(A)    Weight of sphere, $W=\frac{4}{3}\pi \left(\frac{D^3-d^3}{8}\right)\sigma g$

         Buoyant force $F_b=\frac{4}{3}\mathrm{\pi }\left(\frac{{\mathrm{D}}^3}{8}\right){\mathrm{\rho }}_{\mathrm{w}}\mathrm{g}$

         For Just Float, $W=F_b$

        $\frac{4}{3}\mathrm{\pi }\left(\frac{{\mathrm{D}}^3-{\mathrm{d}}^3}{8}\right)\mathrm{\sigma g}=\frac{4}{3}\mathrm{\pi }\left(\frac{{\mathrm{D}}^3}{8}\right){\mathrm{\rho }}_{\mathrm{w}}\mathrm{g}$

        $\Rightarrow 1-\frac{d^3}{D^3}=\frac{{\rho }_w}{\sigma }({\rho }_w=1)$

          $1-\frac{1}{\sigma }={\left(\frac{d}{D}\right)}^3\Rightarrow \frac{D}{d}={\left(\frac{\sigma }{\sigma -1}\right)}^{1/3}$

(3)

$V_1=\text{volume immersed in water}$

$V_2=\text{volume immersed in oil}$

Buoyancy force balance weight at equilibrium

$F_{B_1}+F_{B_2}=mg$

$V_1{\rho }_{w}g+V_2{\rho }_{o}g=(V_1+V_2){\rho }_{c}g$

$V_1+\frac{V_2{\rho }_o}{{\rho }_w}=(V_1+V_2)\frac{{\rho }_c}{{\rho }_w}$

$\Rightarrow V_1+0.8V_2=0.9V_1+0.9V_2$

$\Rightarrow 0.1V_1=0.1V_2\Rightarrow V_1:V_2=1:1\Rightarrow V_1:V_2=1:1$