The three beeps has historically been a RAM issue. However, recently I have seen three 2011-2012 machines where it is not and is the logic board. To test use just one stick of RAM and remove the other. Try it in one slot and then the other slot. On each try take a look at the system configuration and see if it is reading correctly. Do this with the other stick. This will check both the sticks and the slots. Let us know your results. As it is, it look like a GPU issue.
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Let's also ask @danj if he has run into this.
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UPDATE
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@clarkclark The only solution I have found on the GPU issue is a logic board replacement. Due to the cost of the part, I have sent two into Apple requesting a "Flat Rate" depot repair. My total cost for this in Texas is $357. For that they will fix any and all problems if the machine has not received liquid damage or incurred a major impact. Total time for this is four days including the first call day. I call 1-800-MYAPPLE and request it, give them a credit card. Next morning there is an overnight FedEx box on my door step. I box it up and take it to FedEx. one day there, one day to repair and one day to send it back. It's a sweet deal.
The three beeps has historically been a RAM issue. However, recently I have seen three 2011-2012 machines where it is not and is the logic board. To test use just one stick of RAM and remove the other. Try it in one slot and then the other slot. On each try take a look at the system configuration and see if it is reading correctly. Do this with the other stick. This will check both the sticks and the slots. Let us know your results. As it is, it look like a GPU issue.
The three beeps has historically been a RAM issue. However, recently I have seen three 2011-2012 machines where it is not and is the logic board. To test use just one stick of RAM and remove the other. Try it in one slot and then the other slot. On each try take a look at the system configuration and see if it is reading correctly. Do this with the other stick. This will check both the sticks and the slots. Let us know your results. As it is, it look like a GPU issue.