A Carnot engine absorbs an amount $Q$ of heat from a reservoir at an absolute temperature $T$ and rejects heat to a sink at a temperature of $T/3$ . The amount of heat rejected is
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$\because \eta=1-\frac{\mathrm{T}_{2}}{\mathrm{T}_{1}}=\frac{\mathrm{W}}{\mathrm{Q}_{1}}$
where $Q_{1}=$ heat absorbed, $Q_{2}=$ heat rejected
$\Rightarrow 1-\frac{\mathrm{T} / 3}{\mathrm{T}}=\frac{\mathrm{W}}{\mathrm{Q}_{1}} \Rightarrow \frac{2}{3}=\frac{\mathrm{W}}{\mathrm{Q}_{1}}=\frac{\mathrm{Q}_{1}-\mathrm{Q}_{2}}{\mathrm{Q}_{1}}$
$\Rightarrow \frac{2}{3}=1-\frac{Q_{2}}{Q_{1}} \Rightarrow \frac{Q_{2}}{Q_{1}}=\frac{1}{3} \Rightarrow Q_{2}=\frac{Q_{1}}{3}=\frac{Q}{3}$
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Statement $I:$ If heat is added to a system, its temperature must increase.
Statement $II:$ If positive work is done by a system in a thermodynamic process, its volume must increase.
In the light of the above statements, choose the correct answer from the options given below