Let $[x]$ denote the greatest integer function. Then match List-I with List-II:</p> | List-I | List-II | | --- | --- | | (A) $ \vert x - 1\vert + \vert x - 2\vert $ | (I) is differentiable everywhere except at $ x = 0 $ | | (B) $ x - \vert x\vert $ | (II) is continuous everywhere | | (C) $ x - [x] $ | (III) is not differentiable at $ x = 1 $ | | (D) $ x \, \vert x\vert $ | (IV) is differentiable at $ x = 1 $ |

The function $f(x)=\begin{cases}\frac{e^{3x}-e^{-5x}}{x}, & x\neq 0\\ \lambda, & x=0\end{cases}$ is continuous at x=0 for the value of '$\lambda$' as:

If $\sqrt{y+x} + \sqrt{y-x} = a$, $a > 1$, $\frac{d^2y}{dx^2}$ is equal to :

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

If $x=2at^5$ and $y=at^{10}$, then $\dfrac{dy}{dx}=$

Let $y = m\sin rx + n\cos rx$. What is the value of $\frac{d^2y}{dx^2}$?

If $y = x\log_e\left(\dfrac{x}{2+x}\right)$, then $x^3\dfrac{d^2y}{dx^2}$ is equal to: