As shown in figure, when a spherical cavity (centred at O) of radius 1 is cut out of a uniform sphere of radius R (centred at C). The center of mass of remaining (shaded) part of sphere is at G, i.e., on the surface of the cavity.

R can be determined by the equation:

putting the cavity's centre at O and the solid sphere's centre at C.

Origin should be at G.

The sphere's mass is assumed to be m₁, which may be expressed as:

$$\begin{gathered} M_1=\frac{4}{3}{\mathrm{\pi R}}^3\mathrm{\rho } \\ {\mathrm{M}}_2=\frac{4}{3}\mathrm{\pi }{\left(1\right)}^3(-\mathrm{\rho }) \\ {\mathrm{X}}_{\mathrm{COM}}=\frac{{\mathrm{M}}_1{\mathrm{X}}_1+{\mathrm{M}}_2{\mathrm{X}}_2}{{\mathrm{M}}_1+{\mathrm{M}}_2} \\ {\mathrm{X}}_{\mathrm{COM}}=\frac{\left[\frac{4}{3}{\mathrm{\pi R}}^3\mathrm{\rho }\right]0+\left[\frac{4}{3}\mathrm{\pi }{\left(1\right)}^3(-\mathrm{\rho })\right]\left[\mathrm{R}-1\right]}{\frac{4}{3}{\mathrm{\pi R}}^3\mathrm{\rho }+\frac{4}{3}\mathrm{\pi }{\left(1\right)}^3(-\mathrm{\rho })}=-(2-\mathrm{R}) \\ {\mathrm{X}}_{\mathrm{COM}}=\frac{(\mathrm{R}-1)}{({\mathrm{R}}^3-1)}=(2-\mathrm{R}) \\ {\mathrm{X}}_{\mathrm{COM}}=({\mathrm{R}}^2+\mathrm{R}+1)(2-\mathrm{R})=1 \end{gathered}$$

Hence the correct answer is $({\mathrm{R}}^2+\mathrm{R}+1)(2-\mathrm{R})=1.$