Hello aspirant,

Hope you are doing great. So, as per your question, you cannot give the GATE exam soon after completion of JEE exam. After completion of JEE you need to do the Btech course. After completion of Btech course, then you will be eligible for the GATE exam. So, GATE exam in general will be conducted after graduation level.

The subjects for the syllabus will be based upon the branch that you had chosen in your graduation level. However, General Aptitude and Verbal Ability is common for all the papers.

Further, for any information regarding the GATE exam:

https://engineering.careers360.com/articles/gate-syllabus

Hope it helps you,

Thank you.