If $y=\log_e\left(\frac{x^2}{e^2}\right)$, then $\frac{d^2y}{dx^2}$ is equal to:

If $xy\log_e(x+y) = 1$, then $\dfrac{dy}{dx} =$

If $x = t^2$ and $y = t\sqrt{t}$, then $\dfrac{d^2y}{dx^2}$ at $t = \dfrac{1}{4}$ is :

If $f(x) = \begin{cases} \frac{k\cos x}{\pi - 2x}, & x \neq \frac{\pi}{2} \\ 3, & x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$, then k is :

Match the LIST-I with LIST-II

Choose the correct answer from the options given below:

A function defined as $f:R\to R$ satisfies $f(x+y)=f(x)f(y)$ and $f(x)\ne 0$ for all $x,y\in R$, where R is set of real numbers. If $f(x)$ is differentiable at $x=0$ and $f'(0)=2$, then $f'(x)$ is equal to

If $x = a\left(\cos t + \log \tan\frac{t}{2}\right), y = a\sin t$, then value of $\frac{dy}{dx}$ at $t = \frac{\pi}{4}$ is