The correct answer is A) Gulbarga.
The Bahmani Sultanate was a Muslim sultanate that ruled over the Deccan region of India from 1347 to 1518. It was founded by Alauddin Bahman Shah, a former general of the Delhi Sultanate. The capital of the Bahmani Sultanate was Gulbarga, which is now in the state of Karnataka.
The Bahmani Sultanate was a powerful and influential state. It was a major center of Islamic culture and learning. The Bahmani Sultans were patrons of the arts and sciences, and they built many mosques, tombs, and other monuments.
The Bahmani Sultanate declined in the 16th century, and it was eventually conquered by the Mughal Empire.
Option B) Bidar was also a capital of the Bahmani Sultanate, but it was not the first capital. It was the capital from 1422 to 1527.
Option C) Bijapur was also a capital of the Bahmani Sultanate, but it was not the first capital. It was the capital from 1527 to 1619.
Option D) Hyderabad was not a capital of the Bahmani Sultanate. It was founded in 1591 by Muhammad Quli Qutb Shah, the founder of the Qutb Shahi dynasty.