The correct answer is: C. Patna
Patna is the capital of Bihar and the most populous city in the state. It is also the 19th most populous city in India. The population of Patna as per the 2011 Census was 1,084,663. The density of population in Patna was 12,344 per sq. km.
Sheohar is a district in the state of Bihar in India. The district headquarters is Sheohar town. The population of Sheohar as per the 2011 Census was 1,025,854. The density of population in Sheohar was 1,025 per sq. km.
Vaishali is a district in the state of Bihar in India. The district headquarters is Hajipur town. The population of Vaishali as per the 2011 Census was 3,097,825. The density of population in Vaishali was 1,284 per sq. km.
Darbhanga is a district in the state of Bihar in India. The district headquarters is Darbhanga town. The population of Darbhanga as per the 2011 Census was 4,835,215. The density of population in Darbhanga was 1,088 per sq. km.
Therefore, the district of Bihar with the highest density of population (per sq. km) as per the 2011 Census is Patna.