The duty of irrigation water is the amount of water required to irrigate one hectare of land in one second. To calculate the duty of irrigation water, we need to determine the volume of water required for the transplantation of rice and then convert it into cubic meters per second (cumecs).

1. Volume of water required for transplantation:
- The total depth of water required during transplantation is 48 cm, which is equivalent to 0.48 meters.
- The area of land to be irrigated is not mentioned in the question. Let's assume it is one hectare (10,000 square meters).
- The volume of water required for transplantation can be calculated by multiplying the depth of water by the area of land:
Volume = Depth x Area
= 0.48 m x 10,000 m²
= 4,800 cubic meters

2. Effective rainfall during transplantation:
- The effective rainfall is given as 8 cm, which is equivalent to 0.08 meters.
- The volume of water provided by effective rainfall can be calculated by multiplying the effective rainfall by the area of land:
Volume = Effective rainfall x Area
= 0.08 m x 10,000 m²
= 800 cubic meters

3. Total volume of water required for transplantation:
- The total volume of water required for transplantation is the sum of the volume of water required and the volume of water provided by effective rainfall:
Total volume = Volume required + Volume provided
= 4,800 cubic meters + 800 cubic meters
= 5,600 cubic meters

4. Time required for transplantation:
- The transplantation of rice requires 10 days.
- To convert this into seconds, we multiply by the number of seconds in a day (24 hours x 60 minutes x 60 seconds):
Time = 10 days x 24 hours/day x 60 minutes/hour x 60 seconds/minute
= 8,640,000 seconds

5. Duty of irrigation water:
- The duty of irrigation water is the volume of water required per second to irrigate one hectare of land.
- To calculate the duty of irrigation water, we divide the total volume of water required by the time required for transplantation:
Duty of irrigation water = Total volume / Time
= 5,600 cubic meters / 8,640,000 seconds
≈ 0.000648 cubic meters per second
≈ 0.648 liters per second

The duty of irrigation water is approximately 0.648 liters per second. Converting this into hectare/cumecs, we divide by 10000 (one hectare) and multiply by 1000 (to convert cubic meters to liters):
Duty of irrigation water = 0.648 liters per second x (1000 liters per cubic meter) / 10000 square meters
= 0.0648 cubic meters per second
≈ 0.065 cumecs

Therefore, the correct answer is option B) 0.065 cumecs.