If $y(\theta) = \frac{2 \cos \theta + \cos 2 \theta}{\cos 3 \theta + 4 \cos 2 \theta + 5 \cos \theta + 2}$,then at $\theta = \frac{\pi}{2}$,$y'' + y' + y$ is equal to:

$A$ function $f$ satisfies the relation $f(x) = f''(x) + f'''(x) + \dots \infty$,where $f(x)$ is an infinitely differentiable function. If $f(1) = 5$,then the value of $f'(1) + f''(1)$ is equal to:

If $f(x) = (x^2 - 1)^7$, then $f^{(14)}(x)$ is equal to

If $y = \sin(\log_e x)$, then $x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx}$ 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 $y = a{e^{mx}} + b{e^{ - mx}}$,then $\frac{d^2y}{dx^2} - m^2y = $