No, the institutes do not have separate examinations. Which is to say, the admissions can be availed under 2 categories:

1. 15% National Quota
2. 85% State Quota

Now, you also have to go through AIPMT, wherein the seats are assigned to students based upon the rank they have secured, under the 1st category nonetheless.

When talking about the applicants who qualify for the Delhi State Quota, there is a provision of a separate state-wise ranking. You are required to fill a separate form based upon the seats assigned, which most likely is 85% as the state quota.