A sphere having temperature 600 K is losing heat due to radiation. At this temperature its rate of cooling is $R$. The rate of cooling of this sphere at 400 K is $\dfrac{x}{243}$. The value of $x$ is (Temperature of surrounding is 300 K)

A black body is at temperature of 5780 K. The energy of radiation emitted by the body at wavelength 300 nm is $U_1$, at wavelength 500 nm is $U_2$ and that at 900 nm is $U_3$ respectively. Wien's constant $b = 2.89 \times 10^6\ \text{nmK}$. This shows that

The efficiency of Carnot engine which operates between the two temperatures $T_1 = 500$ K and $T_2 = 300$ K is

An ideal gas having pressure $P$, volume $V$ and temperature $T$ is expanded isothermally, to a volume $3V$ and final pressure $P_I$. The same gas is expanded adiabatically to a volume $3V$, the final pressure being $P_A$. The ratio $\dfrac{P_A}{P_I}$ is $\left(\dfrac{C_P}{C_V} = \gamma\right)$