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 = $

For a binomial variate $X$ with $n=6$,if $P(X=2)=9 P(X=4)$,then its variance is

If a wire of length $L$ forms a loop of radius $R$ and has $n$ turns,find the magnetic field at the center of the loop. The current flowing in the loop is $I$.

The network shown in the figure is a part of a complete circuit. If at a certain instant the current $i$ is $5\, A$ and is decreasing at the rate of $10^3\, A/s$,then $V_A - V_B$ is......$V$.

$A$ radioactive nucleus (initial mass number $A$ and atomic number $Z$) emits $3$ $\alpha$-particles and $2$ positrons. The ratio of neutrons to protons in the final nucleus will be:

During an adiabatic process,the pressure of a gas is proportional to the cube of its temperature. The value of $C_p / C_V$ for that gas is