You might be suffering from https://support.apple.com/kb/TS4088 (I had a similar issue with a 15" mid-2010). Take it to apple store and ask them to run the TS4088 test (there's a special test for the issue) and you'll still be covered even if your applecare expied.
Thanks for your response!
I think that my issue is a little different - Mac works normally while playing video... It just shuts down for no reason. Like its lost the power or drained the battery.
It appears that battery is 3 years old and has old 71 cycles. Could the problem be because of the battery?
Try disabling automatic graphics switching (i didnt upgrade to ML, so I dont know if itll work on your machine)
When I had the issue, I used gfxCardStatus and noticed that the machine would go on the fritz only when dynamic switching was on.
As for the battery, you can check the status if you go to Apple -> About This Mac -> More Info -> System Report -> Power and look for the "Condition" line. It should say "Normal"
I'm using only 9600M, not the other one... I installed an app that helps me switch between the two, and it appears that both work normally...
This the info of the battery:
Charge Remaining (mAh): 5543
Fully Charged: No
Full Charge Capacity (mAh): 10609
Cycle Count: 72
Battery Installed: Yes
Amperage (mA): -5233
Voltage (mV): 7364
It does look as if you have a relatively 'healthy' battery, actually. I'm assuming that you're running on battery now since you're at less than a full charge and your battery isn't charging?
AFAIK, the NVIDIA problem with the mid-2010 models was only with the 15" models - however, you will probably need to take your machine into your local Apple Store or an AASP to have them run hardware diagnostics.