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

At $x = 2$, $f(x) = [x]$ is

[where [.] is greatest integer function]

Which of the following functions $f(x)$ are differentiable at $x = 0$?

(A) $|x|$

(B) $|x - 1|$

(C) $[x]$, where $[t]$ denotes the greatest integer $\leq t$

(D) $|x + 1|$

(E) $x^2$

Choose the correct answer from the options given below:

If $xy + \frac{x^2}{y} = x^3y + y$, then $\frac{dy}{dx}$ is equal to

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

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

For the function $f(x) = \min\{|x|, |x-2|\}$ which of the following are TRUE?

A. Function $f(x)$ is continous on $\mathbb{R}$

B. Function $f(x)$ is differentible on $\mathbb{R}$

C. Function $f(x)$ is differentible on $\mathbb{R} \sim \{0,1,2\}$

D. Function is continous on $\mathbb{R}$ but not differentiable at $x = \dfrac{1}{2}$

Choose the correct answer from the options given below: