How can I join Indian Army after completing 12 science?

Yes, you can definitely join Indian Army after completing 12 science .For that you have to qualify the NDA exam which holds by UPSC twice in a year.

Recruitment of applicants to Army, Navy and Air force wings of the National Defence Academy (NDA), is through National Defence Academy Entrance Exam which is held twice a year, generally in the month of March and August.
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Army wings :-
Male candidates are eligible.But the candidate should be unmarried at the time of appearing.
Any stream candidates can apply for this examination like Science or arts or commerce.
Applicants need to complete the 12th examination from a recognized School.
Final year of 12th examination can also eligible.
Applicants age should be 16 1/2 years to 19 years.

Air force :-
Unmarried male candidate are eligible to apply for this examination.
Physics and Mathematics at 12th level examination.
Applicants must be completed the 12th examination from a recognized School or equivalent qualification.
Candidates who are appearing in the 12th examination can also eligible.
Applicants age should be 16 1/2 years to 19 years.
Particular stream candidates are eligible.That means science stream candidates are eligible.

Navy wings :-
The candidate must be Citizen of India.
Physics and Mathematics at 12th level examination.
Final year of 12th examination can also eligible.
Applicants age should be 16 1/2 years to 19 years.
Unmarried male candidate are eligible to apply for this examination.
Applicants need to complete the 12th examination from a recognized School.

Important dates:
Date of notification : 21 Dec 2013
Last date of form submission : 20 January 2014
Date of NDA exam is 20th April 2014