The correct answer is C. 1819 AD.
In 1819, Maharaja Ranjit Singh, the founder of the Sikh Empire, conquered Kashmir from the Afghans. He ruled Kashmir for the next 20 years until his death in 1839. After his death, the Sikh Empire began to decline and Kashmir was eventually annexed by the British in 1846.
Option A is incorrect because Kashmir was not a part of the Sikh Empire in 1640 AD. In that year, Kashmir was ruled by the Mughal Empire.
Option B is incorrect because Kashmir was not a part of the Sikh Empire in 1752 AD. In that year, Kashmir was ruled by the Afghans.
Option D is incorrect because Kashmir was not a part of the Sikh Empire in 1900 AD. In that year, Kashmir was ruled by the British.