Lately my computer shuts down or freezes at random

you mention there is a beeping noise you heard when rebooting your pc... check your unit by unscrew your cover side of your unit and check any hardware installed if its properly connected, specially your video card if you have, your memory, your lancard...before you unplug and connect again your peripherals please unplug the power first from your Ac outlet.

regarding for your suspesion in trojans infection.... try to run a combofix software... you can download that one in bleepingcomputers.com or just tyle the word combofix in your search engine of your browser. download that one. it should be the latest on the net after downloading combofix copy and paste it in your desktop and run it....take note follow the steps carefully appear on the screen.

about your question to properly rid off of computer virues you have a reliable anti-virus internet security installed in your PC/laptop.....and it should be updated...another solution to that one is by freezing your hardrive use the deepfreeze installer.

check memory card and fit it well to the slot...if u use two cards remove one from the slot and let the other one card remain intact fitted in the slot..then switch on your pc..if OS work and it dont display,,check your vga driver or else there is something wrong with your video card..