With reference to art and archaeological history of India, which one among the following was made earliest?

The correct option is Rock-cut Elephant at Dhauli.

To determine which of the given art and architectural monuments was created the earliest, we can examine the historical periods, patron dynasties, and approximate dates associated with each option:
1. Rock-cut Elephant at Dhauli (3rd Century BCE): Located in Odisha, this sculpture is carved directly out of a rock face above the edicts of the Mauryan Emperor Ashoka. Dating back to the Maurya period (circa 272–231 BCE), it is one of the oldest rock-cut stone sculptures in India, representing the early phase of Buddhist art transition from wood and clay to stone.
2. Varaha Image at Udayagiri (5th Century CE): This famous relief sculpture showing the boar incarnation of Vishnu rescuing Goddess Earth is located in Cave 5 of the Udayagiri Caves in Madhya Pradesh. It was carved during the Gupta Empire, specifically dating to the reign of Chandragupta II in the early 5th century CE.
3. Rock-cut Monuments at Mahabalipuram (7th–8th Century CE): These structures (including the famous Shore Temple and Pancha Rathas) in Tamil Nadu were patronized by the Pallava dynasty, primarily under the reigns of Mahendravarman I and Narasimhavarman I during the 7th and 8th centuries CE.
4. Lingaraja Temple at Bhubaneswar (11th Century CE): This temple represents the peak of Kalinga temple architecture in Odisha. It was built primarily by the rulers of the Somavamshi dynasty in the 11th century CE, with later additions by the Ganga rulers.

Comparing these time periods chronologically:
- 3rd century BCE (Rock-cut Elephant at Dhauli) is significantly earlier than the 5th century CE, 7th–8th century CE, and 11th century CE.

Therefore, the Rock-cut Elephant at Dhauli is the earliest artwork among the options listed.