The number of real solutions of `tan^(-1)sqrt(x(x+1))+sin^(-1)sqrt(x^2+x+1)=pi/2` is

Options

A. zero
B. one
C. two
D. infinite

Answer

C

Explanation

Given function is
`tan^(-1)sqrt(x(x+1)) + sin ^(-1)sqrt(x^(2) + x + 1) = (pi)/(2)`
Function is defined, if
(i) `x (x + 1) ge 0`, since domain of square root function.
(ii) `x^(2) + x + 1 ge 0`, since domain of square root function.
(iii) ` sqrt(x^(2) + x + 1) le 1`, since domain of `sin ^(-1)` function.
From (ii) and (iii)`, 0 le x^(2) + x + 1 le 1 nn x^(2) + x ge 0`
`rArr " " 0 le x ^(2) + x + 1 le 1 nn x^(2) + x + 1 ge 1`
`rArr " " x ^(2) + x + 1 = 1 `
`rArr " " x^(2) + x = 0`
`rArr " " x(x+1) =0`
`rArr " " x=0, x =-1`