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Yes, it is possible to take the NEET (National Eligibility cum Entrance Test) even if already having completed BA and meeting the eligibility criteria. A more accurate explanation is here: 

NEET Eligibility Criteria: 

1. Educational Qualification:

You are required to have minimum of 10+2 or an equivalent qualification from any recognised board.

The subjects taken in 10+2 should comprise of Physics, Chemistry, and Biology/Biotechnology, as well as English as mandatory subjects.

2. Minimum Aggregate Percentages Required:

General category: Should not be less than 50% aggregate in Physics, Chemistry and Biology/Biotechnology.

OBC/SC/ST: Aggregate of at least 40%.

PWD: Aggregate percent of 45%.

3. Age Limit:

Minimum Age: The minimum age of the candidate should reach 17 years age by December 31 of the year in which admission is being sought.

Upper Age Limit: At present there does not exist any limit (currently guidelines state this).

4. Additional Points:

Even after completing BA, eligibility for NEET depends on 12th qualifications and fundamental subject requirements.

If you did not do Biology in 12th, possibility of scenario of being eligible becomes nil even for aggregate percentage conditions.

Key Issues:

Ensure that you comply with the subject and marks requirements, both of these come from 12th standard.

In that case, if you are deemed to be eligible, at the time when the applications are out, NTA NEET portal allows candidates to register for NEET. 

Kindly let me know if you wish to have furhter clarification regarding the matter!Hope this helps you