Three designs are proposed for an engine which is to operate between $500 \ K$ and $300 \ K$. Design $A$ claims to produce $150 \ J$ of work per $1000 \ J$ of heat input, design $B$ $450 \ J$ of work per $1000 \ J$, and design $C$ $300 \ J$ of work per $1000 \ J$. Which of the designs would you choose?

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

$A$ Carnot engine operating between temperatures $T_1$ and $T_2$ has an efficiency of $0.2$. When $T_2$ is lowered by $45 \text{ K}$, its efficiency becomes $0.5$. The temperatures $T_1$ and $T_2$ are respectively:

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

This question has Statement $1$ and Statement $2.$ Of the four choices given after the Statements,choose the one that best describes the two Statements.
Statement $1 :$ An inventor claims to have constructed an engine that has an efficiency of $30\%$ when operated between the boiling and freezing points of water. This is not possible.
Statement $2:$ The efficiency of a real engine is always less than the efficiency of a Carnot engine operating between the same two temperatures.

$A$ heat engine operates between a cold reservoir at temperature $T_{2} = 400 \, K$ and a hot reservoir at temperature $T_{1}$. It takes $300 \, J$ of heat from the hot reservoir and delivers $240 \, J$ of heat to the cold reservoir in a cycle. The minimum temperature of the hot reservoir has to be $.... K$.