The charge density $P$ in a nucleus varies with distance $r$ from the centre of the nucleus according to the curve in the figure. Which of the following graphs correctly represents this variation?

The atomic masses of two isotopes of an element are $34.98 \ u$ and $36.98 \ u$ and their relative abundances are $75.4 \%$ and $24.6 \%$ respectively. The average atomic mass of the element is: (in $u$)

Nuclei with magic numbers of protons $Z = 2, 8, 20, 28, 50, 82$ and magic numbers of neutrons $N = 2, 8, 20, 28, 50, 82, 126$ are found to be very stable.
$(i)$ Verify this by calculating the proton separation energy $S_p$ for $^{120}Sn$ $(Z = 50)$ and $^{121}Sb$ $(Z = 51)$. The proton separation energy for a nuclide is the minimum energy required to separate the least tightly bound proton from a nucleus of that nuclide. It is given by $S_p = (M_{Z-1, N} + M_H - M_{Z, N})c^2$. Given:
$^{119}In = 118.9058 \ u, ^{120}Sn = 119.902199 \ u, ^{121}Sb = 120.903824 \ u, ^1H = 1.0078252 \ u$
$(ii)$ What does the existence of magic numbers indicate?

Calculate the height of the potential barrier for a head-on collision of two deuterons. (Radius of deuteron is $2$ fm).

$A$ nucleus splits into two parts. If the ratio of their velocities is $8 : 1$, what is the ratio of their radii?

Consider a nucleus ${ }_{30}^{60} X$. Its approximate density is (Take $1 \text{ amu} = 1.6 \times 10^{-27} \text{ kg}$, $R_0 = 1.2 \times 10^{-15} \text{ m}$)