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Hello Aryan,
I would suggest you to appear for IIT JEE and AIEEE. These two entrance examinations provide admissions to IITs/ NITs and other government colleges, which are cost-effective.
I would also suggest you to appear for individual state examinations (OJEE in Odisha, EAMCET in Andhra Pradesh/Telangana, COMEDK in Karnataka, etc.). These exams also guarantee you good colleges and serve as a backup in case things don't end up the way you've planned for IIT/ AIEEE.
Every state has its own share of good engineering colleges. So, don't underestimate or overlook them. They can prove to be vital. The more the entrance examinations you appear for, the more the options you will have for choosing colleges.
Certain colleges like SRM, KIIT (Odisha), GITAM (Vishakhapatnam), Manipal University and ITER (Odisha) conduct their own entrance examinations. You can opt for these as well.
I hope it helps. If so, kindly, upvote this answer.

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Hello Aditya,
I understand that you need a bit more time compared to others in understanding a topic and I appreciate your desire to crack JEE entrance examination. However, you should not hesitate to join a coaching institute just because of this.
In your case, I would suggest you not to join any commercial entrance coaching institutes. The reason is they have too many students in every batch. Mostly they jump to the next topic as soon as MAJORITY of the students are done with the current topic I.e. they might not pay enough attention to ALL students, particularly the weak ones. So, that would result in wastage of money. You can still have a word with the institute of your choice - FIITJEE and check if they have a batch with less students or any special batch or services for the weak students.
I would rather suggest you to opt for non-commercial tutors who don't teach in institutes. These are mostly teachers teaching in schools, who have a much smaller batch size, cover each topic completely, pays the most attention, mostly teaches in their homes but are very difficult to find.
Take decisions judiciously. All the best!
I hope I have been able to answer your queries to your satisfaction. If so, kindly upvote this answer. If you've any further questions, kindly revert.