IGNOU - Indira Gandhi National Open University

FOR MBA you have to give Open MAT first
If you clear the MAT exam then only you can apply for MBA through IGNOU, although there is no entrance exam required if you apply for distance education through any other university.

Yes, you can apply for the MCA course at IGNOU. If meeting the eligibility criteria as given below.
Eligibility: Any Bachelors's degree of minimum 3(three) year duration from a recognized University AND Mathematics as one of the subjects at 10+2 level or graduation level; else the student is required to pursue and pass the BCS-012 course concurrently with IGNOU MCA.
If you had studied Maths at +2 or graduation level, then you need not register for BCS-012.