The princely state of Kutch had its capital at:

Bhuj
Jamnagar
Rajkot
Gandhinagar

The correct answer is Bhuj.

Bhuj is the capital of Kutch district in Gujarat, India. It is located on the eastern coast of the Rann of Kutch, an arid region of salt marshes and sand dunes. The city has a population of over 400,000 people.

Bhuj was founded in the 16th century by Rao Khengarji I, the ruler of the Kutch kingdom. The city was originally named Bhujyanagar, after the name of the local goddess Bhujia. Bhuj became the capital of the Kutch kingdom in the 17th century.

The city was ruled by the Jadeja Rajput dynasty for over 400 years. The Jadejas were a powerful and influential Rajput clan. They were known for their military prowess and their patronage of the arts.

Bhuj was a prosperous city under the rule of the Jadejas. The city was a major center of trade and commerce. It was also a center of learning and culture.

In 1819, the Kutch kingdom became a British protectorate. Bhuj remained the capital of the Kutch kingdom under British rule.

In 1947, India gained independence from British rule. The Kutch kingdom was merged with the Indian Union. Bhuj became the capital of the Kutch district in Gujarat.

Bhuj is a major tourist destination. The city is home to several historical monuments, including the Bhuj Fort, the Prag Mahal Palace, and the Aina Mahal Palace. The city is also home to several museums, including the Kutch Museum and the Bhuj Heritage Museum.

Bhuj is a vibrant and cosmopolitan city. The city is home to people from all over India and the world. The city is a major center of education and culture. Bhuj is also a major center of trade and commerce.