$A$ sound source is moving towards a stationary observer with $1/10$ of the speed of sound. The ratio of apparent to real frequency is

If $f: \mathbb{R} \rightarrow \mathbb{R}$ is such that $f(x+y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$,$f(1) = 7$,and $\sum_{r=1}^n f(r) = 14112$,then $n = $

If the line $2x + 5y = 12$ intersects the ellipse $4x^2 + 5y^2 = 20$ in two distinct points $A$ and $B$,then the mid-point of $AB$ is

If $a, b, c$ are unit vectors and the maximum value of $|a-b|^2+|b-c|^2+|c-a|^2$ is $k$,then $k(2|a|^2+3|b|^2-4|c|^2)=$

Which of the following substances releases nitrogen gas upon thermal decomposition?

The maximum potential energy due to electrostatic repulsion between two hydrogen nuclei is nearly (radius of the nucleus $= 1.1 \text{ fermi}$). $\left[\frac{1}{4 \pi \varepsilon_0} = 9 \times 10^9 \text{ Nm}^2 \text{C}^{-2}\right]$ (in $MeV$)