Let $f(x)$ be a twice differentiable function such that $f''(x) = -f(x)$, $f'(x) = g(x)$, and $h(x) = \{f(x)\}^2 + \{g(x)\}^2$. If $h(5) = 11$, then $h(10)$ is equal to ...

If $f(x)$ and $g(x)$ are twice differentiable functions on $(0,3)$ satisfying $f^{\prime \prime}(x)=g^{\prime \prime}(x)$, $f^{\prime}(1)=4$, $g^{\prime}(1)=6$, $f(2)=3$, and $g(2)=9$, then $f(1)-g(1)$ is

If $y = (x + \sqrt{1 + x^2})^n$,then $(1 + x^2)\frac{d^2y}{dx^2} + x\frac{dy}{dx}$ is equal to:

If $\log _e y=3 \sin ^{-1} x$,then $(1-x^2) y^{\prime \prime}-x y^{\prime}$ at $x=\frac{1}{2}$ is equal to :

If $\sqrt{1-x^2}+\sqrt{1-y^2}=a(x-y)$, then $\left[\left(1-x^2\right)^2 \frac{d^2 y}{d x^2}+y\left(1-x^2\right)\right] \frac{d y}{d x}=$

Let $f$ and $g$ be twice differentiable functions such that $f(x) \cdot g(x) = 1$ for all $x \in R$ and $f'$ and $g'$ are never zero. Then $\frac{f''(x)}{f(x)} + \frac{g''(x)}{g(x)}$ equals: