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Hindi is spoken extensively in India and outside of (I am relying on personal knowledge here, but will post with more references when better connected) Kerala and Tamil Nadu, there is probably a large percentage of the population that speaks or understands it. English, again IIRC, is the language of choice for various official documents and communication. Gandhi tried to introduce a common language "Hindustani" (again speaking without checking here) that didn't quite take off.
In Tamil Nadu widespread rioting resulted when certain attempts were made to alternately force or insinuate Hindi into local usage. Hindi names on Central Government institutions were blacked out by protestors, etc. The Hindification of Tamil Nadu is being achieved rather trivially these days through TV and pop culture. (That reminds me to post ASAP info on an ongoing controversy regarding an actress and her opinion about female pre-marital sex).
To answer your question: there is no common language. If you took a person from Athoor in Tamil Nadu and put him in Bhatinda [sp?] in Punjab or some quite smaller town in Assam, he would be almost entire unable to communicate.
--Horace Lowell