The value of '$\lambda$' for which the function $f(x) = \begin{cases}\frac{1-\cos 4x}{4x^2}, & \text{if } x \neq 0 \\ \lambda, & \text{if } x = 0\end{cases}$ is continuous at $x=0$, is:

If $x = a\left(t - \frac{1}{t}\right)$, $y = b\left(t + \frac{1}{t}\right)$, then $\frac{dy}{dx} =$

Let $y = \log(x + \sqrt{x^2+1})$, and $a\frac{d^2y}{dx^2} + b\frac{dy}{dx} = c$. Then identify the correct statements about the values of $a$, $b$ and $c$ :

(A) $a = 1 + x^2$

(B) $b = 0$

(C) $c = 0$

(D) $b = x$

(E) $c = 2$

Choose the correct answer from the options given below :

If $x = \sqrt{a^{\tan^{-1}t}}$ and $y = \sqrt{a^{\cot^{-1}t}}$, then $\frac{d^2y}{dx^2}$ is

Let $[x]$ denote the greatest integer function. Then match List-I with List-II:

If $y = x^{(x \sin x)}$ then $\frac{dy}{dx} = ?$

If $e^y(x+1) = 1$, then $\dfrac{d^2y}{dx^2}$ is equal to