Concept:
India has several major river systems that play an important role in agriculture, transportation, irrigation, hydroelectric power generation, and human settlement.
The rivers of India are generally classified into:
• Himalayan rivers
• Peninsular rivers
Among the given options, all are important rivers of India.
Step 1: Know the length of the given rivers.
Approximate lengths of the rivers are:
• Godavari -- \(1465 \, \text{km}\)
• Kaveri -- \(800 \, \text{km}\)
• Mahanadi -- \(858 \, \text{km}\)
• Kosi -- \(730 \, \text{km}\) approximately
Step 2: Identify the longest river among the options.
Comparing the river lengths:
\[
1465 > 858 > 800 > 730
\]
Thus, the Godavari River is the longest among the given options.
Step 3: Know important facts about Godavari.
The Godavari River is:
• The longest river of Peninsular India
• Also called the "Dakshin Ganga"
• Originates from Trimbakeshwar in Maharashtra
• Flows into the Bay of Bengal
Step 4: Understand why other options are incorrect.
• Kaveri -- Important South Indian river but shorter
• Mahanadi -- Major river of Odisha and Chhattisgarh but shorter than Godavari
• Kosi -- Tributary of the Ganga and shorter in length
Step 5: Conclusion.
Therefore, the longest river among the given options is:
\[
\boxed{\text{Godavari}}
\]