$A$ Carnot engine has the same efficiency between $800 \ K$ to $500 \ K$ and $x \ K$ to $600 \ K$. The value of $x$ is ...... $K$. (in $K$)

Three Carnot engines operate in series between a heat source at temperature $T_1$ and a heat sink at a temperature $T_4$. There are two other reservoirs at temperatures $T_2$ and $T_3$. If the three engines are equally efficient,find the values of $T_2$ and $T_3$ in terms of $T_1$ and $T_4$,given that $T_1 > T_2 > T_3 > T_4$.

Given below are two statements: one is labelled as Assertion $A$ and the other is labelled as Reason $R$.
Assertion $A$: Efficiency of a reversible heat engine will be highest at $-273^{\circ} C$ temperature of cold reservoir.
Reason $R$: The efficiency of Carnot's engine depends not only on the temperature of the cold reservoir but it depends on the temperature of the hot reservoir too and is given as $\eta = (1 - \frac{T_2}{T_1})$.
In the light of the above statements,choose the correct answer from the options given below:

$A$ Carnot engine whose low temperature reservoir is at $7\,^{\circ}C$ has an efficiency of $50\%$. It is desired to increase the efficiency to $70\%$. By how many degrees should the temperature of the high temperature reservoir be increased? (in $K$)

$A$ 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 $K$)

The efficiency of a Carnot engine is $25 \%$,when the temperature of the sink is $300 \ K$. The increase in the temperature of the source required for the efficiency to become $50 \%$ is (in $K$)