The number of solution of equation `sin^(-1)x+nsin^(-1)(1-x)=(mpi)/2`,where n >0, `m<=0,` is

Options

A. 3
B. 1
C. 2
D. none of these

Answer

D

Explanation

`sin^(-1) x` is defined if `-1 le x le 1 and sin^(-1) (1- x)` is defined if
`- 1 le 1 -x le 1 rArr 0 le x le 2`
Therefore, `sin^(-1) x + n sin^(-1) (1 -x)` is defined if `0 le x le 1` when `0 le x le 1, " also " 0 le 1 - x le 1`. So,
`0 le sin^(-1) x le (pi)/(2) and 0 le sin^(-1) (1-x) le (pi)/(2)`
`L.H.S. ge 0 and R.H.S. le 0`
Which equally holds if L.H.S. = R.H.S. = 0
But L.H.S. `= 0 " if " sin^(-1) x and n sin^(-1) (1 -x)` are simultaneously zero, which is not possible