JEE Advanced 2026 — Paper 1 — PHY-III solution

JEE Advanced2026Paper 1PHY-III

As shown in the figure, five Carnot engines, each with efficiency $η$ and same number of cycles per unit time, are operating between six heat reservoirs. The amount of heat released per cycle by one engine is completely absorbed by the next engine. Consider $Q_{0}$ to be the amount of heat absorbed per cycle by the first engine and $W$ as the amount of total work done by all the engines per cycle, then the net efficiency of the system is found to be $η_{net} = \frac{W}{Q _{0}} = \frac{211}{243}$ . The value of $η$ is:

Your answer

Solution

Recurrence for heat passed down the chain. Each engine absorbs $Q_{n - 1}$ , does work $W_{n} = η Q_{n - 1}$ , and releases $Q_{n} = (1 - η) Q_{n - 1}$ .

Heat after five stages. $Q_{5} = Q_{0} (1 - η)^{5}$ .

Total work and net efficiency. Total work $W = Q_{0} - Q_{5} = Q_{0} [1 - (1 - η)^{5}]$ , so $η_{net} = 1 - (1 - η)^{5}$ .

Solve for $η$ . $1 - (1 - η)^{5} = \frac{211}{243} \Rightarrow (1 - η)^{5} = \frac{32}{243} = (\frac{2}{3})^{5}$ .

$∴ 1 - η = \frac{2}{3} \Rightarrow η = \frac{1}{3} \approx 0.33$ .

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