How to delete database without using any plugin?

If the database server is available publicly you should be able to use a MySQL client to interact with it.

Sequel Pro is a good one for mac, check this list if you need something for windows or linux.

If the database is only available locally on your web server, you might be able to upload a DB admin script like adminer via filezilla and use that to interact with the database.

If all else fails you will likely have to go back to your instructor and ask him/her to reset the DB for you.