The Madras Presidency was a major administrative division during:

Vijayanagara rule
British colonial rule
Nayak period
Mughal Empire

The correct answer is: b) British colonial rule.

The Madras Presidency was a major administrative division of British India. It was formed in 1639 by the British East India Company and was headquartered in the city of Madras (now Chennai). The presidency covered an area of over 100,000 square miles and had a population of over 20 million people. It was one of the most important provinces of British India and played a major role in the development of the country.

The Vijayanagara Empire was a Hindu empire that ruled over southern India from the 14th to the 16th centuries. It was one of the largest and most powerful empires in Indian history. The Nayak period was a period of political instability in southern India that followed the decline of the Vijayanagara Empire. The Mughal Empire was a Muslim empire that ruled over northern India from the 16th to the 18th centuries. It was one of the largest and most powerful empires in Indian history.