 -1) + cos^(-1) (2 -3) + tan^(-1).(2)/(2 -4) = cos^(-1) k + pi`
or `sin^(-1) 1 + cos^(-1) (-1) + tan^(-1) (-1) = cos^(-1) + pi`
`rArr (pi)/(2) + pi - (pi)/(4) = cos^(-1) k + pi`
`rArr cos^(-1) k = (pi)/(4) " or " k = (1)/(sqrt2)`