Final answer:
To grant execute permission only for the owner of the file 'project1', the correct command is 'chmod u+x project1'. This command modifies the permissions of the file, specifically adding execute permission for the file's owner only.
Step-by-step explanation:
To grant execute permission only for the owner of the file project1 while you are the root user, you would use the command chmod u+x project1. Here, chmod is the command to change file modes or access control lists, u stands for the user who owns the file, + means to add the permission, and x stands for execute. Other options listed, such as chmod +x project1 and chmod a+x project1, would respectively grant execute permission to all users and not just the owner, which is not the required action.