The correct answer is: A) Bangalore.
The British East India Company established a trading post in Bangalore in 1689. In 1799, the British took control of Mysore, the kingdom that ruled over Bangalore. The British made Bangalore the capital of their new province of Mysore. Bangalore remained the capital of Mysore until 1947, when India gained independence. After independence, Bangalore became the capital of the new state of Karnataka.
Option B, Mysore, is incorrect because Mysore was the capital of the kingdom that ruled over Bangalore, not the British administration.
Option C, Madras (Chennai), is incorrect because Madras was the capital of the British Presidency of Madras, which included parts of present-day Tamil Nadu, Andhra Pradesh, Karnataka, and Kerala.
Option D, Bombay (Mumbai), is incorrect because Bombay was the capital of the British Presidency of Bombay, which included parts of present-day Maharashtra, Gujarat, Goa, and Daman and Diu.