Science

You can get admission but you need to check the criteria with your University.

The "Kishore Vaigyanik Protsahan Yojana" (KVPY) is a program started in 1999 by the Department of Science and Technology (DST), Government of India to encourage students who are studying Basic Sciences to take up research career in Science. The aim of the program is to identify and encourage talented and motivated students to pursue career in research. In my opinion IISC is very good option for research in Science & Technology.

in every engineering branch, you must have the basic mathematics knowledge.however, engineering mathematics will be taught to you for a few semesters.mainly that engineering mathematics will come in application.so, it does not matter how well your maths is, if you have an interest in electrical branch,then go for it.

I am not aware of such difficulty.The level of difficulty may vary, depending on the background. If the student has an appropriate subject background, he will naturally feel it easy.Otherwise, he may have to devote more time.

Depends upon your interest.

Yes. Maths is very useful for CAT. One of the two papers in CAT checks your quantitative abilities. However majority of questions are from class 9th to 12th maths. Also you should not base your life decisions on requirements of one examination. If you think your time and energy are useful in other subjects than maths, I think you should take that up.