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Is the test significantly easier than last year, or only a little easier? Estimated E4 cut off?

Re: HSC 2013 4U Marathon 0<p , \ 0 < q p+q < 1 $Prove$ (px+qy)^2 \leq px^2 + qy^2

Re: HSC 2013 2U Marathon $Solve the equation$ x^2 -6x + \frac{3}{x^2-6x+3} = -1

Re: HSC 2013 3U Marathon Thread $By taking the fact that$ (1+x)^n = (1+x)(1+x)(1+x) \dots (1+x) $Prove the Binomial Theorem$

There should be 8 solutions it seems from graphing them in geogebra \cos(\pi/ 2- 4x) = \cos 3x Now we use sum to product formula: -2\sin \frac{3x + (\pi /2 - 4x)}{2} \sin \frac{3x - \pi / 2 + 4x}{2} = 0 Which is easy enough to solve, for the first one: \frac{-\pi}{4} \leq \frac{\pi /2 -...

Try letting z = \cos \theta + i \sin \theta

Re: HSC 2013 3U Marathon Thread If you write it in summation notation it becomes easier to understand, so this is the general sum: S_{n} = \sum_{m=1}^{2^{n} -1 } \frac{1}{m} We need to prove that S_n > \frac{n}{2} For n = k (just like normal) $assume true$ \ \ S_k = \sum_{m=1}^{2^{k}...

Re: HSC 2013 3U Marathon Thread Yep

Re: HSC 2013 3U Marathon Thread Say I have a sequence of terms s_1 = a_1 + a_2 s_2 = a_1 + a_2 + a_3 + a_4 s_3 = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 and so on, so that means: s_n = a_1 + a_2 + \dots + a_{2n-1} + a_{2n} So when we do s_{n+1} then, how many extra terms are we adding on...

I don't know if that is the official jargon for doing: a= b to raise both sides by 10 is 10^a = 10^b But I use those words to explain it Just think about what a logarithm means, a logarithm is the opposite of exponentiation, so to get rid of a logarithm, we simply exponentiate, this is...

just do what you normally do when solving for a \ln a = b What would you do normally? raise both sides by e a = e^{b} Similarly: \ln(\ln x) = 1 \ln x = e and then.......

Alright thanks guys, I'll try to be a bit faster at short answers then if some people are hitting an hour for each!

So the recommended words for an economics essay is 800 words according to the board of studies. Is this just an average? If we want to aim for 18+/20, should we keep it at 800 words to be succinct or should we go to about 1000-1200 words because I usually have a lot of time remaining after...

This probably goes too deep into the definition of a constant and a variable and isn't really something I can explain (maybe someone else could shed some light on this). From what I can tell with my limited knowledge: When we differentiate, f(x) = ax + b, and we get f'(x) = a, why can't we...

Don't think of y is as a variable. Consider the function: f(x) = e^{x-1} - xy + y\ln(y) f'(x) = e^{x-1} - y f'(x) = 0 \ \Rightarrow \ \ x = 1 + \ln y f''(1+ \ln y) = e^{\ln y} = y $Since$ \ \ y > 0 \ \ \therefore \ \ f''(1+\ln y) \ \ $is a minimum$ \therefore \ \ f(x) \geq f(1+\ln y)...

Re: HSC 2013 4U Marathon Ahhh ok ============= Something a little easier: For a triangle ABC, with point P, Q, R on AB, BC, CA respectively Such that, AQ bisects BC, CP bisects AB, and BR bisects CA. These 3 lines intersect at 1 point (a property which you may assume), at a point K, this...

Re: HSC 2013 4U Marathon $We establish$ \frac{1}{2} < \frac{n+2}{2n+2} < \frac{3}{4} \ \ \ (*) $for$ \ \ n \geq 1 $Consider the sequence$ \ \ b_{n+1} = \frac{3}{4} b_{n} + 1 \ \ \ b_0 = 1 $By repeated iteration, we find$ b_{n+1} = 1 + \frac{3}{4} + \dots + \frac{3^n}{4^n} +...

Re: HSC 2013 4U Marathon S_n = \sum_{k=0}^{n} \binom{n}{k}^{-1} S_{n+1} = 1 + \sum_{k=0}^n \binom{n+1}{k}^{-1} \binom{n+1}{k} = \binom{n}{k} \times \frac{n+1}{(n+1) - k} \therefore \ \binom{n+1}{k}^{-1} = \left(1- \frac{k}{n+1} \right ) \binom{n}{k}^{-1} S_{n+1} = 1 + \sum_{k=0}^n...

Re: HSC 2013 3U Marathon Thread $Let$ \ \ r=\frac{p}{q} \ \ $for integers$ \ \ p, q \ \ $with no common divisor except$ \ \ \pm 1 y = x^{p/q} y^q= x^p $Since we have proven$ \ \ \frac{d}{dz} z^n = nz^{n-1} \ \ $and we assume chain rule$ \frac{d}{dx} y^q = \frac{d}{dx} x^p q y^{q-1}...

Re: HSC 2013 4U Marathon Practising for the BOS trial? :P (btw that was HSC 2007 4U 13 marks compressed). $You may use the fact that$ \sum_{k=1}^{\infty} \frac{\sin(k\theta)}{k} = \frac{\pi}{2} - \frac{\theta}{2} $Prove that$ \pi = 3 + \frac{4}{2 \cdot 3 \cdot 4} - \frac{4}{4\cdot 5...