CAT Scores

You have good chances to get admission in a good B-school. You can get new IIMs if your overall profile is good. You can try to convert top private colleges like IMI-D, IMT-G, GIM, MDI-G, Great Lakes, IFMR and TAPMI.

Target a CAT score of above 95 percentile if you want to get into top 25 B-schools in India. Usually, IIMs do not provide specialization and give a diploma in General Management. You will get a PGDM/MBA degree from such colleges. While colleges like S.P Jain gives MBA/PGDM with Marketing/Finance/Operations specialization.

78.19 percentile is a good score but the trend of TAPMI is to take percentile above 80. You can apply with XAT separately. How much did you score in XAT?

Apply in Jaipuria, ITM, Doon Business School, NDIM-Delhi, Amrita Business School, PSG-Coimbatore, Prestige Institute-Indore
Check the below link-
https://www.shiksha.com/colleges-accepting-cat-scores-range-60-to-70-percentile-article-11984-1
Also, check this link-
https://www.shiksha.com/mba/resources/application-forms

Hi Sumit,
DEI conducts a written exam of its own for admission in MBA . Apart from that, a group discussion and interview follows the exam.

You can try for Amity-Noida, Woxsen School of Business, Shiv Nadar University and UPES with your NMAT score.

You can try for ITM-Mumbai, Balaji Institutes-Pune, XIME-Bangalore, Christ University-Bangalore, Alliance University- Bangalore, ICFAI-Hyderabad and ASB-Coimbatore.

There is least possibility of admission with 12 score.
But still, you can try for rounds.

Apeejay is not recommended at all, neither should you take a drop and reappear. I am sending you a list of colleges at your MAT score below. Kindly check their cutoffs and decide accordingly.
http://www.aimamat.in/mat-college-list-percentile-wise/