A proton is fired from very far away towards a nucleus with charge $Q=120 \ e$, where $e$ is the electronic charge. It makes a closest approach of $10 \ fm$ to the nucleus. The de Brogle wavelength (in units of $fm$ ) of the proton at its start is :

(take the proton mass, $m _0=(5 / 3) \times 10^{-27} kg , h / e =4.2 \times 10^{-15} J / s / C ; \frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 m / F ; 1 fm =10^{-15} m$ ) 

The diagram below shows the lowest four energy levels for an electron in a hypothetical atom.The electron is excited to the $-1\,eV$ level of the atom and transitions to the lowest energy state by emitting only two photons. Which of the following energies could not belong to either of the photons ? .....$eV$

The number of completely filled shells for the element ${ }_{16} S ^{32}$ is

Ratio of longest wavelengths corresponding to Lyman and Balmer series in hydrogen spectrum is

The following diagram indicates the energy levels of a certain atom when the system moves from $4E$ level to $E$. A photon of wavelength $\lambda _1$ is emitted. The wavelength of photon produced during it's transition from $\frac{7}{3}E$ level to $E$ is $\lambda_2$. The ratio $\frac{{{\lambda _1}}}{{{\lambda _2}}}$ will be

$\sqrt{d_{1}}$ and $\sqrt{d_{2}}$ are the impact parameters corresponding to scattering angles $60^{\circ}$ and $90^{\circ}$ respectively, when an $\alpha$ particle is approaching a gold nucleus. For $d_{1}=x d_{2}$, the value of $x$ will be ________