$A$ $U$-tube has a left limb with a cross-sectional area one-quarter that of the right limb. The tube contains mercury (density $13.6 \text{ g/cm}^3$). The level of mercury in the narrow (left) limb is at a distance of $36 \text{ cm}$ from the upper end of the tube. What will be the rise in the level of mercury in the right limb if the left limb is filled to the top with water (in $\text{ cm}$)? (Density of water = $1 \text{ g/cm}^3$)

Given below are two statements:
Statement $I$: Pressure of fluid is exerted only on a solid surface in contact as the fluid-pressure does not exist everywhere in a still fluid.
Statement $II$: Excess potential energy of the molecules on the surface of a liquid, when compared to interior, results in surface tension.
In the light of the above statements, choose the correct answer from the options given below.

$A$ drop of water detaches itself from the exit of a tap when $(\sigma=$ surface tension of water, $\rho=$ density of water, $R=$ radius of the tap exit, $r=$ radius of the drop $)$

The Karman line is a theoretical construct that separates the Earth's atmosphere from outer space. It is defined as the height at which the lift on an aircraft flying at the speed of a polar satellite $(8 \, km/s)$ is equal to its weight. Taking a fighter aircraft of wing area $30 \, m^2$ and mass $7500 \, kg$,the height of the Karman line above the ground will be in the range of .............. $km$. (Assume the density of air at height $h$ above the ground to be $\rho(h) = 1.2 e^{-h/10} \, kg/m^3$,where $h$ is in $km$,and the lift force to be $\frac{1}{2} \rho v^2 A$,where $v$ is the speed of the aircraft and $A$ is its wing area.)

Fill in the blanks:
$(i)$ The cohesive force between the molecules of liquid is more than the adhesive force between the molecules of the plate, then the angle of contact obtained is ...... (acute/obtuse) and the free surface has a shape of ...... (concave/convex).
$(ii)$ The cohesive force between the molecules of liquid is less than the adhesive force between the molecules of the plate, then the angle of contact obtained is ...... (acute/obtuse) and the free surface has a shape of ...... (concave/convex).
$(iii)$ $A$ large pressure is exerted on the surface of a liquid having a shape of .......... (concave/convex).

$A$ liquid drop of density $\rho$ is floating half-immersed in a liquid of density $d$. If $T$ is the surface tension,then the diameter of the liquid drop is ($g$ = acceleration due to gravity).