I would have thought that physics is fairly fundamental to understanding climate science. If you don't want to study it, I suggest that you reconsider your major.
Compulsory subjects in a major are there for a reason. Particularly at first year level.
This is not correct. I did a BSc BEd majoring in maths at UNSW and did the higher stream courses the whole way through. Any student can enroll in the higher stream provided their maths WAM is high enough, it doesn't matter if you are in advanced science or not.
When I studied at UNSW, maths teachers only had a single major. From the link you've posted it looks like this is still the case. The situations with science teachers is a bit different, because they need to do a wide variety of science subjects in order to teach general science in years 7-10.
You do occasionally meet people who teach both. But they are rare. I suspect that if you qualify to teach both you will find that most of the time you are teaching maths, as that is where the big shortages are.
No, they haven't. You do not need to do a dip ed after doing the BSc BEd at UNSW. Check with the institute of teachers if you want to be sure. They have a list of approved initial education courses.
'Cause I am feeling lazy, I shall link you to the appropriate wikipedia site.
http://en.wikipedia.org/wiki/Envelope_(mathematics)
It even has a gif annimation
The maths modelling course; Optimisation; Dynamical systems and Chaos; PDEs.
You should also consider doing Higher Analysis, even though it is a Pure maths subject.
Computing, maybe.
Do you mean subjects that you have already done that you can't count towards your degree, or electives that you need to choose for the Advanced Maths degree?
If the former, you can probably apply to have some of your B.Com subjects counted in place of Gen Eds, and if the latter, you just do...
Presumably you have been using the fact that an integer cannot be simultaneously odd and even to do Q1. You do can do the base 3 one by appealing of the Fundamental Theorem of Arithmetic:
http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic
(The odd/even stuff is just a special case...
You could express y as a function of x (or rather two functions of x, as you will have to take a square root somewhere) and then think real hard about what happens as x goes to infinity.
Take eq(1) and complete the square:
5x^2 - y^2 + 4xy = 18
9x^2-(y^2-4xy+4x^2)=18
9x^2-(y-2x)^2=18
Using the difference of two squares, we have
(3x-y+2x)(3x+y-2x)=18
(5x-y)(x+y)=18
Now we can see that neither of the factors on the LHS can be zero since the RHS is not, so 5x-y=0...