The question asks to identify the Governor of India during a specific historical event: the first Anglo-Maratha war. To answer this, we first need to know when this war took place.
The first Anglo-Maratha war occurred between the British East India Company and the Maratha Empire from $1775$ to $1782$.
During this period, the highest British official in India was the Governor-General of Bengal (the British administrative headquarters). We need to find who held this position between 1775 and 1782.
Warren Hastings served as the Governor of Bengal from $1772$ to $1785$. This entire period completely encompasses the duration of the first Anglo-Maratha war ($1775$-$1782$). Therefore, Warren Hastings was the Governor of India (Governor-General of Bengal) at the time of the first Anglo-Maratha war.
His administration was deeply involved in the political and military events leading up to and during the war. He played a key role in shaping the British East India Company's policies towards the Marathas during this critical time.
Let's look at why the other options are incorrect:
Based on the timelines, only Warren Hastings was the Governor of India during the first Anglo-Maratha war.
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