Let $f(x)=\begin{cases}\frac{1-\cos4x}{x^2}, & \text{if } x<0\\2\lambda, & \text{if } x=0\\\frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & \text{if } x>0\end{cases}$ such that $f$ continuous at $x=0$. Then, the value of $\lambda$ is equal to:

The value of derivative of the function $\cot^{-1}\{(\cos 2x)^{1/2}\}$ at $x = \frac{\pi}{6}$ is

The function $f(x) = e^{|x|}$ is

(a) continuous everywhere on $R$

(b) not continuous at $x = 0$

(c) Differentiable everywhere on $R$

(d) not differentiable at $x = 0$

(e) continuous and differentiable on $R$

Choose the most appropriate answer from the options given below :

If $x = t^3$, $y = t^2$ then $\frac{d^2y}{dx^2}$ is equal to:

If $y = \cot^{-1}\left(\frac{1-x}{1+x}\right)$ then $\frac{dy}{dx}$ equals

If $f(x+y) = f(x) + f(y)$ for all $x, y \in R$ (set of real numbers) and $f(x)$ is continuous at $x = 0$, then identify the correct statement ?

If $f(x)$, defined by $f(x)=\left\{\begin{array}{lll}k x+1 & \text { if } & x \leq \pi \\ \cos x & \text { if } & x>\pi\end{array}\right.$ is continuous at $x=\pi$, then the value of $k$ is :