iMac keeps shutting down after new RAM installation

Screenshot: it shows disk error and that it could not be verified. So, you need to try to repair your hard drive - the RAM issue shouldn't be related, it may just be the drive, but at this point, it is hard to tell.

So, first try to repair from within Disk Utility. If that does not work, boot into internet recovery, once there, go to Disk Utility and repair the drive. It will show you if it did successfully or not. Booting into internet recovery means you'll see a rotating globe while connecting to Apple's servers - some key combos here:

https://support.apple.com/en-us/HT204904

After that, run the extended hardware test again.

macs are pretty picky with memory which is why most buy from crucial and others who test modules with each module to verify compatibility

also memory modules are easy to fry with static electricity when installing, they are based on a technology which can die from a 50volt difference and statics can be 5K volts

Ok, what brand RAM please (and is it "value" branded)?. Even though the specs may be the same, as was already mentioned, Macs are super particular about their RAM. The two most recommended here are either OWC or Crucial. Personally, I would not buy any other.

If it is one of those or we can't figure out anything else obvious, then it could be the RAM is faulty, not seated correctly, or there is a problem with the slots. Did you have any in the "new" slots previously?

Next step would be to run the Apple hardware test with the new RAM installed - have you done that yet? Run the extended test; it is by no means 100% foolproof, but may give an indication.

Depending on the outcome, next step is to contact the RAM vendor and get a replacement. If that still doesn't do it...

Last step would be to take it to Apple (Genius Bar appointment) and have them run diagnostics - their equipment is much better.

As for your screenshot, sorry, but I can't make out anything (too small).

Well then - for now - I'd assume that the entire problem is disk related - the RAM may have nothing to do with it - the drive may have decided to fail right at that time.

So, first thing: back up your files and/or make a bootable clone of your system. It appears that your hard drive is failing so you will need to either take it in or (since it's out of warranty) replace it yourself. I'd take it for a free diagnostic test at the Genius Bar first to make sure. You can also ask how much they'd want for a replacement. If that is too pricey, check with an AASP - they may be less expensive.

If you want to do it yourself, here is a good source for drives (or SSDs) and video instructions:

https://eshop.macsales.com/search/?q=late%202012%20imac%20upgrades&filter.catidp ath=4081

https://eshop.macsales.com/installvideos/imac_27inch_late_2012/

If you want to try - fine; however, there is no guarantee. The disk error(s) may be fixable or not. All hard drives fail at some point - some sooner than later. (By the way, I had the very same iMac - great machine; sold it while it was still covered by Applecare). I have DW, so I'd probably try that first, but the diagnostic equipment at the Genius bar is very good and they'll check it for free.

You did not tell us what RAM you installed and what RAM configuration/slots. And, you did not tell us what model Mac you put them in. Both are important to troubleshooting.