Am I eligible for admission without taking GMAT or CAT?

Yes. IIMB does require the CAT exam, but the rest of the top B-schools in Bangalore also take in candidates through the state-level CET, ie, the Karnataka PGCET exam, or through MAT or CMAT. 

Note: All info. is from official sites and can change.  

IIM Calcutta has only mentioned accepting GMAT score for admission into EMBA courses.

However, CAT is commonly accepted by other similar institutes for both MBA and MBAEx courses. Thus, candidates are advised to ask admission desk for confirmation on this question.

It is not possible to determine a candidate's admission in IIM Calcutta MBA course through CAT score alone. In Stage 1, a score 85% in CAT will be enough for a general category candidate to be selected for PI+WAT admission stage.

Final admission cutoffs are based on many factors other than CAT score.

No, IIM Calcutta cannot get into MBA courses without appearing CAT first. CAT is a mandatory entrance exam and it is conducted every year for admissions into all MBA courses offered by this institute.

However, candidates who are applying for MBAEx course will be required to appear in GMAT for admissi

To get a seat at IFMR Chennai, students have to first appear for any one of the entrance exams accepted by the institute for the MBA/PGDM course admission. CAT, XAT, NMAT, CMAT, Andhra Pradesh ICET, IPAMT and SAT India are the entrance exams accepted by this institute for admission to the management