Exam

You can appear for JEE main thrice and JEE Advance twice that too in the consecutive years of completing your 12th so you cant appear for JEE after completing B.Tech agriculture

Among your choices, DTU IT is a very good option unless you want IIT brand attached in your CV.

Yes, you can appear for NMAT exam.

You can get NIT, Patna.

You are eligible for some colleges.

No, you cannot appear. As per the rules, student appearing in improvement should not be engaged in any higher education. For more details you can visit improvementexam.com.

I think Self study is the best tool.
I would recommend that you try out the study material from IMS / TIME / Career Launcher as your first resource.
Some of the Books recommended for CAT are-
1.How to prepare for Data Interpretation for the CAT   TMH / Arun Sharma
2.Pearson Guide to Quantitative Aptitu

Self study is all that matters ! I would suggest you that - write a mock CAT to figure out your strengths and weak areas. Depending on your score you can figure out when to start preparing. Its possible to score well in CAT by preparing rigorously for 4-5 months if your basics are good.
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Hi,
These are some of the best institutes which offer CAT coaching :
1.Career Launcher
2. IMS
3. TIME
4. Bulls Eye
5. PT Education
They offer excellent coaching for CAT in India.