The correct answer is option 3, Gulbadan Begum.
Gulbadan Begum was the daughter of Emperor Babur and the sister of Emperor Humayun. She wrote a memoir called Humayun Nama, which is considered to be the first known Mughal biography. The book chronicles the life and reign of her brother, Humayun. It provides valuable insights into the political and personal events of that time.
Option 1, Roshanara Begum, was the daughter of Emperor Shah Jahan and a prominent figure in the Mughal court. She was known for her piety and patronage of Islamic scholars and Sufi saints.
Option 2, Ruquaiya Sultan Begum, was the wife of Emperor Akbar and the mother of Emperor Jahangir. While she held political influence, there is no evidence to suggest that she authored any books.
Option 4, Gauhara Begum, was the daughter of Emperor Aurangzeb and a poet herself. However, there is no record of her authoring the Humayun Nama.
In conclusion, out of the given options, Gulbadan Begum was the author of the Humayun Nama.