Alarm goes off opening trunk with key

I recently bought a 2002 330ci vert. The keyfob doesn't work for starters. I have already tried replacing the rechargeable battery for the keyfob but it still doesn't work. That's another problem.



I lock everything using the key in the driver's door lock, if I unlock the trunk using the key, the alarm goes off. The alarm doesn't go off if I unlock the doors using the key in the door lock. Any suggestions? Thanks.