Dear Applicant,
I hope you are doing well, most of the councilling processes have already started in Karnataka for the year 2020. Since the official rank list has been out for neet and kcet so yes only after the list is out councilling will start. You need to check the official sites to proceed with the admission process. Since this year due to the pandemic everything is done online so keep in touch with the official site of Colleges etc..
To apply for Christian minority u will have to check according to your state some offer a form and make a minority certificate or some take self Declaration.
Hope this helps!!
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