The Mahe district is surrounded by which State

Mahe is one of the four districts of the Union Territory of Puducherry. It is a small coastal enclave located on the western coast of India. Although it is part of Puducherry administratively, it is geographically situated within the state of Kerala.

More specifically, Mahe is surrounded by the Kannur and Kozhikode districts of Kerala. It lies along the Arabian Sea and is completely enclosed by Kerala on the landward side. This makes Mahe a unique geographical and administrative feature, as it functions as part of a Union Territory while being embedded within a different state.

The other options are incorrect:
Tamil Nadu surrounds the Puducherry and Karaikal districts, not Mahe.
Karnataka is located to the north of Kerala but does not border Mahe.
Goa is a separate state on the western coast and is far from Mahe.

Therefore, Mahe district is surrounded by the state of Kerala.