A solid cube with coefficient of linear expansion $\alpha$ floats on a liquid whose coefficient of volume expansion is $\gamma$, when both the solid and the liquid are heated
A) the solid sinks more
B) the solid does not sink (or) lift
C) the solid lifts up
d) $\gamma <3\alpha$
e) $\gamma >3\alpha$
f) $\gamma =3\alpha$

Let the initial temperature be $0^0\mathrm{C}$, density of body at $0^0\mathrm{C}$ be ${\left({\rho }_b\right)}_0$and that of liquid at $0^0\mathrm{C}$ be ${\left({\rho }_l\right)}_0$ Fraction of volume of body inside the liquid is 

$f=\frac{{\left({\rho }_b\right)}_0}{{\left({\rho }_l\right)}_0}$

Let the temperature be increased by $∆\theta$

$f\text{'}=\frac{{\rho }_b}{{\rho }_l}=\frac{{\left({\rho }_b\right)}_0}{1+{\gamma }_b∆\theta }\frac{1+{\gamma }_l∆\theta }{{\left({\rho }_l\right)}_0}=\frac{1+{\gamma }_l∆\theta }{1+{\gamma }_b∆\theta }\frac{{\left({\rho }_b\right)}_0}{{\left({\rho }_l\right)}_0}$

If ${\gamma }_l={\gamma }_b (or \gamma =3\alpha ), \mathrm{then} f\text{'}=f$,  the solid does not sink (or) lift.    B-f

If ${\gamma }_l>{\gamma }_b (or \gamma >3\alpha ), \mathrm{then} f\text{'}>f$,  the solid sinks more.      A-e

If ${\gamma }_l<{\gamma }_b (or \gamma <3\alpha ), \mathrm{then} f\text{'}<f$, the solid lifts up.    C-d