Why can't I delete this file as root?

From root user check attributes of files

# lsattr 

if you notice i (immutable) or a (append-only), remove those attributes:

# man chattr
# chattr -i [filename]
# chattr -a [filename]

The most common reason for rm complaining that you don't have permission to delete a file, is that the permissions on the directory forbid you from deleting the file. In order to delete a file, you need write permission on the directory. The permissions on the file are irrelevant (rm without -f prompts for confirmation before deleting a read-only file, but that's just a confirmation, not a limitation). On some Unix variants such as OSX (but not Linux), the ACL on a file can prevent its deletion; ls -l would show @ at the end of the permission field if there was an ACL entry on the file.

Access as root bypasses permissions, so root can delete files even in a read-only directory.

The output from ls -l shows a . at the end of the permission column. This indicates that the file has an SELinux security context. Unlike basic permissions and ACL, the SELinux security context on a file can control who is allowed to delete it. Furthermore SELinux cannot always be bypassed by root (it's possible to have a process running as user ID 0 but with as few rights as the SELinux policy designer chose). To see what the SELinux context allows you to do, run ls -lZ . exam_a.

Another thing that can prevent a file from being deleted is if it or the directory that contains it has the append-only or immutable Linux attribute. Run lsattr -d . exam_a to view the Linux attributes. If the a or i attribute is on, you'll need to remove it (chattr -a -i . exam_a) in order to delete the file; only root can do that. Root cannot bypass these attributes to delete a file, the attributes have to be turned off first.

Yet another thing that prevents a file from being deleted is if the filesystem is mounted read-only, but you'd get a different error message in that case.