A Carnot engine $(E)$ is working between two temperatures $473\ \text{K}$ and $273\ \text{K}$. In a new system, two engines are used: engine $E_1$ works between $473\ \text{K}$ and $373\ \text{K}$, and engine $E_2$ works between $373\ \text{K}$ and $273\ \text{K}$. If $\eta_{12}$, $\eta_1$ and $\eta_2$ are the efficiencies of the engines $E$, $E_1$ and $E_2$, respectively, then
Answer: (D) $\eta_{12} < \eta_1 + \eta_2$
$$\eta_{12} = 1 - \frac{273}{473} = \frac{200}{473} \approx 0.423$$
$$\eta_1 = 1 - \frac{373}{473} = \frac{100}{473} \approx 0.211, \qquad \eta_2 = 1 - \frac{273}{373} = \frac{100}{373} \approx 0.268$$
$\eta_1 + \eta_2 \approx 0.479 > \eta_{12}$.
(In general $1 - \eta_{12} = (1-\eta_1)(1-\eta_2)$, so $\eta_{12} = \eta_1 + \eta_2 - \eta_1\eta_2 < \eta_1 + \eta_2$.)
Solution by Sreeraj P, M.Sc Physics