Hi Raj,

Apart from  JEE Mains and JEE Advance several universities and states conduct their own entrance exams for the  admissions:                                                        Here is a list of a few top exams for private institutes :

BITSAT : Birla Institute of Technology conducts an entrance exam exclusively. The university conducts the examination for  all three of it's campuses i.e  BITS Pilani, BITS Goa and BITS Hyderabad. The Exam is divided in four parts : Physics ,Chemistry , Mathematics and English proficiency and logical reasoning .

Last date for Application: 29th May 2021

Date of Exam: 24th-30th June (Tentative)

MET : MET is for Manipal Academy for higher education. It is for admission to 16 engineering courses. It consists of questions from Physics Chemistry Mathematics , English and general Aptitude. Last Day of Application:15th March 2021

Date of Exam: In the month of May

VITEEE: The VIT university conducts exam for 4  institutes i.e VIT Chennai, VIT Vellore , VIT Andhra Pradesh and VIT Bhopal. The exam includes question from Physics, Chemistry, Mathematics , Aptitude and English.

Last Day of Application: Isn't announced yet

Date of Exam: 18th to 26th June (Tentative)

SRMJEE: SRM Institute of Science and Technolog y conducts exam  for all eligible aspirants.  The entrance examination consists of 125 questions and the applicants will have to qualify the exam for admissions in following colleges: SRM University Kattankulathur, SRM University Ramapuram, SRM University Ramapuram Par – Vadapalani and SRM University Ghaziabad.

Last Date of Application: 31st May 2021

Date of Exam: 11th to 16th June

SITEEE:

Symbiosis Institute of Technology conducts examination exclusively. It also accepts MHC-NET and JEE scores.

Last Date of Application: 4th June 2021

Exam Date: 27th June

KITEEE:

Kalinga Institute of Industrial Technology conducts entrance exams foe admission in B.Tech courses it offers. The exam focuses on Physics, Chemistry and Mathematics.

Last day of Application: 31st May 2021

Date of Exam: 16th June to 23rd June (tentative)

For more details on other examination too you can use this link:

https://engineering.careers360.com/articles/top-engineering-entrance-exams-in-india

I hope this helped you.