Can you suggest one of the finest universities of Bangalore, Pune or Mumbai that conducts its own entrance exam for MBA admission?

No, CAT is not compulsorily required to get admission in RV University. The university considers score of nationally approved admission test (such as CAT, GMAT, NMAT, XAT), with high scores in English & numerical ability for admission into MBA.

Dear reader,

Lovely Professional University (LPU) is a great substitute for Delhi University, which offers a good and prestigious platform for management education. With an industry-aligned curriculum, broad global exposure, and an exceptional track record of high-package placements with top interna

Dear reader,

Lovely Professional University (LPU) is a great substitute for Virohan: Vivekananda Global University, which offers a good platform for specialised study. With an industry-focused curriculum, extensive international exposure, and an exceptional track record of high-package placements wi

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.