Step 1: Understand the Index of Industrial Production (IIP).
IIP is a measure of industrial production in a country, including various sectors like manufacturing, mining, and electricity. The manufacturing sector has a significant weight in the IIP.
Step 2: Evaluate each option.
Option (A): Correct, the manufacturing sector's weight in the IIP during 2020-2021 is approximately 78%.
Option (B): Incorrect, the correct value is 78%, not 81%.
Option (C): Incorrect, the manufacturing sector's weight is not 65%.
Option (D): Incorrect, the correct value is not 71%.
Thus, the correct answer is (A).