Match List-I with List-II regarding a solid sphere of radius r...

According to Stokes' law, the viscous force on a spherical body is $F = 6\pi\eta r v$.
For a constant given velocity $v$, the viscous force $F \propto r$. Thus, (A) matches with (III).

The terminal velocity of a sphere falling under gravity is given by $v_t = \dfrac{2r^2(\rho - \sigma)g}{9\eta}$.
Therefore, terminal velocity $v_t \propto r^2$. Thus, (B) matches with (IV).

The viscous force at terminal velocity is $F = 6\pi\eta r v_t$.
Substituting $v_t \propto r^2$, we get $F \propto r(r^2) = r^3$. Thus, (C) matches with (I).

An ideal fluid is non-viscous ($\eta = 0$). Therefore, the viscous drag is zero. Thus, (D) matches with (II).

The correct matching is (A)-(III), (B)-(IV), (C)-(I), (D)-(II).

Answer: (A)-(III), (B)-(IV), (C)-(I), (D)-(II)