Correct answer: A. Chalukya
Explanation: The direct answer is A, Chalukya. Badami, in present-day Karnataka, was an important centre of the Early Chalukyas, also called the Badami or Vatapi Chalukyas. Its cave temples were cut into sandstone cliffs and include Hindu, Jain and Buddhist images, but the monuments are principally associated with Chalukya-period architecture. Option A is correct because the dynasty developed Badami as a major capital and sponsored these cave temples. Option B, Mughal, is wrong because Mughal architecture belongs mainly to a much later period and is not the dynasty linked with Badami’s cave temples. Option C, Lodi, refers to a Delhi Sultanate dynasty, not the rulers of Badami. Option D, Slave, refers to the Mamluk or Slave dynasty of the Delhi Sultanate, also unrelated to the origin of these caves. The fact chain is: Badami caves → Karnataka → Early Chalukyas. Memory cue: Badami and Pattadakal are key Chalukya architecture sites.