During the time of which Mughal Emperor did the east India company establish its first factory in India?

The correct option is Jahangir.

Historical Context & Step-by-Step Explanation:

1. In 1608, Captain William Hawkins arrived at the court of the Mughal Emperor Jahangir as a representative of the English East India Company to seek permission for trade and factory establishment in India.
2. After initial negotiations, Emperor Jahangir issued a royal decree (farman) in 1612–1613 allowing the English East India Company to establish its first official trading post (factory) at Surat.
3. The Mughal Emperor Akbar had passed away in 1605 before the Company established its presence in India, and emperors Shah Jahan and Aurangzeb ruled much later.
4. Therefore, it was during the reign of Emperor Jahangir (who ruled from 1605 to 1627) that the East India Company established its first factory in India.