To solve this question, we need to apply the principles of the Carnot reversible engine and the efficiency formula for heat engines. Let's break down the problem and solve it step by step:

1. Given information:
- The Carnot reversible engine converts 1/6 of heat input into work.
- When the temperature of the sink is reduced by 62 K, the efficiency of Carnot’s cycle becomes 1/3.

2. Efficiency of a Carnot engine:
The efficiency (η) of a Carnot engine is given by the formula:
η = 1 - (Tc / Th)
where Tc is the temperature of the cold sink and Th is the temperature of the hot source.

3. Let's assume the initial temperature of the hot source is Th1 and the initial temperature of the cold sink is Tc1.

4. From the given information, we can set up the following equations:
- Work output / Heat input = 1/6
=> Work output = (1/6) * Heat input
- New efficiency after reducing the sink temperature:
η_new = 1/3
- Temperature difference after reducing the sink temperature:
ΔT = 62 K

5. Using the efficiency formula, we can write the following equation:
1/3 = 1 - ((Tc1 - ΔT) / Th1)

6. Rearranging the equation, we get:
(Tc1 - ΔT) / Th1 = 2/3

7. Since we are given the ratio of Tc1 to Th1, we can assume any values for Tc1 and Th1 as long as their ratio is maintained. Let's assume Tc1 = 2 and Th1 = 3.

8. Plugging in the values, we get:
(2 - 62) / 3 = 2/3
-60 / 3 = 2/3
-20 = 2/3 (which is not true)

9. Therefore, our initial assumption was incorrect. Let's try another set of values. Let's assume Tc1 = 20 and Th1 = 30.

10. Plugging in the values, we get:
(20 - 62) / 30 = 2/3
-42 / 30 = 2/3
-7/5 ≠ 2/3

11. Again, our assumption was incorrect. Let's try another set of values. Let's assume Tc1 = 300 and Th1 = 400.

12. Plugging in the values, we get:
(300 - 62) / 400 = 2/3
238 / 400 = 2/3
0.595 ≠ 2/3

13. Once again, our assumption was incorrect. Let's try another set of values. Let's assume Tc1 = 310 and Th1 = 372.

14. Plugging in the values, we get:
(310 - 62) / 372 = 2/3
248 / 372 = 2/3
0.666 = 2/3 (which is true)

15. Therefore, the correct answer is option 'A': 372 K (source temperature)