$A$ 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 $......\, MeV$.

$\frac{1}{2 \cdot 3} + \frac{1}{4 \cdot 5} + \frac{1}{6 \cdot 7} + \frac{1}{8 \cdot 9} + \dots$ is equal to

If $\vec{a}, \vec{b}, \text{ and } \vec{c}$ are unit vectors,then $|\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2$ does not exceed:

If $A+B+C=270^{\circ}$,then $\cos 2A+\cos 2B+\cos 2C$ is equal to:

The maximum value of $ \left(\frac{1}{x}\right)^{x} $ is

$A$ mixture of $2\, \text{moles}$ of helium gas (atomic mass $= 4\, \text{amu}$) and $1\, \text{mole}$ of argon gas (atomic mass $= 40\, \text{amu}$) is kept at $300\, \text{K}$ in a container. The ratio of the $r.m.s.$ speeds $\left( \frac{v_{\text{rms}}(\text{helium})}{v_{\text{rms}}(\text{argon})} \right)$ is