Dual

Download college brochure from the link provided below for application dates to apply for Integrated B.Sc + M.Sc in mathematics from the Central University of Kashmir.
Admission Process
Admission is through counseling based on the score obtained in Central Universities Common Entrance Test (CUCET).
https://www.shiksha.com/science/course/integrated-b-sc-m-sc-in-mathematics-central-university-of-kashmir-srinagar-268725

Yes, there is. All you have to do is check out the HRD ministry website or visit the HRD department. Certain states like MP and Rajasthan offer really good scholarships for students who get selected in IIM or IIT, regardless of their percentage.
All the best.

Please provide your passing mark sheet of first year and NOC from current institute. You can call our toll free number 18001026004 for more details.

The course is called IPM - integrated programme in management. Apply for IPM AT after completing 12th standard.

Yes, you can pursue Mass communication after completing 10+2 from a recognized board of education. You will have to appear for an entrance test, which is a general aptitude test that includes quantitative aptitude test, Logical reasoning, English Language test and General Knowledge. The General Knowledge section includes current affairs and media awareness questions. You rank in this test will determine your admission. You can see the best colleges of Mass Communication on this link. https://www.shiksha.com/mass-communication-media/articles/top-10-mass-communication-colleges-in-india-blogId-13552.

It depends upon which course you want to pursue in mathematics. See, different IITs provide different Master's programs in mathematics. So you have to check the eligibility required for each of it. To check the eligibility criteria you can visit: https://www.eduncle.com/iit-jam-exam

IIM Indore has been offering an integrated MBA programme of 5-years duration. In my opinion, it is the best integrated MBA in India. Certain universities also gave B.Tech and MBA integrated programme as well. The scope depends largely on candidates interest and the brand equity of the college. BBA followed by work experience is also a good option before MBA. So, it all depends what you're trying to achieve in a quantum of time.