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Re: HSC 2013 4U Marathon I just solved it: http://i981.photobucket.com/albums/ae294/sy08071996/Image4-2.jpg However I believe that extends it to non-zero complex numbers as well. Brief explanation of solution: - Let polynomial have roots x, y, z Find co-efficients of b, c...

Re: HSC 2013 3U Marathon Thread If I were to give this a topic it would be 'Harder 2U' However you need to use a technique that you probably would not have seen if you haven't done polynomials. However you do not need it, if you can notice something special about f'(x), and notice how it can...

I'm no mathematician nor a school teacher. But I read through the document and I agree with some points, on others however I disagree greatly. He seems to be greatly concerned about the difficulty of the courses. In my opinion mathematics should be difficult, it should challenge students and...

Re: HSC 2013 3U Marathon Thread $Find a general quadratic function such that if $ f(x) \\ $ is the quadratic function then $ \\ \\ f(f'(x))=f'(f(x)) I feel like it belongs here more than it does in the 4U marathon.

http://lichess.org/g8tcob6d EDIT: I think Ill go do english heh. Ill play against you tomorrow if you are up for it :)

Unlike its 3U counterpart in my opinion the 4U book is horrible, the questions are horrible, the excercises are just tedious and boring. Barely anything original. For 3U though I recommend it.

http://lichess.org/nchj7ok7

http://lichess.org/o2i6z6es

still up for one?

Every complex number can be expressed within the rectangular co-ordinate system, if it cant be expressed its some higher order uni thing (qauternion?). Whether we can humanly find this though is another story, for instance. Is it possible for us to compute (IN EXACT FORM)...

Re: HSC 2013 4U Marathon $Define the function $ f(x)=(\cos x + i\sin x) \cdot e^{-ix} $By considering $ f'(x) $ Prove the famous result $ \\ \\ e^{i\pi}=-1 (Do not worry about whether it is actually possible to take exponentials of complex numbers)

Re: Mole calculations In our given amount of substance, in this substance if there is an 'n' number of AlCl3 molecules, since there is only 1 Al atom for every 1 AlCl3 molecule because this Al atom is PART of the AlCl3 molecule itself, therefore there must be an 'n' number of Al atoms. What...

Re: Mole calculations From what I interpreted from the question: Find the number of moles in AlCl3 2.80 x 10^-3 By definition of a mole, one mole has the avagadro's constant amount of particles in it. Hence the number of molecules of AlCl3 2.80 \times 10^{-3} \times N_A= 2.80 \times...

Re: HSC 2013 4U Marathon The proof by induction is quite easy, do you mean the derivation of the result?

Re: HSC 2013 4U Marathon The Fibonacci sequence is a sequence of numbers whereby each term of the sequence is the sum of the previous two terms, the sequence starts off with two terms 1, 1 Then it goes on following the Fibonacci rule: 1, 1, 2, 3, 5, 8, 13, 21 ... Notice how each...

Re: HSC 2013 4U Marathon Lets cheat and use the result from the previous question I asked. Although they are reciprocals of the question I asked, the result still applies. We can prove that if they have modulus 1 (which u, v, w all do when solving the equation), then they are all purely...

Re: HSC 2013 4U Marathon b) On our sketch, notice that: \vec{OM}=0.5(a+b) \vec{QN}=0.5a-b Now, the point R lies on both vectors. Indeed for OM, it lies on the vector therefore there must exist some real constant k such that we reduce the modulus of \vec{OM} Such that it equals to...

This is formally known as the triangle inequality however indeed it is quite logical. Can I not establish the fact that the sum of any 2 sides of a triangle are greater than the third side? Now lets apply this to our complex plane. I advise you to draw this out as I say it so you can...

Re: HSC 2013 4U Marathon z \in \mathbb{C} \ \ \ \ \ |z|=1 $Prove $ \ \ \ \ $Re$(\left\frac{1-z}{1+z})\right=0

Re: HSC 2013 4U Marathon Note the nth roots of unity. Lets call a general kth one of the nth roots of the unity: \alpha_k=\cos \frac{2\pi}{n}k + i \sin \frac{2\pi}{n}k If we fix j, we can still put it into alpha and it will still be a root since 'jk' will cover all the possible roots. Hence...