Match List-I with List-II

List-I	List-II
(A) $\frac{d}{dx}(\sqrt{\sin x})$	(I) $\frac{-\sin x}{2\sqrt{\cos x}}$
(B) $\frac{d}{dx}(\sqrt{\cos x})$	(II) $-\sin 2x$
(C) $\frac{d}{dx}(\sin^2 x)$	(III) $\frac{\cos x}{2\sqrt{\sin x}}$
(D) $\frac{d}{dx}(\cos^2 x)$	(IV) $\sin 2x$

Choose the correct answer from the options given below:

If $y = \dfrac{\log_e x}{x}$, then $\dfrac{d^2y}{dx^2}$ is :

Let $[x]$ denote the greatest integer $\leq t$ and $a \mathbb{Z} = [ax: x \in \mathbb{Z}, a \in \mathbb{R}]$ (where $\mathbb{Z}$ is set of integer and $\mathbb{R}$ is set of real number). The set of points of discontinuity of the function $f(x) = [2x]$ is given by

If $f(x) = \begin{cases} \frac{\tan(\pi/4 - x)}{\cot 2x}, & x \neq \pi/4 \\ k, & x = \pi/4 \end{cases}$ is continuous at $x = \pi/4$, then the value of k is

Match List-I with List-II

Where $\mathbb{R}$ is set of real numbers

Choose the correct answer from the options given below:

If $x=a(\theta+\sin\theta)$, $y=a(1-\cos\theta)$, then $\dfrac{d^2y}{dx^2}=$

Let

$f(x)=\begin{cases} |x|+3 & \text{if } x\le -3 \\ -2x & \text{if } -3<x<3 \\ 6x+2 & \text{if } x\ge 3 \end{cases}$

Then, which of the following is true?