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Wait a minute, I thought when people meant UNSW comp, they meant ICAS........ I didn't do the UNSW comp then, I didn't even know it existed. fail

Re: MX2 Integration Marathon Yep, well done.

Re: HSC 2013 3U Marathon Thread Ah yep, I remember that, oops.

Re: MX2 Integration Marathon \int \frac{x^2-1}{x^4+x^2+1} \ dx

Re: MX2 Integration Marathon \int \frac{dx}{(x+\sqrt{x^2-1})^2}

Re: MX2 Integration Marathon x=\tan \theta \Rightarrow I=\int_{0}^{\pi/2} \ln(\tan \theta) \ d\theta = \int_{0}^{\pi /2} \ln(\cot \theta) \ d\theta 2I = \int_{0}^{\pi /2} \ln(\tan \theta) + \ln(\cot\theta) \ d\theta = \int_0^{\pi /2} \ln(1) \ d\theta = 0 \Rightarrow I = 0

Re: HSC 2013 4U Marathon $Let$ P(x) = x^6-x^5-x^3-x^2-x Q(x) = x^4-x^3-x^2 -1 z_1, z_2, z_3, z_4 \ \ $are roots of the polynomial$ \ \ Q(x) $Prove that$ P(z_1)+P(z_2)+P(z_3)+P(z_4) = 6

haha yep :P

I hope its on Wednesday, I completely forgot about it until you made this thread.

Re: MX2 Integration Marathon x=\tan u Get an integral of secant^5 u, integration by parts twice to arrive at the answer. ==== \int_0^{\infty} \frac{\ln x}{1+x^2} \ dx

Re: HSC 2013 4U Marathon Maybe you should try to find 1+xcis theta in mod-arg form, so you can use De Moivere's formula.

Re: HSC 2013 3U Marathon Thread $i) Explain why$ 1-x+x^2-x^3+\dots+(-1)^{n-1}x^{n-1} = \frac{1}{1+x} + (-1)^{n+1} \frac{x^n}{x+1} \ \ \ \ \fbox{1} $ii) Hence prove that$ 1 - \frac{1}{2} + \frac{1}{3} - \dots + \frac{(-1)^{n-1}}{n} = \ln 2 + (-1)^{n+1} \int_0^1 \frac{x^n}{x+1} \ dx \...

Yes you are correct, it is indeed 127/7

For 1). First find the number of numbers that we can make with only the digits (1, 2, 3, 4). How many 1 digit numbers? 4 How many 2 digit numbers? 4*3 = 12 How many 3 digit numbers? 4*3*2 = 24 How many 4 digit numbers? 4*3*2*1 = 24 Total is 64. Now find the number of numbers, that only have...

Re: HSC 2013 4U Marathon $Solve the following equation for$ \ \ x \sum_{k=0}^{n} \binom{n}{k}x^k \cos(k\theta) = 0

Re: HSC 2013 4U Marathon $i) Prove by induction or otherwise, that for positive real numbers$ \ \ a_k \frac{a_1+a_2+ \dots + a_n}{n} \geq \sqrt[n]{a_1 \cdot a_2 \cdot a_3 \dots a_n} $And state the conditions for equality$ $ii) Hence prove for some integer$ \ \ n \ \ $ that$...

Re: MX2 Integration Marathon Well then there you go lol, I did make a mistake.

Re: HSC 2013 4U Marathon Can you please explain this part?

Re: HSC 2013 4U Marathon Alright, now the challenge is to find the average area of the triangles. It can be done easily if we fix P somewhere, but I can't seem to find a good way of finding the averages at a point in general. (I have a method in mind, but it feels very brute force-y) If no...

Re: HSC 2013 4U Marathon Yep nice work I get the first one, but using my method you get the factorised version first, n^2(n-1)(n-1) is that the same for yours? EDIT: I don't know if its possible to get this but: $For the above problem, find the average area of the triangles$