$A$ Carnot engine having efficiency $60 \%$ receives heat from a source at a temperature $600 \text{ K}$. For the same sink temperature, to increase its efficiency to $80 \%$, the temperature of the source is (in $\text{ K}$)

$A$ Carnot heat engine works with an ideal diatomic gas and an adiabatic volume expansion ratio of $32$. Then its efficiency is ....... $\%$

Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures $T_{1}$ and $T_{2}$. The temperature of the hot reservoir of the first engine is $T_{1}$ and the temperature of the cold reservoir of the second engine is $T_{2}$. $T$ is the temperature of the sink of the first engine,which is also the source for the second engine. How is $T$ related to $T_{1}$ and $T_{2}$,if both engines perform equal amounts of work?

State Carnot's theorem.

Determine the efficiency of a Carnot cycle if,in the adiabatic expansion,the volume becomes $3$ times its initial value and $\gamma = 1.5$.

The efficiency of a Carnot engine operating between reservoirs maintained at temperatures $27^{\circ}C$ and $-123^{\circ}C$ is ...... $\%$