Two Carnot engines A and B are operated in series. Engine A receives heat from a reservoir at 600 K and rejects heat to a reservoir at temperature T. Engine B receives heat rejected by engine A and in turn rejects it to a reservoir at 100 K. If the efficiencies of the two engines A and B are represented by ηA and ηB respectively, then what is the value of ηB/ηA?

Efficiency is given by ηA = 1 − T/600 (for engine A)
Efficiency is given by ηB = 1 − 100/T (for engine B)
The work produced by engine A and B is same so
intermediate temperature T = √(T1*T3)
So T = √(600 * 100) = √60000 = 244.95 ≈ 245 K
On putting value in efficiency we get
ηA = 1 - 245/600 = 355/600 = 0.5917
ηB = 1 - 100/245 = 145/245 = 0.5918
ηB/ηA = 0.5918/0.5917 ≈ 1
However, if we assume that the work done by both engines are equal, then:
W_A = Q_H(1 - T/600) = Q_C(T/100 -1)
W_B = Q_C(1 - 100/T)
If W_A = W_B
Q_H(1 - T/600) = Q_C(1 - 100/T)
Since Q_C = Q_H(1 - T/600), then
Q_H(1 - T/600) = Q_H(1 - T/600)(1 - 100/T)
1 = 1 - 100/T
This equation is incorrect. Let's use the fact that the heat rejected by A is the heat absorbed by B.
Let Q_H be the heat absorbed by A. Then the heat rejected by A is Q_H (1 - ηA) = Q_H (T/600).
This is the heat absorbed by B. The work done by B is then Q_H (T/600)(1 - 100/T) = Q_H(T/600 - 100/600) = W_B
The work done by A is W_A = Q_H(1 - T/600)
If W_A = W_B, which is a reasonable assumption for this problem (though not explicitly stated), then
Q_H(1 - T/600) = Q_H(T/600 - 100/600)
1 - T/600 = T/600 - 100/600
1 = 2T/600 - 100/600
600 = 2T - 100
700 = 2T
T = 350 K
ηA = 1 - 350/600 = 250/600 = 5/12
ηB = 1 - 100/350 = 250/350 = 5/7
ηB/ηA = (5/7)/(5/12) = 12/7 ≈ 1.714
This is not one of the options. There must be some error in the problem statement or the given options.