Give me a situation where bet/folding is better than check/calling

I may be incorrect but I believe that this river needs to be value bet, however the correct line would proably be bet fold because there is virtually no way I'm beating a raise.

Party Poker 2.00/4.00 Hold'em <font color="#0000FF">(10 handed)</font> link

Preflop: Hero is UTG with Q[img]/images/graemlins/diamond.gif[/img], Q[img]/images/graemlins/heart.gif[/img].
<font color="#CC3333">Hero raises</font>, <font color="#666666">4 folds</font>, MP3 calls, <font color="#CC3333">CO 3-bets</font>, <font color="#666666">3 folds</font>, <font color="#CC3333">Hero caps</font>, MP3 calls, CO calls.

Flop: (13.50 SB) 6[img]/images/graemlins/heart.gif[/img], 3[img]/images/graemlins/spade.gif[/img], 6[img]/images/graemlins/diamond.gif[/img] <font color="#0000FF">(3 players)</font>
<font color="#CC3333">Hero bets</font>, MP3 calls, CO calls.

Turn: (8.25 BB) 3[img]/images/graemlins/heart.gif[/img] <font color="#0000FF">(3 players)</font>
<font color="#CC3333">Hero bets</font>, MP3 calls, CO folds.

River: (10.25 BB) A[img]/images/graemlins/diamond.gif[/img] <font color="#0000FF">(2 players)</font>
<font color="#CC3333">Hero bets</font>, MP3 calls.

Final Pot: 12.25 BB.