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 the Indian state of Gujarat. It is located on the Rann of Kutch, a large salt marsh in the northwestern part of the state. Bhuj was the capital of the princely state of Kutch from the 16th century until its merger with India in 1947.

Jamnagar is a city in the Indian state of Gujarat. It is the headquarters of Jamnagar district and Jamnagar division. Jamnagar is the second-largest city in Gujarat after Ahmedabad. It is located on the Gulf of Kutch, about 240 kilometers (150 mi) from Ahmedabad.

Rajkot is a city in the Indian state of Gujarat. It is the headquarters of Rajkot district and Rajkot division. Rajkot is the third-largest city in Gujarat after Ahmedabad and Surat. It is located on the banks of the Aji River, about 250 kilometers (160 mi) from Ahmedabad.

Gandhinagar is the capital of the Indian state of Gujarat. It is located on the banks of the Sabarmati River, about 100 kilometers (62 mi) from Ahmedabad. Gandhinagar was founded in 1960 as the new capital of Gujarat, replacing Ahmedabad.