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 $x\sqrt{1+y}+y\sqrt{1+x}=0, x>-1$ and $x\neq y$, then $(1+x)^2\frac{dy}{dx}$ is equal to

Let $x=at^2, y=2at$. If $\frac{d^2y}{dx^2}+\frac{dy}{dx}=0$ at $t=3$, then $a$ is equal to:

If $x=a(\theta+\sin\theta)$, $y=a(1-\cos\theta)$, then $\dfrac{d^2y}{dx^2}=$

If $y = \frac{1}{x+1}$, then $\frac{d^2y}{dx^2}$ at $x = 2$ is:

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 $y^{1/m} + y^{-1/m} = 2x$, then the value of $(x^2 - 1)\frac{d^2y}{dx^2} + x\frac{dy}{dx}$ is: