Step 1: Understanding the Question
The question asks for the name of the new faith or ethical system introduced by the Mughal emperor Akbar in 1582.
Step 2: Evaluating the Options
(A) Mansabdhari: This was the administrative system introduced by Akbar to rank government officials and military generals. It was not a religion.
(B) Din-i Ilahi: Translating to "Religion of God," this was a syncretic religion or ethical system propounded by Akbar. It drew elements from Islam, Hinduism, Christianity, Jainism, and Zoroastrianism, with the aim of uniting the different communities of his empire.
(C) Sir-i-Akbar: This means "The Great Secret" and is the title of the Persian translation of the Upanishads completed by Akbar's great-grandson, Dara Shikoh.
(D) Dabestan-e Mazaheb: This is a Persian text from the 17th century that studies and compares various religions of South Asia. It was not founded by Akbar.
Step 3: Final Answer
The religion founded by Akbar in 1582 was Din-i Ilahi. Therefore, option (B) is the correct answer.