At $T \text{ K}$,the ratio of kinetic energies of $4 \text{ g}$ of $H_{2(g)}$ and $8 \text{ g}$ of $O_{2(g)}$ is

If $-\frac{\pi}{4} < x < \frac{\pi}{4}$,then the general solution of the differential equation $\cos ^2 x \cdot \frac{d y}{d x}-(\tan 2 x) y=\cos ^4 x$ is

If $\frac{x-4}{x^2-5x-2k} = \frac{2}{x-2} - \frac{1}{x+k}$,then $k$ is equal to

$A$ light string passing over a smooth light pulley connects two blocks of masses $m_1$ and $m_2$ (vertically). If the acceleration of the system is $\frac{g}{8}$,then the ratio of masses $m_2 : m_1$ is:

The tyre of a bicycle has a volume of $2 \times 10^{-3} ~m^3$. Initially,the tube is filled to $75 \%$ of its volume with air at an atmospheric pressure of $10^5 ~Nm^{-2}$. When a rider is on the bicycle,the area of contact of the tyre with the road is $24 \times 10^{-4} ~m^2$. The mass of the rider with the bicycle is $120 ~kg$. If a pump delivers a volume of $500 ~cm^3$ of air in each stroke,then the number of strokes required to inflate the tyre is $\left(g=10 ~ms^{-2}\right)$.

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