MBA Entrance Exam

No, you can't write because it asks all your documents.

NMAT score varies from 80% - 90% based on the campus. For Mumbai campus candidate need to score 208 marks i.e., 94%.

If 3.83 is the CGPA out of 4, it is good score in CAT application, so you have 98.9%.

Top MBA colleges with budget fees-
RV institute of management -75000. Directorate of distance education 8230. Dr Ambedkar institute of technology 3700
Dayanand sagar college of management and information technology 63750. BMS College of engineering 25300. The Oxford college of engineering 70000. Xavie

Any student from any degree in their undergraduate can apply for CAT exam.

It is better to pursue a regular MBA degree rather than a distance degree. Because MBA is one of those degrees which require a lot of regular networking and working. You would not get all of those from a distance degree.

Yes, you have to crack the CAT exam in order to make it into IIMs.

The simple answer is 'no'. If you wish to pursue M. B.A. in I. MS then you have to clear the CAT exam. GATE exam is for those who wants to do M.Tech.