At the top of a mountain,a thermometer reads $7^{\circ}C$ and a barometer reads $70 \ cm-Hg$. At the base of the mountain,the thermometer reads $27^{\circ}C$ and the barometer reads $76 \ cm-Hg$. The ratio of the density of air at the top of the mountain to the density of air at the base is:

At a temperature of $300 \ K$,the average translational kinetic energy and $rms$ speed of a sample of oxygen gas are $6.21 \times 10^{-21} \ J$ and $484 \ m/s$ respectively. At $600 \ K$,these values will be respectively: (Assume ideal gas behavior)

Two boxes containing different ideal gases are placed on a table. Box $A$ contains $1 \text{ mole}$ of nitrogen gas at temperature $T_o$,and box $B$ contains $1 \text{ mole}$ of helium gas at temperature $(7/3) T_o$. If they are brought into thermal contact such that both gases reach a final common temperature $T_f$,what is the final temperature $T_f$ in terms of $T_o$? (Neglect the heat capacity of the boxes.)

This question has Statement-$1$ and Statement-$2$. Of the four choices given after the Statements,choose the one that best describes the two Statements.
Statement-$1$: The internal energy of a perfect gas is entirely kinetic and depends only on the absolute temperature of the gas and not on its pressure or volume.
Statement-$2$: $A$ perfect gas is heated keeping pressure constant and later at constant volume. For the same amount of heat,the temperature rise of the gas at constant pressure is lower than that at constant volume.

$A$ molecule in a gas container hits a horizontal wall with speed $200 \; m s^{-1}$ at an angle of $30^{\circ}$ with the normal,and rebounds with the same speed. Is momentum conserved in the collision? Is the collision elastic or inelastic?

Given below are two statements:
Statement $I$: The temperature of a gas is $-73^{\circ} C$. When the gas is heated to $527^{\circ} C$,the root mean square speed of the molecules is doubled.
Statement $II$: The product of pressure and volume of an ideal gas is equal to the translational kinetic energy of the molecules.
In the light of the above statements,choose the correct answer from the options given below: