I have been through this all before.. Except I was in your DS position..I never cared enough about school or had the motivation to do homework and assignments.. although I never lied about it. I just simply said I didn't do it.. Which I think annoyed my parents/teachers more.. They used to say that I cared so little I couldn't even be bothered to lie.. Which is true.
I can't say for sure the reason why.. I guess because I had a very up and down mixed up childhood that structure never really worked for me.. I didn't like being told what to do.

I ended up leaving school, and I was immediately relieved.. I got a few casual jobs for about 6 months while I settled into life outside of school and then got my first full time job at a law firm and have been in law ever since.. I now manage team within the firm and get to boss other people around (much more to my liking).. I am dedicated, responsible and very successful I what I do..

School just really does not suit some children..
Let him struggle through his HSC and then make him go out and get a trade.. Carpentry, electrics etc..
I found that the work place suited me much better.. I was giving and getting something in return. I was praised for my hard work and rewarded in ways that aren't possible at school..
My DF sounds exactly like your son at the same age.. He is now a successful builder who runs his own company..

Just because he isn't good at school doesn't be he won't succeed in life

I know it would be hard for you to think from his perspective but just try to... He has obviously fallen behind, the work has become too hard. He feels like it is hopeless trying to catch up and doing assignments because he is probably going to fail anyway, then he goes to school and has teachers on his back and then he comes home and has you on his back and it just makes school this huge unenjoyable stress in his life that is just a constant source of anxiety, which will perpetuate that feeling of just giving up.. And which will make him lie to you and his teachers and come up with excuses because he wants some reprieve from the constant nagging.. I'm sure as far as he is concerned his HSC is a lost cause and no matter what he does now he will still fail.. you can battle with him to try and change this mindset.. Maybe get him a one on one tutor and try and talk to him about what will make his life easier I.e no chores of a night so long as he is studying.. If he can prove he is doing his homework you will give him and allowance..

And just know that this is not uncommon at all..

I really hope that everything works out. I'm almost positive it will. A little understanding goes along way