I want to give permission to specific role to create new contact .lets say Role A can now only search contacts but is not able to create new contact.So I want to show that create new button for Role A , so that It can create new contact as well. How to achieve that?
If you view the "Functional Access" tab for the Role what is the Access level for "Contact"? If they cannot create contacts today it's likely "View", you should be able to change to "Modify".
You may also need to validate their data partitions to confirm nothing would prevent them from editing contacts.