$A$ fireman of mass $60 \, kg$ slides down a pole. He is pressing the pole with a force of $600 \, N$. The coefficient of friction between the hands and the pole is $0.5$. With what acceleration will the fireman slide down (in $, m/s^2$)? (Take $g = 10 \, m/s^2$)

$A$ thermally insulated vessel contains an ideal gas of molecular mass $M$ and ratio of specific heats $\gamma$. It is moving with speed $v$ and is suddenly brought to rest. Assuming no heat is lost to the surroundings,its temperature increases by

The minimum force required to move a body up an inclined plane is three times the minimum force required to prevent it from sliding down the plane. If the coefficient of friction between the body and the inclined plane is $\frac{1}{2 \sqrt{3}}$,then the angle of the inclined plane is (in $^{\circ}$)

The ionization potential of a hydrogen atom is $13.6 \ eV$. Hydrogen atoms in the ground state are excited by monochromatic radiation of photon energy $12.1 \ eV$. The number of spectral lines emitted by the hydrogen atoms according to Bohr's theory will be

Let $\overline{a}, \overline{b}$ and $\overline{c}$ be three vectors having magnitudes $1, 1$ and $2$ respectively. If $\overline{a} \times(\overline{a} \times \overline{c})+\overline{b}=\overline{0}$,then the acute angle between $\overline{a}$ and $\overline{c}$ is

For the hyperbola $\frac{x^2}{\cos^2 \alpha} - \frac{y^2}{\sin^2 \alpha} = 1$,which of the following remains constant as $\alpha$ varies?