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

Mechanical engineering skill set and knowledge will be one of the biggest driving force behind product development, innovation, Intellectual property ( patents) and quality of manufacturing in India. Now a days the scope of mechanical engineering is expanding beyond its traditional boundaries and is more of inter disciplinary in nature. Mechanical engineers are focusing their attention towards new areas such as nanotechnology, development of composite materials, biomedical applications, and environmental conservation. With this shift in paradigm the new job avenues have been opened for the mechanical engineers. Big players are going to setup their manufacturing plants in India. List of some of the Top and best private colleges in India are the following
LNM Institute of Information Technology, Jaipur
Vellore Institute of Technology, Vellore
Manipal Institute of Technology
SRM Institute of Science and Technology, Chennai
Dayanand Sagar college of engineering, Bangalore
Geeta University, Panipat, RV college of engineering, Bangalore
BMS college of engineering, Bangalore