Step 1: Understanding the Concept:
Indian riverine systems and inland fisheries geography: the Godavari River (Dakshin Ganga) originates at Trimbakeshwar in the Western Ghats near Nashik, Maharashtra.
Step 2: Detailed Explanation:
Geographical origin of major Indian riverine systems:
1. The River Godavari (A): The second longest river in India ($1,465\text{ km}$); originates at an elevation of $1,067\text{ m}$ at Trimbakeshwar (Triambak) near Nashik in the Brahmagiri Deolali hills of Maharashtra and flows eastward across the Deccan plateau into the Bay of Bengal.
2. The River Mahanadi: Originates in Sihawa hills, Chhattisgarh.
3. The River Krishna: Originates near Mahabaleshwar, Maharashtra.
4. The River Narmada: Originates from the Amarkantak plateau, MP.
Step 3: Final Answer:
Hence, the river is The River Godavari, matching option (A).