Under which ideal condition does the efficiency of a Carnot engine become $100 \%$?

Suppose that two heat engines are connected in series,such that the heat exhaust of the first engine is used as the heat input of the second engine as shown in the figure. The efficiencies of the engines are $\eta_1$ and $\eta_2$,respectively. The net efficiency of the combination is given by

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?

$A$ Carnot engine whose sink is at $300\, K$ has an efficiency of $40\%.$ By how much should the temperature of the source be increased so as to increase its efficiency by $50\%$ of its original efficiency? (in $K$)

$A$ Carnot engine with efficiency $50\%$ takes heat from a source at $600\text{ K}$. To increase the efficiency by $20\%$, keeping the temperature of the sink the same, the new temperature of the source will be: (in $\text{ K}$)

State Carnot's theorem.