The correct answer is: a) Hyderabad State.
Hyderabad State was the largest princely state within the present-day boundaries of Andhra Pradesh during British rule. It was founded in 1724 by Nizam-ul-Mulk Asaf Jah I, a Mughal nobleman who was appointed as the viceroy of the Deccan by the Mughal emperor Muhammad Shah. The Nizams of Hyderabad were the hereditary rulers of the state, and they ruled it until it was merged with the Indian Union in 1948.
Hyderabad State was a vast and populous state, with a population of over 16 million people in 1941. It was also a wealthy state, with a strong economy based on agriculture, trade, and industry. The capital of Hyderabad State was the city of Hyderabad, which is now the capital of the Indian state of Telangana.
The other options are incorrect because they were not princely states within the present-day boundaries of Andhra Pradesh during British rule. Vizianagaram was a princely state in the northern part of Andhra Pradesh, Bobbili was a princely state in the southern part of Andhra Pradesh, and Gadwal was a princely state in the western part of Andhra Pradesh.