If

$f(x) = \begin{cases}\frac{1- \tan x}{4x-\pi}, & x \neq \frac{\pi}{4} \\ k, & x = \frac{\pi}{4}\end{cases}$

is continuous at $x = \frac{\pi}{4}$, then the value of k is

If $x = a\sec^3 \theta$, $y = a \tan^3 \theta$, then $\frac{dy}{dx}$ at $\theta = \frac{\pi}{3}$ is

$f(x) = [x]$, where $[\,]$ represents greatest integer function.

(A) For $2 \leq x < 3$, $[x] = 3$

(B) For $2 \leq x < 3$, $[x] = 2$

(C) Right hand derivative of f at $x = 2$ is not defined

(D) Left hand derivative of f at $x = 2$ is zero

(E) $f(x)$ is not differentiable at $x = 2$

Choose the correct answer from the options given below :

if $x=\sqrt{a^{\tan^{-1}t}}$ and $y=\sqrt{a^{\cot^{-1}t}}$ then which one of the following is true?

For what value of $\alpha$, the function $f$ defined by $f(x) = \begin{cases} \alpha(x^2 - 2x + 1), & \text{if } x \leq 0 \\ 2x + 1, & \text{if } x > 0 \end{cases}$ is continuous at $x = 0$?

If the function $f(x)=\begin{cases} x^2, & x\le c \\ ax+b, & x>c \end{cases}$ is differentiable at $x=c$, then the relation between $b$ and $c$ is:

If $y = x^3\log(3x^2+6)$, then $\dfrac{dy}{dx} =$