One mole of an ideal gas expands adiabatically from an initial state $$(T_A, V_0)$$ to final state $$(T_f, 5V_0)$$. Another mole of the same gas expands isothermally from a different initial state $$(T_B, V_0)$$ to the same final state $$(T_f, 5V_0)$$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $$\gamma$$. What is the ratio $$T_A/T_B$$?

For an ideal gas undergoing an adiabatic process, the relation between temperature and volume is
$$T\,V^{\gamma-1}= \text{constant} \quad -(1)$$

Case 1: Adiabatic expansion
Initial state $$\left(T_A,V_0\right)$$ → final state $$\left(T_f,5V_0\right)$$.
Using $$(1):$$
$$T_A\,V_0^{\gamma-1}=T_f\,(5V_0)^{\gamma-1}$$
$$\Longrightarrow\; T_A=T_f\,5^{\gamma-1} \quad -(2)$$

Case 2: Isothermal expansion
Initial state $$\left(T_B,V_0\right)$$ → final state $$\left(T_f,5V_0\right)$$.
For an isothermal process, temperature is constant, so
$$T_B=T_f \quad -(3)$$

Dividing $$(2)$$ by $$(3):$$
$$\frac{T_A}{T_B}= \frac{T_f\,5^{\gamma-1}}{T_f}=5^{\gamma-1}$$

Hence the required ratio is $$T_A/T_B = 5^{\gamma-1}$$.

Option A which is: $$5^{\gamma-1}$$