As the mass number A increases, which of the following quantities related to a nucleus do not change?

Both the mass and volume of an atom is directly proportional to the mass number.

i.e. Mass = (mass of 1 proton)A 

Volume = kA              , $\left[\mathrm{where}, \mathrm{k} = \frac{4\mathrm{\pi }}{3}{{\mathrm{r}}_0}^3\right]$

density = $\frac{(\mathrm{mass} \mathrm{of} 1 \mathrm{proton})\mathrm{A}}{\mathrm{kA}} = \frac{\mathrm{mass} \mathrm{of} 1 \mathrm{proton}}{\mathrm{k}}$

Hence, the density of nucleus is independent of its mass number.