$A$ Carnot heat engine has an efficiency of $50 \%$. The temperature of the sink is maintained at $500 \ K$. To increase the efficiency up to $80 \%$, the required increment in the source temperature is: (in $K$)

Show the four steps of a Carnot engine in a $P-V$ graph,write the equation for each step,and obtain the work done by the system. Also,derive the efficiency of a Carnot engine.

The sink temperature of a Carnot engine is $27^{\circ}C$ and its efficiency is $25\%$. What is the temperature of the source in $^{\circ}C$?

$A$ reversible engine has an efficiency of $\frac{1}{4}$. If the temperature of the sink is reduced by $58^{\circ}C$,its efficiency becomes double. Calculate the temperature of the sink in $^{\circ}C$.

$A$ Carnot engine has an efficiency of $\frac{1}{6}$. When the temperature of the sink is lowered by $57 \ K$,its efficiency becomes $\frac{1}{3}$. The temperature of the source is: (in $K$)

Two Carnot engines $A$ and $B$ are operated in series. The engine $A$ receives heat from the source at temperature $T_1$ and rejects the heat to the sink at temperature $T$. The second engine $B$ receives the heat at temperature $T$ and rejects to its sink at temperature $T_2$. For what value of $T$ are the efficiencies of the two engines equal?