For the function $(ex)^y=e^x$;

Match the LIST-I with LIST-II

LIST-I	LIST-II
A. $y$ is equal to	I. $\dfrac{1-\log_e x}{x(1+\log_e x)^3}$
B. value of $\dfrac{dy}{dx}$	II. $\dfrac{x}{1+\log_e x}$
C. value of $\dfrac{d^2y}{dx^2}$	III. $\dfrac{\log_e x}{(1+\log_e x)^2}$
D. $x^y$ is equal to	IV. $e^{x-y}$

Choose the correct answer from the options given below:

If $f(x) = \begin{cases} \frac{x^2+3x-10}{x-2}, & x \neq 2 \\ 2k+1, & x = 2 \end{cases}$ is continuous at $x = 2$, then the value of $k$ is

If $x=\log_e t$ and $y=\dfrac{1}{t}$, then $\dfrac{d^2y}{dx^2}$ at $t=0.1$ is

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

If $y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \cdots + \infty}}}$ then $\frac{dy}{dx}$ equals to

If $y = \sqrt{x + \sqrt{x + \sqrt{x + ...\ ...\ ...}}}$, then

If $x = 2\sin\theta$ and $y = 2\cos\theta$, then the value of $\frac{d^2y}{dx^2}$ at $\theta = 0$ is