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I have connected an external second screen on imac 27 late 2013 with a broken screen, but the imac does not open.  I do not see the logo and then the gray screen.  But if I isolate the broken imac screen by disconnecting its cables, then on the second screen I have a normal image and I see everything but the fans are full. Is there a way not to hear the fans? What should I do?  Thanks and sorry for my English

Your system is designed with sensors to monitor your systems health. So when you disconnect your internal display screen, SMC looses access to the sensors within it. When it does it goes into CPU Safe mode! This is done to protect your system from over-heating. Ramping up the fans RPM as well as lowering the CPU’s clocking. You can’t alter this. Basically, you need to replace your internal display. Here’s the needed part: 27” iMac LCD Display Assembly, Apple P/N 661-7169