Explanation:

The correct answer is (1) United Provinces of Agra and Awadh. This was the official administrative name for the region that largely constitutes modern-day Uttar Pradesh during the British Raj, specifically from 1902 to 1937. It was formed by the amalgamation of the North-Western Provinces and Oudh (Awadh). Option (2) Kingdom of Banaras and Tehri-Garwal refers to princely states that existed within or near the region, but they were not the name for the entire British province. Option (3) United Provinces of Allahabad is incorrect; while Allahabad was a significant city and administrative center, it was not the full official designation for the entire province. Option (4) Awadh Pradesh is an unofficial and incomplete term; 'Awadh' was a part of the province, but 'Awadh Pradesh' was not the official name.