I'm going to say there is hope for General Maths students out there. Currently, tutoring a General Maths student and I seriously think the kid , (Band 6) at the moment could have and would be able to do Engineering or Science related maths course. I have tutored a student in the past who also said to me he wanted to do Engineering , with only a General Maths background. There are bridging courses, Youtube links out there that enable that knowledge to be gained.
I said this before in another thread , if you are in Year 9 or Year 10 5.3 maths class, you should without any hesitation choose 2 Unit maths or higher in Year 11. The standard is no more difficult than General.
General Maths is only slightly behind in content in comparison to the IB Mathematical Studies. (In studies they cover the basics of differentiation which is more than catchable in about 4-6 weeks)
I also am not sure how strict universities , over here are with pre-requisites and what not. But I do get the feeling if you believe you have good reason to do a certain course, the university will let you do it. (i.e you did a bridging course like
http://sydney.edu.au/science/fstudent/undergrad/entry/bridging.shtml where if you showed you have taken some assessments and have sufficient background THEY will let you do a course. )