The expression $1+x \log _e a+\frac{x^2}{2 !}(\log _e a)^2+\frac{x^3}{3 !}(\log _e a)^3+\ldots$ for $a>0, x \in R$ is equal to:

In the adjoining circuit,the battery $E_1$ has an emf of $12 \, V$ and zero internal resistance,while the battery $E_2$ has an emf of $2 \, V$. If the galvanometer $G$ reads zero,then the value of the resistance $X$ (in $\Omega$) is

Three masses $700\,g, 500\,g$ and $400\,g$ are suspended at the end of a spring as shown and are in equilibrium. When the $700\,g$ mass is removed,the system oscillates with a period of $3\,s$. When the $500\,g$ mass is also removed,it will oscillate with a period of .... $s$.

$A$ $10 \ kW$ transmitter emits radio waves of wavelength $500 \ m$. The number of photons emitted per second by the transmitter is of the order of:

The half-life of a radioactive substance is $20 \, min$. The approximate time interval $(t_2 - t_1)$ between the time $t_2$ when $\frac{2}{3}$ of it has decayed and the time $t_1$ when $\frac{1}{3}$ of it has decayed is ......... $min$.

$A$ gamma ray photon creates an electron-positron pair. If the rest mass energy of an electron is $0.5\, MeV$ and the total kinetic energy of the electron-positron pair is $0.78\, MeV$,then the energy of the gamma ray photon must be $......\, MeV$.