Apart from IIMs, can you suggest some good private institutes which accept CAT score and give efficient placements?

Yes, JAGSoM accepts CAT scores as part of its multi-stage admission process. Candidates are shortlisted for selection rounds based on their entrance exam scores.

No, CAT scores are not important for admission to the MBA course at the School of Business, Sushant University. As per the info. taken from the website of the university, it conducts its own entrance exam for enrollment into the MBA course.

RGIPT allows admission without CAT/XAT/GMAT/CMAT/MAT/CUET-PG score under two circumstances. First, you should be a company-sponsored candidate. Second, if you belong to part time registration category. You will still have to fulfil the remaining eligibility criteria. 

Candidates with a valid score of CAT/ GMAT/ GRE score are exempted from the written test. These candidates still need to complete the form by paying the application fees. However, these aspirants may get called for the IIM Trichy - Chennai campus PGPBM interview round. 

CAT is one of the accepting exams for admission to PGDM at IPE Hyderabad. Candidates with a good score in CAT can also get a scholarship. However, CAT is not mandatory. Aspirants with a score in MAT, XAT, ATMA, GMAT or CMAT exams can also get admission in the PGDM course.