$50 \, g$ of ice at $0^{\circ} C$ is dropped into a calorimeter containing $100 \, g$ of water at $30^{\circ} C$. If the thermal capacity of the calorimeter is zero,then the amount of ice left in the mixture at equilibrium is .......... $g$.

If a gas contains only three molecules that move with velocities of $100, 200, 500 \ ms^{-1}$. What is the $rms$ velocity of that gas in $ms^{-1}$?

$A$ light wave has a frequency of $4 \times 10^{14} ~Hz$ and a wavelength of $5 \times 10^{-7} ~m$ in a medium. The refractive index of the medium is:

An ideal gas expands isothermally from a volume $V_1$ to $V_2$ and is then compressed to the original volume $V_1$ adiabatically. The initial pressure is $P_1$ and the final pressure is $P_3$. The total work done is $W$. Then:

$A$ bag contains $2n+1$ coins. It is known that $n$ of these coins have a head on both sides,whereas the remaining $n+1$ coins are fair. $A$ coin is picked up at random from the bag and tossed. If the probability that the toss results in a head is $\frac{31}{42}$,then $n$ is equal to

If the cold junction is held at $0^{\circ} C$,the thermo emf $V$ of a thermocouple varies as $V = 10 \times 10^{-6} t - \frac{1}{40} \times 10^{-6} t^2$,where $t$ is the temperature of the hot junction in $^{\circ} C$. The neutral temperature and the maximum value of thermo emf are respectively: