With which princely state was the Kathak dance primarily associated during the 19th century?

The correct answer is Awadh. Kathak flourished in the 19th century in the court of the Nawabs of Awadh (Oudh), whose capital was Lucknow, in present-day Uttar Pradesh. The Lucknow gharana of Kathak developed in this courtly setting.

The greatest patron of Kathak in Awadh was Nawab Wajid Ali Shah (ruled 1847-1856), the last Nawab of Awadh. He was himself a poet, musician and dancer, and he wrote plays and composed thumris. He supported Kathak masters such as Thakur Prasad, who is said to have been his dance teacher, and the brothers Kalka Prasad and Bindadin Maharaj, who are regarded as the founders of the modern Lucknow gharana. Under this patronage, Kathak became known for its delicate, expressive abhinaya, its graceful movements and its close relationship with the thumri. After the British annexed Awadh in 1856, Wajid Ali Shah was exiled to Matiaburj near Kolkata, where he continued to support the arts. Pandit Birju Maharaj, the most famous Kathak dancer of modern times, belonged to the family of Kalka Prasad and Bindadin.

About the other options:

• Mysore was an important centre of Carnatic music and of Bharatanatyam (in its Mysore style) under the Wodeyar kings.
• Travancore was an important centre of Kerala's arts, such as Kathakali and Mohiniyattam, especially under Maharaja Swathi Thirunal.
• Hyderabad, ruled by the Nizams, was known for Urdu literature, Deccani culture and Hindustani music, but it is not the main centre linked with Kathak.

The other gharanas of Kathak are Jaipur (known for its powerful footwork and rhythmic complexity), Banaras (known for its balance of footwork and expression) and Raigarh (which grew under Raja Chakradhar Singh of Raigarh, in present-day Chhattisgarh).