Yes, in the final year of your graduation you will be eligible for UPSC examination. Coming to your course, BA is all time suggested and one of the best course for cracking UPSC examination. Once you clear your UPSC examination your graduation and the College or your course are not consider. So no matter where you have done your graduation, you will get a job only if you crack it.
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If you are going for any company rather then UPSC your graduation, College and your score will be given preference

Greetings Ritik! Yes, you can apply for UPSC during your third year but you will have to produce other valid documents from the college. The course you have chosen doesn't matter for the appearance of UPSC exam. The main eligibility criteria is YOU SHOULD HOLD ANY BACHELOR'S DEGREE (No matter Arts, Science, Engineering or Medicine).

Your BA Course certificate will be helpful only for the eligibility. You cannot depend on your course for the exam unless you are studying History and Political Science or Geography. You must go through the Daily Newspaper to be thorough with the latest Indian politics and law, and also should prepare using other textbooks. The main syllabus of UPSC exams includes History of India, Geography of India, Language and Aptitude test.

Along with your degree, you require preparation to appear for the UPSC Exams.

For further details : https://competition.careers360.com/articles/upsc-ias-exam-pattern

Hope this helps!