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

If $y = \log_x(x^2+1).\log_{(x^2+1)} x$, then $\frac{dy}{dx}$ is

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 $\sin y = x \cos(a + y)$, then $\frac{dy}{dx}$ is equal to

If $x=\log_e t$ and $y=\dfrac{1}{t}$, then $\dfrac{d^2y}{dx^2}$ at $t=0.1$ 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:

$\frac{d^2}{dx^2} \left\{ \det \begin{bmatrix} x^3 & x \\ 2 & e^x \end{bmatrix} \right\}$ equals