JEE

You can get admission through Alliance entrance ( They have their entrances held all over India ). Second round of admissions have started. You can register at their website before its too late. (Their entrance is much much easier then mains).

Your marks should be above 120+.

Please use Shiksha's college predictor:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor

Of course, you can apply JEE Mains.

Hello,
I hope you are doing well. So, you are asking that whether we can give JEE having less than 75%. So, let me clear you that you can give JEE Mains with any percent rate but for JEE Advanced, you need to get 75% in 12th examination and there will be no other way to appear in advance without having 75% in 12th.
All the best.

That would be also from Rajasthan if you have substantial proof of your residing in Rajasthan namely Domicile Certificate, Electricity Bill, Water Bill.

Hello Anika, Let me explain the concept of cutoff. JEE Mains is conducted for two reasons. To screen students who are allowed to appear for JEE Advanced and for entry into NIT's. This time about 10L students appeared for JEE Mains. Now everybody has been given rank but only about 2.3L students are declared to be above cutoff. But only less than 10% of these 2.3 L are likely to get admission. So what to say about those who have scored less than cut off? Its simply a seive to allow only reasonable number of students to progress to next round. But yes, those below cutoff are simply discarded.

With this rank, you might get admission in any good college. But if you want admission you can get admission in any private college but they will take you large amount of money from you. And your 12th percentage does not take that much importance in getting a good college. Or, you can take a drop and can try for getting admission.

GATE Exam is not looking for any syllabus completion from the candidate. A good understanding of the basic concepts and their application is required. By understanding, it is implied that candidate is not supposed to just be able to mug up & explain but rather this exam needs candidate to have a feel/common sense.