The correct answer is option 3, the Treaty of Bassein. This treaty marked the end of the independent rule of Peshwa Baji Rao II. Now, let`s dissect the other options to understand why they are not correct.

Option 1, the Treaty of Purandhar, was signed in 1776 between the Maratha Empire and the British East India Company. This treaty did not directly impact the rule of Peshwa Baji Rao II.

Option 2, the Convention of Wadgaon, was a significant victory for the Maratha Empire against the British in 1779. It did not result in the end of Baji Rao II`s rule.

Option 4, the Treaty of Salbai, was signed in 1782 between the Maratha Empire and the British East India Company. While it led to some territorial adjustments, it did not bring an end to Peshwa Baji Rao II`s independent existence.

In conclusion, the Treaty of Bassein is the correct answer as it ended the independent existence of Peshwa Baji Rao II.