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 the engines perform equal amounts of work?
Answer: (B) $T = \dfrac{T_1 + T_2}{2}$
For a Carnot engine $Q \propto T$. Let the first engine take $Q_1$ at $T_1$ and reject $Q$ at $T$; the second takes this $Q$ and rejects $Q_2$ at $T_2$.
$$W_1 = Q_1 - Q \propto T_1 - T, \qquad W_2 = Q - Q_2 \propto T - T_2$$
Equal work: $T_1 - T = T - T_2 \Rightarrow T = \dfrac{T_1 + T_2}{2}$.
Solution by Sreeraj P, M.Sc Physics