The district known for its ancient rock-cut cave temples is:

Madurai
Tirunelveli
Kanchipuram
Mahabalipuram

The correct answer is Mahabalipuram.

Mahabalipuram is a town in Tamil Nadu, India, known for its ancient rock-cut cave temples. The temples were built in the 7th and 8th centuries AD by the Pallava dynasty. They are a UNESCO World Heritage Site.

The temples are carved out of the rock face and are decorated with intricate sculptures. They are dedicated to Hindu gods and goddesses. The most famous temple is the Shore Temple, which is located on the shore of the Bay of Bengal.

The other options are incorrect. Madurai is a city in Tamil Nadu, India, known for its temples and palaces. Tirunelveli is a city in Tamil Nadu, India, known for its temples and churches. Kanchipuram is a city in Tamil Nadu, India, known for its temples and silk sarees.