Explanation:

 . First statement is incorrect. The Badami Chalukya architecture was a temple building idiom that evolved in the 5th – 8th centuries in the Malaprabha river basin, in present-day Bagalkot district of Karnataka state, under the Chalukya dynasty. This style is sometimes called the Vesara style and Chalukya style, a term that also includes the much later Western Chalukya architecture of the 11th and 12th centuries.
Second is correct. The Badami cave temples represent some of the earliest known examples of Hindu temples in the Deccan region. They are composed of four caves, all carved out of the soft Badami sandstone on a hill cliff in the late 6th to 7th centuries.
Caves 1 to 4 are in the escarpment of the hill in soft Badami sandstone formation, to the south-east of the town.
• In Cave 1, among various sculptures of Hindu divinities and themes, a prominent carving is of the Tandava-dancing Shiva as Nataraja.
• Cave 2 is mostly similar to Cave 1 in terms of its layout and dimensions, featuring Hindu subjects of which the relief of Vishnu as Trivikrama is the largest.
• The largest cave is Cave 3, featuring Vishnu-related mythology, and it is also the most intricately carved cave in the complex.
• Cave 4 is dedicated to revered figures of Jainism.
Third statement is correct. Of the total four temples, three of those belonging to the Hindu faith are manmade caves while the fourth one, also the only natural cave in the vicinity belongs to the Jain religion. The Badami cave temples are composed of four caves all carved out of the soft Badami sandstone on a hill cliff in the late 6th to 7th centuries.
Fourth statement is incorrect. The Badami cave temples are a complex of four Hindu, a Jain and possibly Buddhist cave temples located in Badami, a town in the Bagalkot district in northern part of Karnataka, India.