Energy of a vacant atom is higher than that of a neutral atom.
Hence, option $(a)$ is incorrect.
$K\ X-$ray is emitted when an electron makes a jump to the $K$ shell from some other shell. As a result, a positive charge hole is created in the outer shell. As the electron continuously moves to the $K$ shell, the hole moves from the $K$ shell to some other shell.
Hence, option $(b)$ is correct.
$K\ X-$ray is emitted due to the transition of an electron from the $L$ or $M$ shell to the $K$ shell and $L\ X-$ray is emitted due to the transition of an electron from the $M$ or $N$ shell to the $L$ shell. The energy involved in the transition from the $L$ or $M$ shell to the $K$ shell is higher than the energy involved in the transition from the $M$ or $N$ shell to the $L$ shell. Since the energy is inversely proportional to the wavelength, the wavelength of the $K$ $X-$ray is smaller than the wavelength of the $L$ $X-$ray of the same material. Hence, option $(c)$ is correct.
If $E_K, E_L$ and $E_M$ are the energies of $K, L$ and $M$ shells, respectively, then the wavelength of $\text{K}_\alpha$ $X-$ray $(\lambda_1)$ is given by
$\lambda_1=\frac{\text{hc}}{\text{E}_\text{K}-\text{E}_\text{L}}$
Here,
$h =$ Planck's constant
$c =$ Speed of light
Wavelength of the $\text{K}_\beta\ x-$ray $(\lambda_2)$ is given by
$\lambda_2=\frac{\text{hc}}{\text{E}_\text{K}-\text{E}_\text{M}}$
As the difference of energies $(E_K - E_M)$ is more than $(E_{K }- E_L),$ $\lambda_2$ is less than $\lambda_1$.
Hence, option $(d)$ is not correct.
