Answer
$\mathrm{a}.$ equal
$\mathrm{b}.$ I
Work Step by Step
Archimedes' Principle:
An object completely immersed in a fluid experiences an upward buoyant force equal in magnitude to the weight of fluid displaced by the object.
$\mathrm{a}.$
Assuming the Bathysphere retains its volume due to its sturdy build, which resists water pressure, it displaces the same volume of water at any depth. Water also can not be compressed, so the volume displaced has the same mass (and weight), regardless of depth.
By Archimedes' principle, the buoyant force remains constant after the Bathysphere is fully immersed.
$\mathrm{b}.$
I is correct.
The fact that the pressure in a fluid increases with depth leads to a net upward force, a buoyant force, does not necessarily mean that the buoyant force changes with depth. So, II and III are not correct.
