If the function $f(x)=\begin{cases}\frac{x^3-1}{x-1}, & x<1\\a, & x=1\\\frac{b\sin(x-1)}{x-1}, & x>1\end{cases}$ is continuous at $x=1$, then which of the following statements are correct ?

A. $a=b$

B. $a+b$ is a multiple of 4

C. $2a-b$ is a multiple of 3

D. $a+4b$ is divisible by 5

Choose the correct answer from the options given below:

If $y = \left(\log\left(x + \sqrt{x^2+a^2}\right)\right)^2$ and $x \neq \frac{1-a^2}{2}$, then $(x^2+a^2)\frac{d^2y}{dx^2} + x\frac{dy}{dx}$ is equal to:

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

If the function $f(x)=\begin{cases}\frac{e^{2\log_e x^2}-1}{x-1}, & x\ne 1\\ a, & x=1\end{cases}$ is continuous at x=1, then the value of a is:

The value of k, for which the function $f(x) = \begin{cases} \frac{\sin kx}{x} + 3\cos x, & x \neq 0 \\ 7, & x = 0 \end{cases}$ is continuous at $x = 0$, is

If $f(x) = \begin{cases} \frac{k\cos x}{\pi - 2x}, & x \neq \frac{\pi}{2} \\ 3, & x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$, then k is :

If $x\sqrt{1+y}+y\sqrt{1+x}=0, x\neq y$, then $\frac{dy}{dx}$ is equal to