Hey there
5 year integrated course in Jadavpur University is good. And if you want to take your gate exam then ofcourse you can if you are in 5 year integrated course at Jadavpur University. But you can apply only when you are in fourth year or higher during your course. Gate is a exam you can give lifetime.
Check about gate
https://www.google.com/url?sa=t&source=web&rct=j&url=https://engineering.careers360.com/exams/gate/articles&ved=2ahUKEwiLx8z8o8PkAhWEbysKHfdEB2EQFjACegQICBAB&usg=AOvVaw2-QGV2tuHAKrilOlbVZqlc
Good luck
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