The spring balance $A$ reads $2 \, kg$ with a block $m$ suspended from it. $A$ balance $B$ reads $5 \, kg$ when a beaker filled with liquid is put on the pan of the balance. The two balances are now arranged such that the hanging mass is inside the liquid as shown in the figure. In this situation:

An ornament weighs $10\, g$ in air and $6\, g$ in water. The density of the material of the ornament is $20\, g/cc$. The volume of the cavity in the ornament is ........ $cc$.

$A$ cubical block of wood of specific gravity $0.5$ and a chunk of concrete of specific gravity $2.5$ are fastened together. The ratio of the mass of wood to the mass of concrete which makes the combination to float with its entire volume submerged under water is

$A$ body of density $d_1$ is counterpoised by $M$ of weights of density $d_2$ in air of density $d$. Then the true mass of the body is

$A$ piece of metal weighs $45 \ g$ in air and $25 \ g$ in a liquid of density $1.5 \times 10^3 \ kg \ m^{-3}$ kept at $30^{\circ} C$. When the temperature of the liquid is raised to $40^{\circ} C$,the metal piece weighs $27 \ g$. The density of the liquid at $40^{\circ} C$ is $1.25 \times 10^3 \ kg \ m^{-3}$. The coefficient of linear expansion of the metal is:

$A$ 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