Dhruv there is no shortcut to getting into JEE by skipping/messing up 11th class syllabus because mostly in examination syllabus from both the class are of almost equal weightage.

So, to get done all the syllabus of 11th and 12th class is to join a coaching which is running a droppers batch. This batch starts in the month of April (right now) after your board examination and the teachers are skilled enough to teach you everything right from the basics so you can catch up with all the things you have actually missed and go through it. And of course you know the basic funda better the coaching better will be the faculty and they will transfer the whole knowledge with all the exam tricks in a much better way.

And believe me, it is not at all an impossible thing, the only thing you will need is complete dedication towards your study. You will need to get rid of all the distractions (TV, Computer, Games, Movies etc... hope you understand) and for the whole year, your only aim needs to be to clear JEE exam. This is the very basic rule to get through JEE. Let me tell you even I messed up with my 11th class due to some personal reasons, took a drop, cleared JEE examination and got admission in IIT Roorkee. So, everything is possible if you really give your 100% in studies.

Feel free to ask if you have any more queries. All the best!!