I have the same computer and virtually the same problem. I took our computer with us to the Mac Store at Pacific Centre in Vancouver BC. The tech said since the computer was over 5 years old and couldn't run a scan on it. He said it was probably the 'logic board' and wasn't worth fixing.
He did offer to sell us a new computer and transfer our files for free. My suggestion is to do what we did, buy another brand and threaten to throw it into the garbage. Actually, when we returned from Vancouver, it shuts down much less often than before (it got to be about every 10 minutes at one point) but I guess we are now using our new computer much more and our Mac is only a back up.
It is likely the last one we will ever buy.