Given below are two statements:
Statement-$I$: The hot water flows faster than cold water.
Statement-$II$: Soap water has higher surface tension as compared to fresh water.
In the light of the above statements,choose the correct answer from the options given below:

$A$ spray gun is shown in the figure where a piston pushes air out of a nozzle. $A$ thin tube of uniform cross section is connected to the nozzle. The other end of the tube is in a small liquid container. As the piston pushes air through the nozzle,the liquid from the container rises into the nozzle and is sprayed out. For the spray gun shown,the radii of the piston and the nozzle are $20 \ mm$ and $1 \ mm$ respectively. The upper end of the container is open to the atmosphere.
$1.$ If the piston is pushed at a speed of $5 \ mm \ s^{-1}$,the air comes out of the nozzle with a speed of
$(A)$ $0.1 \ m \ s^{-1}$ $(B)$ $1 \ m \ s^{-1}$ $(C)$ $2 \ m \ s^{-1}$ $(D)$ $8 \ m \ s^{-1}$
$2.$ If the density of air is $\rho_{a}$ and that of the liquid is $\rho_{\ell}$,then for a given piston speed,the rate (volume per unit time) at which the liquid is sprayed will be proportional to
$(A)$ $\sqrt{\frac{\rho_{a}}{\rho_{\ell}}}$ $(B)$ $\sqrt{\rho_{a} \rho_{\ell}}$ $(C)$ $\sqrt{\frac{\rho_{\ell}}{\rho_{a}}}$ $(D)$ $\rho_{\ell}$

$A$ cylindrical capillary tube of $0.2 \ mm$ radius is made by joining two capillaries $T_1$ and $T_2$ of different materials having water contact angles of $0^{\circ}$ and $60^{\circ}$,respectively. The capillary tube is dipped vertically in water in two different configurations,case $I$ and $II$ as shown in the figure. Which of the following option$(s)$ is(are) correct?
(Surface tension of water $= 0.075 \ N/m$,density of water $= 1000 \ kg/m^3$,take $g = 10 \ m/s^2$)
$(1)$ The correction in the height of the water column raised in the tube,due to the weight of water contained in the meniscus,will be different for both cases.
$(2)$ For case $I$,if the capillary joint is $5 \ cm$ above the water surface,the height of the water column raised in the tube will be more than $8.75 \ cm$. (Neglect the weight of the water in the meniscus)
$(3)$ For case $I$,if the joint is kept at $8 \ cm$ above the water surface,the height of the water column in the tube will be $7.5 \ cm$. (Neglect the weight of the water in the meniscus)
$(4)$ For case $II$,if the capillary joint is $5 \ cm$ above the water surface,the height of the water column raised in the tube will be $3.75 \ cm$. (Neglect the weight of the water in the meniscus)

The temperature of an air bubble while rising from the bottom to the surface of a lake remains constant, but its diameter is doubled. If the pressure on the surface is $h$ meters of mercury column and the relative density of mercury is $\rho$, then the depth of the lake is: (in $\rho h$)

Two identical cylindrical vessels are kept on the ground and each contain the same liquid of density $d.$ The area of the base of both vessels is $S$ but the height of liquid in one vessel is $x_{1}$ and in the other,$x_{2}$. When both cylinders are connected through a pipe of negligible volume very close to the bottom,the liquid flows from one vessel to the other until it comes to equilibrium at a new height. The change in energy of the system in the process is

$A$ pump motor is used to deliver water at a certain rate from a given pipe. To obtain twice as much water from the same pipe in the same time, the power of the motor has to be increased to: (in $times$)