Two heat engines $X$ and $Y$ of same efficiency are connected in series in such a way that the sink of $X$ works as the source of $Y$. $X$ receives heat at $900 \,K$ and rejects some heat to its sink at $T \,K$, and in turn, $Y$ rejects heat to its sink at $400 \,K$. Then the temperature $T$ is: (in $\,K$)

$A$ Carnot engine $(E)$ is working between two temperatures $473 \ K$ and $273 \ K$. In a new system,two engines are used: engine $E_1$ works between $473 \ K$ and $373 \ K$,and engine $E_2$ works between $373 \ K$ and $273 \ K$. If $\eta_{12}$,$\eta_1$,and $\eta_2$ are the efficiencies of the engines $E$,$E_1$,and $E_2$ respectively,then:

An engine has an efficiency of $1/6$. When the temperature of the sink is reduced by $62^{\circ}C$,its efficiency is doubled. The temperature of the source is ....... $^{\circ}C$.

$A$ Carnot engine takes $3 \times 10^6 \, \text{cal}$ of heat from a reservoir at $627^{\circ}C$ and gives it to a sink at $27^{\circ}C$. The work done by the engine is:

Consider a Carnot cycle operating between $T_1 = 500\, K$ and $T_2 = 300\, K$ producing $1\, kJ$ of mechanical work per cycle. Find the heat transferred to the engine by the reservoirs.

When the temperature difference between the source and sink increases,the efficiency of the heat engine