$A$ slender homogeneous rod of length $2L$ floats partly immersed in water,being supported by a string fastened to one of its ends,as shown. The specific gravity of the rod is $0.75$. The length of the rod that extends out of water is:

The vessel shown in the figure has two sections. The lower part is a rectangular vessel with area of cross-section $A$ and height $h$. The upper part is a conical vessel of height $h$ with base area $A$ and top area $a$,and the walls of the vessel are inclined at an angle $30^o$ with the vertical. $A$ liquid of density $\rho$ fills both the sections up to a height $2h$. Neglecting atmospheric pressure:

$A$ small air bubble is at the inner surface of the bottom of a beaker filled with cold water. Now,the water in the beaker is heated. The size of the bubble increases. The reason for this may be:

Consider the following two statements $A$ and $B$,and identify the correct choice in the given answers.
$A :$ The excess pressure inside a small liquid drop is more than that of a big drop.
$B :$ As the aeroplane moves fast on the runway,the pressure is more on the upper surface of its wings and less on the bottom surface of the wings.

Fill in the blanks:
$(i)$ The surface tension of water at critical temperature is ......
$(ii)$ Bernoulli's equation is based on the conservation of ......
$(iii)$ The energy would be ...... if $1$ large drop of water splits into $8$ small drops of water.
$(iv)$ When strong wind passes over a building,the force acting on the building is in the ......

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.)