A Carnot engine has efficiency $\frac{1}{6}$. It becomes $\frac{1}{3}$, when the temperature of $\operatorname{sink}$ is lowered by

57 K . The temperature of the source is

The r.m.s. speed of gas molecules at 800 K will be

If a black body at 400 K surrounded by atmosphere at 300 K has rate of cooling ' $\mathrm{R}_0$ ', the same body at 900 K , surrounded by same atmosphere, will have rate of cooling nearly

The temperature of an ideal gas is increased from 100 K to 400 K . If ' $x$ ' is the R.M.S. velocity of its molecules at 100 K , it becomes