Which are the good colleges offering admissions with 50 to 60 CAT percentile?

The minimum marks or CAT cut off to appear for IIM selection round is different for every institute. Tier 1 colleges select candidates with CAT score above 95 percentile, Tier 2 colleges select candidates with score between 80-90 percentile and Tier 2 colleges select candidates with score lower than 80 percentile.

There is no CAT PI shortlist for IIMs. Candidates have to simply login with their CAT credentials and fill the PI shortlist form.

There is no minimum qualifying marks for CAT. Anybody scoring above 0 in the exam is eligible for admission, however the B-Schools select candidates on the basis of cut offs which are 60 and above percentile.

The final CAT cutoff are not released publically. However, candidates can check qualifying cut offs in the admission criteria document released by IIMs and other institute before the exam.

CAT cut off is prepared by all accepting colleges individually. There is no one authorized body to do so. Even the IIMs determine their admission cut offs separately.