Hook it up to an external monitor to test. There was a repair program on the 2011 GPU on your series of machine and we can address that after you have checked it.
Please give us the last four figures of your serial number so I can check it out for this:
iMac (27-inch): AMD Radeon 6970M Video Card Replacement Program
Apple has determined that some AMD Radeon HD 6970M video cards used in 27-inch iMac computers with 3.1GHz quad-core Intel Core i5 or 3.4GHz quad-core Intel Core i7 processors may fail, causing the computer’s display to appear distorted, white or blue with vertical lines, or to turn black. iMac computers with affected video cards were sold between May 2011 and October 2012.
If your iMac has an AMD Radeon HD 6970M video card and is exhibiting any of the issues described above, choose one of the following options to arrange to have your iMac evaluated:
Before you go in for service, please back up your data. Learn more about backup options.
If the iMac (27-inch) meets these requirements, Apple will replace the video card free of charge for four years after the first retail sale of the computer.