Statement 1 is correct. The Buddhists in Deccan region were the first to excavate rock-cut Chaityas, which were prayer halls for Buddhist monks. These Buddhist Chaityas were carved out of solid rock and date back to as early as the 1st century BC. These rock-cut Chaityas were later imitated by the Vaishnavas, Shaivas, and Jainas in the later centuries.

Statement 2 is incorrect. While it is true that the Vaishnavas, Shaivas, and Jainas also built temples, they did not necessarily excavate them at sites far distant from rock-cut Chaityas. The location of the temples built by these religious groups varied, and they could be found in proximity to rock-cut Chaityas or even in different regions. Therefore, statement 2 is not correct.

In conclusion, option 1 is correct as it states that only statement 1 is correct. It correctly acknowledges the fact that the Buddhists in Deccan excavated rock-cut Chaityas and that other religious groups imitated this practice in later centuries.