In the Battle of Vatapi, the Pallava ruler Narsimhavarman I defeated and killed Pulakesin II and captured the Chalukyan capital, Badami in 642 AD. Thus, he assumed the title of ‘Vatapikonda’ (Conqueror of Vatapi).
First statement is incorrect because though the traces of Pallava and Chalukya temple architecture are available, but its construction is generally attributed to the 8th century Rashtrakuta dynasty. The second statement is also incorrect because the temple is a complete dravida building