The kings of Vijaynagara believed to rule on the behalf of which God?

Virupaksha

The correct answer is Option 2 - Virupaksha

Vijayanagara kings claimed to rule on behalf of the god Virupaksha.

According to local tradition, these hills sheltered the monkey kingdom of Vali and Sugriva mentioned in the Ramayana . Other traditions suggest that Pampadevi, the local mother goddess, did penance in these hills in order to marry Virupaksha, the guardian deity of the kingdom, also recognised as a form of Shiva. To this day this marriage is celebrated annually in the Virupaksha temple. Among these hills are found Jaina temples of the pre-Vijayanagara period as well. In other words, this area was associated with several sacred traditions.

It is likely that the very choice of the site of Vijayanagara was inspired by the existence of the shrines of Virupaksha and Pampadevi. In fact the Vijayanagara kings claimed to rule on behalf of the god Virupaksha. All royal orders were signed “Shri Virupaksha”, usually in the Kannada script. Rulers also indicated their close links with the gods by using the title “Hindu Suratrana”. This was a Sanskritisation of the Arabic term Sultan, meaning king, so it literally meant Hindu Sultan.