IIT

Hello Yamini,
All the IIT JEE coaching institutes take entrance test before admitting the student, and there are many benefits of the entrance tests - one being that you get fee relaxation up to a certain amount on scoring good marks in the test. For the institues click on the link below:
https://www.google.co.in/search?q=IIT+JEE+coaching+classes+in+thane+with+low+fee&rlz=1C1CHZL_enIN732IN732&oq=IIT+JEE+coaching+classes+in+thane+with+low+fee&aqs=chrome.69i57.17648j0j4&sourceid=chrome&ie=UTF-8
Hope this helps! All the best!

Hello Ricky,
Use the rank predictor in the link given below, once you get your approximate rank, check out the cut - off list of different IITs.
https://www.shiksha.com/b-tech/resources/jee-advanced-rank-predictor
Hope this helps.
All the best.

Core courses are often much helpful for future education. You can learn in new IIT. After all it is an IIT though it is new. Other course often leads to problem in our further education.

Yes, According to Josaa Counselling Process, It is NE to score 75% in HSC in order to get admission in IITs NITs and IIITS for B.Tech.
Best of Luck and Work.

AAT exam is for you to get into IITs to pursue a course in Architecture. There is, of course, not any compulsion if you are not interested in B.Arch, whereas, JEE Advanced is for students who are interested in pursuing B.Tech from IITs.
Hope this was helpful.

Kindly use the college predictor app. Hope this will resolve your query. https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor
All the best.

To qualify for IITs, you have to give JEE Advance and secure a good rank. 87 marks will not give you good rank. So, you cannot qualify for any of IITs.

Find out which rank will you get by entering your JEE Advanced marks in the link given below-
https://www.shiksha.com/b-tech/resources/jee-advanced-rank-predictor
Once you know the rank, check out the colleges cut-off ranks for mechanical branch and compare with the following scores:
https://www.shiksha.com/b-tech/articles/jee-advanced-2016-expected-cut-off-released-blogId-12688
and
https://www.shiksha.com/b-tech/articles/revised-cut-offs-for-jee-advanced-2015-blogId-11007.
Hope this helps! All the best!

If you are general or OBC then chances are very less but if you are from SC or ST category then you may get a seat in IIT. Although we can only make predictions so wait till result.