$$100 \mathrm{~g}$$ of ice at $$0^{\circ} \mathrm{C}$$ is mixed with $$100 \mathrm{~g}$$ of water at $$100^{\circ} \mathrm{C}$$. The final temperature of the mixture is

[Take, $$L_f=3.36 \times 10^5 \mathrm{~J} \mathrm{~kg}^{-1}$$ and $$S_w=4.2 \times 10^3 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1} \text { ] }$$

The $$p$$-$$V$$ diagram of a Carnot's engine is shown in the graph below. The engine uses 1 mole of an ideal gas as working substance. From the graph, the area enclosed by the $$p$$-$$V$$ diagram is [The heat supplied to the gas is 8000 J]

The speed of sound in an ideal gas at a given temperature $$T$$ is $$v$$. The rms speed of gas molecules at that temperature is $$v_{\text {rms }}$$. The ratio of the velocities $$v$$ and $$v_{\text {rms }}$$ for helium and oxygen gases are $$X$$ and $$X^{\prime}$$ respectively. Then, $$\frac{X}{X^{\prime}}$$ is equal to

Pressure of ideal gas at constant volume is proportional to .........