Harihara and Bukka established an independent state in Karnataka and established the capital city Vijayanagar on the banks of river _____ in 1336.

Tungabhadra

The correct answer is option 2- Tungabhadra.

Harihara and Bukka established the Vijayanagara Empire in 1336, and the capital city of Vijayanagara was located on the banks of the Tungabhadra River in modern-day Karnataka, India.

The other options are incorrect:

• The Betwa River flows through central India, not Karnataka.
• The Mahanadi River flows through eastern India, not Karnataka.
• The Tapi River flows through western India, not Karnataka.