$\sqrt{d_{1}}$ 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 ________

Consider the following statements:

$I$. All isotopes of an element have the same number of neutrons.

$II$. Only one isotope of an element can be stable and non-radioactive.

$III$. All elements have isotopes.

$IV$. All isotopes of carbon can form chemical compounds with oxygen-$16$.

Choose the correct option regarding an isotope.

Give the relationship between impact parameter and scattering angle. 

The transition from the state $n = 4$ to $n = 3$ in a hydrogen like atom results in ultraviolet radiation. Infrared radiation will be obtained in the transition

Match List $- I$ (Experiment performed) with List $-II$ (Phenomena discovered/associated) and select the correct option from the options given the lists

Particles used in the Rutherford's scattering experiment to deduce the structure of atoms