The wavenumber of the first spectral line of the Lyman series of $He^{+}$ ion is $x \ m^{-1}$. What is the wavenumber (in $m^{-1}$) of the second spectral line of the Balmer series of $Li^{2+}$ ion?

The radius of the $2^{nd}$ orbit of $Li^{2+}$ is $x$. The expected radius of the $3^{rd}$ orbit of $Be^{3+}$ is:

The ratio of energy of a hydrogen atom for the first and second excited states is ......

For the Balmer series in the spectrum of $H$ atom,$\bar{v}=R_{H}\left\{\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right\}$,the correct statements among $(I)$ to $(IV)$ are:
$(I)$ As wavelength decreases,the lines in the series converge.
$(II)$ The integer $n_{1}$ is equal to $2$.
$(III)$ The lines of longest wavelength corresponds to $n_{2}=3$.
$(IV)$ The ionization energy of hydrogen can be calculated from wave number of these lines.

The threshold frequency of a metal is $10^{15} \ s^{-1}$. The ratio of maximum kinetic energies of the photoelectrons, when the metal is irradiated with radiations of frequencies $1.5 \times 10^{15} \ s^{-1}$ and $2.0 \times 10^{15} \ s^{-1}$ respectively, is

Number of spectral lines obtained in $He^{+}$ spectra,when an electron makes transition from fifth excited state to first excited state will be