Whatever reason you have for hiding/removing your Last name on Facebook I think you should be able to do so if you want it BUT, this is not a straightforward process to do.
I'll describe a trick you can do to achieve this result (at least it works as on 15/01/2017).
*You can click on the links in provided steps not to get lost on the "mbasic" Facebook's version.
1. Configure your browser of choice to use a proxy from Mumbai
2. Go to https://mbasic.facebook.com and Log In into your account (you'll probably have to confirm through email or phone that you're the one trying to Log In from India)
3. Change your Location to Mumbai
4. Change your Language to "Tamil"
5. Change your name as desired and submit the form (you might need to confirm by entering your password)
7. Remove your proxy configurations and use Facebook as normal