certificate of assumed name, it shows the name of the corporation,limited partnership, or limited liability is:
lets say....bear inc.
then it says the assumed name under which business is to be transacted is: cavern industries
my question is, do we have to title the signature card and resolution, bear inc.dba caven industries?? and does the reporting have to be under cavern industries

The account title should be "Bear Inc., DBA Cavern Industries" and carry Bear Inc.'s EIN. (Most corporations are not subject to information reporting. If your bank suppresses returns on corporations, the specific account titling may make little difference.)

The resolutions should be from Bear Inc. and specfically authorize the opening of the account in the name of the DBA and authorize signatories.

For both questions, the answers are driven by the fact that Cavern Industries is not a legal entity, only a name that Bear Inc. has borrowed.