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

If $x = \frac{1-t}{1+t}$ and $y = \frac{3t}{1+t}$, then $\frac{d^2y}{dx^2}$ is equal to

For the greatest integer function $f(x)=[x]$, which of the following statements are correct?

A. It is continuous at all real numbers.

B. It is continuous at $x=2$ and 3.

C. It is continuous and differentiable for $x\in(2,3)$

D. It is continuous and differentiable at $x=4.5$

Choose the correct answer from the options given below:

If the function

$f(x) = \begin{cases} ax + 2, & x \leq 1 \\ x^2 + 3x + b, & x > 1 \end{cases}$

is differentiable at $x = 1$, then the value of $(2a + b)$ is

If $f(x) = \begin{cases} ax - 1 & {if } x \ > 1\\ \ 2x + 1 & {if } x < 1 \end{cases}$ is continuous at $x = 1$, then $a$ equals

If $x = 2\sin\theta$ and $y = 2\cos\theta$, then the value of $\frac{d^2y}{dx^2}$ at $\theta = 0$ is

If $x = at^2$ and $y = 2at$, then the value of $\frac{d^2y}{dx^2}$ is

( where t is a parameter )