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Re: HSC 2014 4U Marathon Here's that inequality: $Let a, b, c and d be non-negative numbers such that $ a+b+c+d=4. $ Prove that $\frac{4}{abcd}\ge\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}. And one for 2014'ers (I think I've given enough to make it reasonable). $Given that $...

From part i) you get two possible conditions (y=0 or x^2 + y^2 = 1) after splitting the real and imaginary parts. Sub these into the real part (relating x, y and k). Then, for the first part it's a simple inequality which you can use x+\frac{1}{x}\ge 2 by a^2+b^2\ge2ab. For the second part...

Re: HSC 2014 4U Marathon This isn't too bad actually and a rather nice question, once you realise that cis\theta\times cis\alpha=cis(\theta+\alpha). Taking the real part of the geometric sum you get S_n=Re\left(cis\theta\left(\frac{cis^n\alpha-1}{cis\alpha-1}\right)\right). Make the...

Use a subtraction of the y coordinates ie y1 - y2 to find the length, since the x value is the same. Getting a quadratic from this, you can manipulate and graph instead of differentiating. My answer is 7/2.

You may have used the wrong intercepts. When I used (0,4) instead of (0,4+a^2) as the y intercept, I got a=2. Try it, remembering that the point is in the first quadrant.

I haven't run into any worked solutions other than those people have physically written in their books. Extension questions are often above the HSC course but are great especially if you have a healthy interest in math. There are people here who would be more than willing to help out with...

I would also like to know. All I found for it was https://en.wikipedia.org/wiki/Quaternion (see the table on the right). "i^2 = j^2 = k^2 = ijk = −1,"

I'd go with harder 3U though polys can get really tricky (see past HSC q15/16s). Does the auxilary angle for R<0 have a different angle theta or something which reduces down through odd/even/trig properties to give the same answer? Question: how do you find the volume of a solid if the curve...

This question has horrendous algebra and I'm sure there's a cleaner way. Pretty sure you've got the method right, and if you want to do it yourself check the bottom of the second page. Neat question though. https://drive.google.com/folderview?id=0B4D3wNobxSuvdEVYOThhSFlWc1U&usp=sharing Sorry for...

It's a pretty standard question, just tricky wording and nothing too tricky really. Here is a diagram oriented nicely (yes MS paint) to clear things up and each piece of tape goes over all four sides. Simultaneous, sub in one and differentiate to find the minimum and that should be it.

Expansion is the way to go. Usually with this question and its variants, there is a previous part where you explain why k(k+1) is even.

Re: HSC 2013 4U Marathon That's the one.

Re: HSC 2013 4U Marathon Mechanics already! I'm not sure if I'm missing a force or my diagram is wrong, but I get \tan\theta=\frac{\omega^2R}{g} which is missing h. Graphing this for w>0, it has an inverse function (autonomous increase is the phrase I think) and so there is only one value...

English Extension 1 (EE1) and lower do not require a major project. I don't remember what we did in year 11 but for year 12 you have two hours to write one essay and one creative (generally 1.2-1.5k words each). That being said, it helps to do research to really understand the module and then be...

It looks like you've got the geometric sum down pat (2n+1 - 2) So then it's just a matter of the arithmetic sum and -1s. My guess is that you missed the multiplication by 2 since 2+4+\dots+2n=2(1+2+\dots+n)=2\times\frac{n(1+n)}{2}. Stuff cancels out and it should give you the answer.

Myself, I would work through the Cambridge from the development questions (the extension questions are typically above the 4u level) for 3u, and the questions in the 4u textbook. Make sure you understand the examples and maybe do some of them without checking the answers. Going through questions...

No problem! Ah, that makes sense then, which means w^3 = 1 ie w is the cube root of unity.

You can factorise using w^3-1=0 \to (w-1)(w^2+w+1). Since w =/= 1 then w^2+w+1=0

Since 1/x -> 0 as x -> infinity, you can use this to describe what happens to y (and dy/dx) at the extremes of the graph. In this case you get y -> x and dy/dx -> 1 which means there's an asymptote. Then you can use all this to sketch the graph (including intercepts, asymptotes, stationary...

Roots of unity is only for z^n=1. Roots of unity questions are all based on this and typically use factorisation or de Moivre's theorem. Sorry for any earlier confusion.