Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

Equation of parabola with focus $(-\frac{1}{2}, 0)$ and directrix $y=-\frac{1}{2}$ is:

${\left(x+\frac{1}{2}\right)}^2=\left(y+\frac{1}{4}\right)$

$\Rightarrow y=\left(x^2+x\right)$

Put $f\left(x\right)=y=x+x^2$ in $S$,

${\tan }^{-1}\sqrt{x\left(x+1\right)}+{\sin }^{-1}\sqrt{x^2+x+1}=\frac{\pi }{2}$

$\Rightarrow 0\le x^2+x+1\le 1$

$\Rightarrow x^2+x\le 0 \dots \left(1\right)$

Also,

 $x^2+x\ge 0 \dots \left(2\right)$

By $\left(1\right) \& \left(2\right)$, we have

$x^2+x=0\Rightarrow x=0,-1$

$S$ contains $2$ element.