The most important river in Tamil Nadu, originating in Karnataka, is the:

Kaveri (Cauvery)
Vaigai
Palar
Thamirabarani

The correct answer is (a) Kaveri (Cauvery).

The Kaveri (Cauvery) is the longest river in the Indian state of Tamil Nadu. It originates in the Western Ghats of Karnataka and flows through the states of Karnataka, Tamil Nadu, and Puducherry. The Kaveri is a perennial river and is the primary source of water for irrigation and drinking water in Tamil Nadu. The river is also a major source of hydroelectric power.

The Vaigai is a river in the Indian state of Tamil Nadu. It originates in the Western Ghats and flows through the districts of Madurai, Theni, Dindigul, and Sivagangai. The Vaigai is a major source of irrigation and drinking water in Madurai district. The river is also a major source of hydroelectric power.

The Palar is a river in the Indian state of Tamil Nadu. It originates in the Western Ghats and flows through the districts of Dharmapuri, Krishnagiri, Vellore, Tirupattur, and Ranipet. The Palar is a major source of irrigation and drinking water in these districts. The river is also a major source of hydroelectric power.

The Thamirabarani is a river in the Indian state of Tamil Nadu. It originates in the Western Ghats and flows through the districts of Tirunelveli, Thoothukudi, and Ramanathapuram. The Thamirabarani is a major source of irrigation and drinking water in these districts. The river is also a major source of hydroelectric power.

The Kaveri is the most important river in Tamil Nadu because it is the primary source of water for irrigation and drinking water in the state. The Kaveri is also a major source of hydroelectric power.