$A$ gas is allowed to expand in an insulated container against a constant external pressure of $10 \ atm$ from an initial volume $4.2 \ L$ to $5.2 \ L$. The change in internal energy $\Delta E$ of the gas will be $:-$

The ratio of the specific heat capacities of a gas is $1.5$. When the gas undergoes an adiabatic process, its volume is doubled and pressure becomes $P_1$. When the gas undergoes an isothermal process, its volume is doubled and pressure becomes $P_2$. If $P_1 = P_2$, the ratio of the initial pressures of the gas when it undergoes adiabatic and isothermal processes is

If $z = \frac{y}{x} \left[ \sin \frac{x}{y} + \cos \left( 1 + \frac{y}{x} \right) \right]$,then $x \frac{\partial z}{\partial x}$ is equal to

Which of the following compounds is optically active?

$A$ uniform rod of mass $M$ and length $L$ is placed on a smooth horizontal surface. $A$ horizontal force $F$ is applied at one end of the rod perpendicular to its length. The acceleration of the centre of mass of the rod is:

The ratio of most probable velocity $(u_{mp})$,average velocity $(u_{av})$,and root mean square velocity $(u_{rms})$ is .......