$A$ tub is filled with water and a wooden cube $10 \ \text{cm} \times 10 \ \text{cm} \times 10 \ \text{cm}$ is placed in the water. The wooden cube is found to float on the water with a part of it submerged in water. When a metal coin is placed on the wooden cube, the submerged part is increased by $3.87 \ \text{cm}$. The mass of the metal coin is . . . . . . gram. (Take water density as $1 \ \text{g/cm}^3$ and density of wood as $0.4 \ \text{g/cm}^3$)

$A$ soft plastic bottle,filled with water of density $1 \text{ g/cc}$,contains an inverted glass test-tube with some air (ideal gas) trapped inside,as shown in the figure. The test-tube has a mass of $5 \text{ g}$,and it is made of thick glass with a density of $2.5 \text{ g/cc}$. Initially,the bottle is sealed at atmospheric pressure $P_0 = 10^5 \text{ Pa}$,such that the volume of the trapped air is $V_0 = 3.3 \text{ cc}$. When the bottle is squeezed from the outside at a constant temperature,the pressure inside increases and the volume of the trapped air decreases. It is observed that the test-tube begins to sink at a pressure $P_0 + \Delta P$ without changing its orientation. At this pressure,the volume of the trapped air is $V_0 - \Delta V$.
Let $\Delta V = X \text{ cc}$ and $\Delta P = Y \times 10^3 \text{ Pa}$.
$(1)$ The value of $X$ is
$(2)$ The value of $Y$ is

$A$ container of large surface area is filled with a liquid of density $\rho$. $A$ cubical block of side edge $a$ and mass $M$ is floating in it with four-fifths of its volume submerged. If a coin of mass $m$ is placed gently on the top surface of the block,it is just submerged. $M$ is:

$A$ cylindrical block of area of cross-section $A$ and of material of density $\rho$ is placed in a liquid of density one-third of the density of the block. The block compresses a spring,and the compression in the spring is one-third of the length of the block. If the acceleration due to gravity is $g$,the spring constant of the spring is:

Equal masses of three liquids are kept in three identical cylindrical vessels $A$, $B$ and $C$. The densities are $\rho_A$, $\rho_B$ and $\rho_C$ with $\rho_A$ < $\rho_B$ < $\rho_C$. The force on the base will be maximum in vessel:

$A$ boat carrying steel balls is floating on the surface of water in a tank. If the balls are thrown into the tank one by one,how will it affect the level of water?