The frequency of the $1^{st}$ line of the Balmer series in the ${H_2}$ atom is ${\nu _0}$. What is the frequency of the corresponding line emitted by a singly ionized ${He^+}$ atom?

Consider that an electron is revolving in an excited state of Hydrogen atom with velocity $\sqrt{25.6} \times 10^5 \text{ m s}^{-1}$. The radius of the orbit is $x \times 10^{-9} \text{ m}$. The value of $x$ is : [Take the mass of electron to be $9 \times 10^{-31} \text{ kg}$, charge of electron = $-1.6 \times 10^{-19} \text{ C}$ and $\frac{1}{4\pi \epsilon_0} = 9 \times 10^9 \text{ N m}^2 \text{ C}^{-2}$]

Kinetic energy of an electron in one of the orbits of a hydrogen atom is $x$. Then, its total energy is . . . . . . .

$A$ photon of wavelength $300 \ nm$ interacts with a stationary hydrogen atom in the ground state. During the interaction, the whole energy of the photon is transferred to the electron of the atom. State which possibility is correct. (Consider, Planck constant $= 4 \times 10^{-15} \ eV \cdot s$, velocity of light $= 3 \times 10^{8} \ m/s$, ionisation energy of hydrogen $= 13.6 \ eV$)

Which of the following graphs in the figure shows the speed $(v)$ of an electron in a hydrogen atom as a function of the principal quantum number $(n)$?

$A$ sample of hydrogen atoms in its ground state is radiated with photons of $10.2 eV$ energy. The radiation emitted from the sample is absorbed by excited ionized $He^{+}$ ions. Which of the following statement$(s)$ is/are true?