If $y = (ax + b) \cos x$,then $y_2 + y_1 \sin 2x + y(1 + \sin^2 x) = $

If $y=f(x)$ is a twice differentiable function such that at a point $P$,$\frac{dy}{dx}=4$ and $\frac{d^2y}{dx^2}=-3$,then $\left(\frac{d^2x}{dy^2}\right)_P$ is equal to:

If $f(x)=1+nx+\frac{n(n-1)}{2}x^2+\frac{n(n-1)(n-2)}{6}x^3+\ldots+x^n$,then $f''(1)$ is equal to

If $x^{2} y^{5}=(x+y)^{7}$,then $\frac{d^{2} y}{d x^{2}}$ is equal to

Let $f : S \rightarrow S$ where $S =(0, \infty)$ be a twice differentiable function such that $f(x+1) = xf(x)$. If $g : S \rightarrow R$ is defined as $g(x) = \log_{e} f(x)$,then the value of $|g''(5) - g''(1)|$ is equal to:

If $y=a \cos (\log x)+b \sin (\log x)$, where $a, b$ are parameters, then $x^2 y^{\prime \prime}+x y^{\prime}$ is equal to