If $f(x) = \begin{cases} mx + 1,\ x \geq \pi/2 \\sin x + n, x \leq \pi/2, & \end{cases}$ is continuous at $x = \pi/2$, where $m \in \mathbb{Z}$ (set of integers), then $\sin 2n =$

The derivative of $|x|$ with respect to $x$, where $x \in R \sim \{0\}$ is

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

A function defined as $f:R\to R$ satisfies $f(x+y)=f(x)f(y)$ and $f(x)\ne 0$ for all $x,y\in R$, where R is set of real numbers. If $f(x)$ is differentiable at $x=0$ and $f'(0)=2$, then $f'(x)$ is equal to

If $y = 500 e^{7x} + 600 e^{-7x}$ and $\frac{d^2 y}{dx^2} = ky$, then the value of k is :

If $y=7^x\log(5x^3)$, then $\dfrac{dy}{dx}$ is equal to: (consider $\log_e x = \log x$)

If $y = 5e^{2x} + 4e^{3x}$, then $\frac{d^2y}{dx^2}$ equals: