Cutoff during boot up?

You could have more than one issue. Best to take a look at the diagnostic LEDs and let us know your finding. It may be advisable to boot from a know good system on an external drive. You may have a partially failing power supply. Also you may have a corrupted hard drive. Let us know your finding.

Update

This one is strange because I would expect #4 to be with #3 off. Look at the scree (when it is booted up) with a flash light held at an angle and tell us if you can discern the desktop. (inverter issue)

LED 1 - Indicates that the trickle voltage from the power supply is detected by the main logic board. This LED will remain ON while the iMac is connected to the AC power. The LED will remain on even when the computer has been shut down or put to sleep. The LED will turn off only if the AC power is disconnected or the power supply is faulty.

LED 2 - Indicates that the main logic board has detected proper power from the power supply when the computer is turned on. This LED will be ON when the computer is turned on and the power supply is working correctly.

LED 3 - Indicates that the computer and the video card are communicating. This LED will be ON when the computer is communicating properly with the video card. If LEDs 1 and 2 are ON and you heard the startup sound, but LED 3 is OFF, then the video card might be installed incorrectly or need replacement.

LED 4 - Indicates that the computer and the LCD display panel are communicating. This LED will be ON when the computer is turned on and video signal is being generated. If the LED is ON and there is no image on the LCD display panel, the LCD display panel or inverter might be installed incorrectly or need replacement.