$A$ Carnot engine operates between a source and a sink. The efficiency of the engine is $40 \%$ and the temperature of the sink is $27^{\circ} C$. If the efficiency is to be increased to $50 \%$,then the temperature of the source must be increased by: (in $K$)

The efficiency of an ideal heat engine working between the freezing point and boiling point of water is ........ $\%$

$A$ Carnot engine having an efficiency of $40\%$ takes heat from a source maintained at a temperature of $500 \ K$. If the efficiency is to be increased to $60\%$ while keeping the sink temperature constant,what should be the new source temperature in $K$?

$A$ Carnot engine whose efficiency is $40\%$ takes in heat from a source maintained at a temperature of $500 \ K$. It is desired to have an engine of efficiency $60\%$. Then,the intake temperature for the same exhaust (sink) temperature must be (in $K$)

$A$ Carnot engine whose efficiency is $40 \%$, receives heat at $500 \ K$. If the efficiency is to be $50 \%$, the source temperature for the same exhaust temperature is (in $K$)

Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures $T_{1}$ and $T_{2}$. The temperature of the hot reservoir of the first engine is $T_{1}$ and the temperature of the cold reservoir of the second engine is $T_{2}$. $T$ is the temperature of the sink of the first engine,which is also the source for the second engine. How is $T$ related to $T_{1}$ and $T_{2}$,if both engines perform equal amounts of work?