$A$ stream-lined body falls through air from a height $h$ on the surface of a liquid. Let $d$ and $D$ denote the densities of the materials of the body and the liquid respectively. If $D$ > $d$, then the time after which the body will be instantaneously at rest is

$A$ non-conducting body floats in a liquid at $20^{\circ} C$ with $\frac{2}{3}$ of its volume immersed in the liquid. When the liquid temperature is increased to $100^{\circ} C$,$\frac{3}{4}$ of the body's volume is immersed in the liquid. Then the coefficient of real expansion of the liquid is (neglecting the expansion of the container of the liquid):

$A$ container of liquid is in free fall. Without any liquid spilling out, does this container obey the principle of Archimedes?

$A$ $20 \,g$ copper block is suspended by a vertical spring causing $1 \,cm$ elongation over the natural length of the spring. If a beaker of water is placed below the block so that the copper block is completely immersed in the liquid, the elongation of the spring is (Density of copper = $9000 \,kg/m^3$, Density of water = $1000 \,kg/m^3$, $g = 10 \,m/s^2$) (in $\,cm$)

$A$ boy has a mass of $60\, kg$. He wants to swim in a river with the help of a wooden log. If the relative density of the wood is $0.6$,what is the minimum volume of the wooden log required? (Density of river water is $1000\, kg/m^3$)

$A$ 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