The Red Fort (Lal Qila) in Delhi, built by the Mughal emperor Shah Jahan, contains several important structures.
The "Diwan-i-Khas" (Hall of Private Audience) was a special hall where the emperor would meet with his most important courtiers, nobles, state guests, and conduct important state affairs of a more exclusive nature. It was distinct from the "Diwan-i-Aam" (Hall of Public Audience), where the emperor would address the general public and conduct more routine administrative tasks.
The Diwan-i-Khas was known for its ornate decoration, including the famous (now lost) Peacock Throne. The inscription "If there is a paradise on earth, it is this, it is this, it is this" is associated with it.
Let's evaluate the options:
(a) The emperor's private audience hall: This is the correct purpose of the Diwan-i-Khas. It was where the emperor held selective audiences with high-ranking officials, ambassadors, and other dignitaries.
(b) The living quarters of the royal family: The living quarters were typically in separate palaces or sections within the fort, such as the Khas Mahal, Rang Mahal, Mumtaz Mahal, etc.
(c) The administrative centre of the Mughal Empire: While important administrative matters were discussed in the Diwan-i-Khas (and Diwan-i-Aam), it wasn't the sole administrative center; various offices and departments supported the administration. Diwan-i-Aam handled more general administrative functions and public grievances.
(d) The treasury for storing precious stones and gold: The treasury (Toshkhana) would have been a separate, secure location within the fort complex.
Therefore, the specific purpose of the Diwan-i-Khas was as the emperor's private audience hall.
\[ \boxed{\text{The emperor's private audience hall}} \]