The correct answer is A) Travancore.
The Paliath Achans were a family of Nair ministers who held the position of hereditary Prime Ministers in the Kingdom of Travancore from the 16th century to the 19th century. They were appointed by the Maharaja of Travancore and were responsible for the day-to-day running of the kingdom. The Paliath Achans were a powerful and influential family and played a major role in the history of Travancore.
Option B) Venad is incorrect because the Paliath Achans did not hold the position of hereditary Prime Ministers in Venad. Venad was a kingdom in Kerala that was later absorbed into Travancore.
Option C) Cochin is incorrect because the Paliath Achans did not hold the position of hereditary Prime Ministers in Cochin. Cochin was a kingdom in Kerala that was later absorbed into Travancore.
Option D) Kozhikode is incorrect because the Paliath Achans did not hold the position of hereditary Prime Ministers in Kozhikode. Kozhikode was a city in Kerala that was not a kingdom.