A free hydrogen atom after absorbing a photon of wavelength $$\lambda $$_a gets excited from the state n = 1 to the state n = 4. Immediately after that the electron jumps to n = m state by emitting a photon of wavelength $$\lambda $$_e. Let the change in momentum of atom due to the absorption and the emission be $$\Delta {p_a}$$ and $$\Delta {p_e}$$, respectively. If $${{{\lambda _a}} \over {{\lambda _e}}} = {1 \over 5}$$, which of the option(s) is/are correct? [Use hc = 1242 eVnm; 1 nm = 10^-9 m, h and c are Planck's constant and speed of light in vacuum, respectively]

A mixture of ideal gas containing 5 moles of monatomic gas and 1 mole of rigid diatomic gas is initially at pressure P₀, volume V₀, and temperature T₀. If the gas mixture is adiabatically compressed to a volume $${{{V_0}} \over 4}$$, then the correct statement(s) is/are

(Given, 2^1.2 = 2.3; 2^3.2 = 9.2; R is a gas constant)

Suppose a $$_{88}^{226}Ra$$ nucleus at rest and in ground state undergoes $$\alpha $$-decay to a $$_{86}^{222}Rn$$ nucleus in its excited state. The kinetic energy of the emitted $$\alpha $$ particle is found to be 4.44 MeV. $$_{86}^{222}Rn$$ nucleus then goes to its ground state by $$\gamma $$-decay. The energy of the emitted $$\gamma $$ photon is ............ keV.

[Given : atomic mass of $$_{86}^{226}Ra$$ = 226.005 u, atomic of $$_{86}^{222}Rn$$ = 222.000 u, atomic mass of $$\alpha $$ particle = 4.000 u, 1 u = 931 MeV/e², c is speed of the light]

A monochromatic light is incident from air on a refracting surface of a prism of angle 75$$^\circ $$ and refractive index $${n_0} = \sqrt 3 $$. The other refracting surface of the prism is coated by a thin film of material of refractive index n as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of $$\theta \le 60^\circ $$. The value of n² is .............