Imacs 27” (mid 2011) have a known issue with some AMD Radeon series where the GPU gets soldering problems.
The “easy" way to solve this is by baking the graphics card in the oven for 8 minutes at 200 deg. celsius.
I repaired mine twice this way (it will last for nearly a year). But last time (a couple weeks ago) I decided to do an upgrade. I've installed a GeForce GTX 780M from a Dell laptop and now I have an awesome performance on games.
You need to use nvidia web drivers in order to work instead of the Apple official one.
Since PC cards doesn't have the Apple EFI on it's EEPROM, you will lose the boot startup screen and the brightness control. If you don't care about that, your old machine will become a very decent beast again!