An odd one. If the microswitch in the lock barrel has stuck, normally the dash will tell you the key is in even when it isn't when the door is opened. It will also cause a mislock beep when locking. I suppose it is possible that it has failed open circuit so it doesn't detect the key in the ignition and that could set the alarm off when the ignition is switched on, but not as soon as the key is put in the ignition (as it wouldn't know the key has been put in the ignition). The coil around the lock barrel is there to cause the fob to transmit the unlock code as soon as the key is put in the ignition so it resets the immobiliser if passive immobilisation is still enabled. As far as I know it is only the immobiliser that comes on with passive and not the alarm too. Also, if passive is still enabled and the coil has failed so the unlock code isn't being sent, you would get the Engine Disabled message on the dash.