Concept:
The Vijayagara Empire ruled over Southern India from its capital at Hampi from 1336 CE until the late seventeenth century. This powerful state was governed sequentially by four distinct dysties: the Sangama, Saluva, Tuluva, and Aravidu dysties. Resolving this "odd-one-out" question requires checking the dystic affiliations of each ruler.
Step 1: Identifying rulers of the Vijayagara Empire.
Let us map out the historical coordites of options (A), (B), and (D):
• (A) Harihara: Along with his brother, Harihara I co-founded the Vijayagara Empire in 1336 CE. He was the initial morch of the Sangama Dysty and established the foundation of its administrative machinery.
• (D) Bukka: Bukka Raya I was the co-founder and second prominent ruler of the Sangama Dysty, succeeding his brother Harihara. He consolidated the empire's boundaries across South India.
• (B) Rama Raya: A powerful statesman of the Aravidu Dysty, Rama Raya served as the de facto ruler and prime minister of Vijayagara during the mid-16th century. His aggressive diplomatic maneuvers led to the fateful Battle of Talikota (1565 CE), which resulted in the sacking of Hampi.
Step 2: Identifying the historical profile of Rudradaman.
• (C) Rudradaman I was a famous ruler of the Western Kshatrapa (Saka) Dysty, which ruled parts of Western India (Saurashtra and Malwa).
• He ruled during the second century CE (c. 130–150 CE), more than a thousand years before the Vijayagara Empire was ever established.
• He is celebrated for his Jugadh rock inscription, which is the first major long inscription written in chaste classical Sanskrit, documenting the repair of the Sudarsha Lake.
Because Rudradaman belongs to an ancient North-Western dysty and an entirely different historical epoch, he is clearly the odd one out.