Consider using a database for this or some kind of server-side check. When the student opens the test for the first time, the time should be stored in a database and the shown time should be calculated. Otherwise refreshing the page would reset the timer. At the end if the time of submitTime-startTime < 60 seconds then he/she made it in time.
This means that you need PHP for example. It should call a method which stores the current time if a field in database if it isn't allready stored and then do the same after submit. If the time of submit is present in the database, you shouldn't allow him/her to do the test again.
JavaScript should only be used for the display of time, not for validity checking alone without any server-side checking.