Dual

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.

The duration of B.Tech+M.Tech in LNMIIT is of 5 years. The fee structure for one year is Rs 3 lakh including mess and hostel and fees will increase by 10% per year.

SPJIMR Global Management programme (GMP), MDI-Gurgaon PGP IM programme, IMT-Ghaziabad DCP (Dual Country programme) and XIMB MBA in Global Management.

Please refer to the link below to get the entrance exam syllabus: http://www.mdurohtak.ac.in/info/entrance_Question_Booklet.html

Placement, exposure and learning in an MBA program depends largely on how the peer group is and how driven the candidate is towards the program. If you happen to make it to the Ivy League schools of India or abroad, you will certainly get a cohort that will have a pedigree and will push you to improve yourself. Schools like IIMs, FMS, XLRI, IIFT, MDI are NITIE are all the schools that have operations and finance as the majors available. If you make it to these, placement opportunities in MNCs and PSUs await you.
All the best.

DTU is a good option.