The Kalyani Chalukyas established their capital at:

The correct answer is: B) Kalyani.

The Kalyani Chalukyas were a dynasty of South Indian rulers who ruled from the 11th to the 13th centuries. They were founded by Tailapa II, who established his capital at Kalyani in 1000 CE. The Kalyani Chalukyas were a powerful dynasty who ruled over a large part of South India. They were patrons of art and literature, and their reign was a golden age for South Indian culture.

The other options are incorrect:

• Warangal was the capital of the Kakatiya dynasty.
• Bidar was the capital of the Bahmani Sultanate.
• Gulbarga was the capital of the Adilshahi Sultanate.