B.Sc Eligibility

Hello Aakash, For GATE you need four year's Bachelors course. If your B.Sc. Is of four years then you are eligible.
I am attaching a link that will give you all the required information.
http://www.gate.iitg.ac.in/?page_id=49.

There is no B.sc course in Law. You can pursue LLB which is a dedicated course in Law. Eligibility for a course in LLB is mentioned below.
LLB Eligibility Criteria
The minimum eligibility criteria for L.L.B. course is listed below:
Successful completion of graduation in any discipline.
Minimum 45% marks in graduation are required for general category and 40% for reserved categories.
There is also a criterion for minimum marks required in graduation which may vary from institute to institute.
There is no age limit for taking admission in this course.

Yes. You are eligible.

Passed 10 plus 2 or equivalent system from any recognised Central/State Board with Maths (with minimum 45% marks) as compulsory subject with the minimum aggregate of 45% or CGPA of 5.0 for Gen/OBC and 40%or CGPA of 4.5 for SC/ST category.