The Palar River flows through Tamil Nadu and:

Karnataka
Andhra Pradesh
Kerala
Puducherry

The correct answer is: a) Karnataka.

The Palar River is a river in southern India. It originates in the Western Ghats in Karnataka and flows through the states of Tamil Nadu and Puducherry. The river is 260 kilometers long and its drainage basin covers an area of 10,000 square kilometers. The Palar River is a major source of irrigation for the agricultural lands in Tamil Nadu and Puducherry. The river is also a popular destination for fishing and boating.

Karnataka is a state in southern India. It is the third-largest state in India by area and the ninth-largest by population. The capital of Karnataka is Bangalore. The state is bordered by Goa to the west, Maharashtra to the north, Andhra Pradesh to the east, Tamil Nadu to the south, and Kerala to the southwest.

The Palar River is one of the major rivers in Karnataka. It originates in the Western Ghats and flows through the state before joining the Bay of Bengal. The river is a major source of irrigation for the agricultural lands in Karnataka. It is also a popular destination for fishing and boating.

The other options are incorrect because the Palar River does not flow through those states.