If $f(x)$ is a function such that $f^{\prime \prime}(x)+f(x)=0$ and $g(x)=[f(x)]^{2}+[f^{\prime}(x)]^{2}$ and $g(3)=8$,then $g(8)$ is equal to

If $f(x)=\tan ^{-1}\left[\frac{\log \left(\frac{e}{x^{2}}\right)}{\log \left(e x^{2}\right)}\right]+\tan ^{-1}\left[\frac{3+2 \log x}{1-6 \log x}\right]$ then the value of $f^{\prime \prime}(x)$ is equal to

If $x=\sin \theta$ and $y=\sin^3 \theta$,then the value of $\frac{d^2 y}{dx^2}$ at $\theta=\frac{\pi}{2}$ is:

If $y=e^{\sqrt{x \sqrt{x} \sqrt{x} \ldots}}, x>1$,then $\frac{d^2 y}{d x^2}$ at $x=\log _e 3$ is

If $x^m + y^m = k$ where $(m \neq 1)$ and $\frac{d^2y}{dx^2} = \frac{ax^b}{y^c}$ such that $a + b + c = 0$, then the value of $k$ is...

If the function $y = \sin^{-1} x$,then $\left(1-x^2\right) \frac{d^2 y}{d x^2}$ is equal to