B.Tech

Dear Student, The Amrita Vishwa Vidyapeetham Coimbatore is a NAAC-accredited A-grade teaching and research institution that conducts a national-level engineering entrance examination, the Amrita Engineering Entrance Exam AEEE, to offer admissions to candidates in the institute's four-year BTech degree programme. According to information given by you, your estimated rank might be around 6,000, and with state quotas, you may easily get a seat in reputed branches like Computer Science and Engineering, Civil Engineering, Electronics and Communication Engineering, Mechanical Engineering, and Aerospace Engineering since the state rank cutoff for many of the courses usually remains over 10,000.

Sorry dear with this rank I don't think you will get any B.TECH college with CSE unless until you belong to the ST and SC's class. Try to improve your percentile and rank in next JEE Mains exam to clear top IIT's and NIT's college of India. There are some college that take admission with 12 marks score you can SEE that option too. Coming back to your college fee structure under 10 Lacs I think Jadavpur University and many more government funded college came under the 10lakhs fee structure work hard to get in it.
The top computer science engineering colleges having fee below 10 lacs:
1. Manipal Institute of Technology, Manipal
2. SRM Institute of Science and Technology, Chennai
3. College of Engineering, Pune
4. Dayananda SAGAR College of Engineering, Bangalore
5. RV College of Engineering, Bangalore
6. Vellore Institute of Technology, Vellore
7. PSG college of technology, Coimbatore
8. Maharaja Sayajirao University of Baroda, Vadodara
9 Kalinga institute of technology
10. JIIT COLLEGE OF Engineering