$A$ body of density $\rho$ is dropped from rest at a height $h$ into a lake of density $\delta$ $(\delta > \rho)$. Neglecting all dissipative forces,find the maximum depth to which the body sinks before returning to float on the surface.

$A$ cylindrical block floats vertically in a liquid of density $\rho_{1}$ kept in a container such that the fraction of volume of the cylinder inside the liquid is $x_{1}$. Then some amount of another immiscible liquid of density $\rho_{2} (\rho_{2} < \rho_{1})$ is added to the liquid in the container so that the cylinder now floats just fully immersed in the liquids with $x_{2}$ being the fraction of volume of the cylinder inside the liquid of density $\rho_{1}$. The ratio $\rho_{1} / \rho_{2}$ will be

$A$ cubical block is floating in a liquid with half of its volume immersed in the liquid. When the whole system accelerates upwards with an acceleration of $g/3$, the fraction of volume immersed in the liquid will be

$A$ hollow sphere has an inner volume equal to half of its outer volume. If $4/5$ of its volume is submerged when placed in water,what is the density of the material of the sphere? (Density of water = $10^3 \ kg/m^3$)

$A$ balloon of mass $m$ contains water of mass $M$. If it is completely immersed in water, find the apparent mass of the balloon with water in it.

$A$ jar is filled with two non-mixing liquids $1$ and $2$ having densities $\rho_1$ and $\rho_2$ respectively. $A$ solid ball,made of a material of density $\rho_3$,is dropped in the jar. It comes to equilibrium in the position shown in the figure. Which of the following is true for $\rho_1, \rho_2$ and $\rho_3$?