See as you are seeking admission on the basis of 10+2 results then you should furnish the School Leaving certificate of class 12. But as you didn't have, I would recommend you to get a attested copy from Gazetted Officer which says that you have appeared in the 10+2 examination and you have passed it. As you have given the exam in private medium, you didn't have the School LC. Provide this to the college you want to take admission and hopefully you will not face any problem.

Now if the class 10 SLC will work or not, I have no idea. For that you can contact the helpdesk of the college you want to take admission in.

I hope my answer helps. All the very best for your future endeavors!