$CH_4$ diffuses two times faster than a gas $X$. The number of molecules present in $32 \ g$ of gas $X$ is ($N$ is Avogadro number).

The major product in the following reaction is:

Which of the following has the lowest boiling point?

$A$ point $P$ moves in a counter-clockwise direction on a circular path as shown in the figure. The movement of $P$ is such that it sweeps out a length $S = t^3 + 5$,where $S$ is in meters and $t$ is in seconds. The radius of the path is $20 \, m$. The acceleration of $P$ when $t = 2 \, s$ is nearly ......... $m/s^2$.

Two rigid boxes containing different ideal gases are placed on a table. Box $A$ contains one mole of nitrogen at temperature $T_0$,while box $B$ contains one mole of helium at temperature $(7/3) T_0$. The boxes are then put into thermal contact with each other,and heat flows between them until the gases reach a common final temperature (ignore the heat capacity of the boxes). The final temperature of the gases $T_f$ in terms of $T_0$ is:

If $a_1, a_2, \dots, a_n$ are positive real numbers whose product is a fixed number $c$,then the minimum value of $a_1 + a_2 + \dots + a_{n-1} + 2a_n$ is