JEE

Yes, you can join somaiya vidyavihaar university without JEE.

There are others various ways to get admission in the university like,  

SAT (Scholastic Assessment Test)

CUET

PERA CET

MHT CET

And also by your 12th percentage.

You can get good colleges with this percentile. 

IET LUCKNOW

KNIT SULTANPUR

BIET JHASI 

If you are able to pay fee then choose universities with less workload, big campus, good college life but low placement

Lucknow University

Bundelkhand University

MJPRU

The school of Engineering and Applied Science, Ahmedabad University allocated seats on the basis of merit of the candidate as determined by the ACPC (Academic committee for professional courses ).

Yes, you can get admissions in many of the colleges with 83 percentile in JEE Mains such as 

NIT Agartala

NIT Meghalaya

Uttaranchal University Dehradun

Lovely Professional University

Dream institute of technology Kolkatta

You can enrol at the BNCA College of Architecture because your JEE and NATA scores are good (71 percentile and 99 respectively).

It will be a hard race for you because at this percentile, chances are low but Merit rank also matters, you should try for it but not confirmed seat. 

There is fair chance of getting newly opened NITs, IIITs in third or spots round. So, opt colleges in Josaa counselling very carefully and if you are not getting any college in any round then wait for spot round and apply for it.

Thank You