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Re: HSC 2014 4U Marathon \\ $Let the series of complex numbers$ \ z_k = 1+ ik \ $for$ \ k=-n, -n+1, \dots, 0 , \dots, n-1, n \ $be in the Argand diagram$ \\ $Find the function$ \ f \ $so that the complex numbers$ \ f(z_k) \ $represent the corners of a$ \ (2n+1) \ $sided regular polygon$

Re: HSC 2014 4U Marathon Hmmm yea that looks right I guess, though your wording in the last sentence of your first answer is a little vague. Rather I would say: Ce^x = Ae^x + Be^x \Rightarrow \ Ce^{cx} = Ae^{cx} + Be^{cx} Thus, the only solutions (a,b) such that Ae^{ax} + Be^{bx} =...

Re: HSC 2014 4U Marathon - Advanced Level \\ $Find$ \ \lim_{n \to \infty} \left(\frac{(2n)!}{n^n n!} \right)^{1/n}

Re: MX2 Integration Marathon \int_0^{\pi/8} \frac{(\cos x + \sin x)^{\frac{3}{2}} - (\cos x - \sin x)^{\frac{3}{2}}}{\sqrt{\cos 2x}} \ dx

Re: HSC 2014 4U Marathon - Advanced Level \\ $Every number between$ \ 1 \ $and$ \ 10^n \ $not inclusive can be represented by the sum$ \\ \\ 10^k a_k + 10^{k-1}a_{k-1} + \dots + 10a_1 + a_0 \\ \\ $for$ \ k=0,1,2,\dots,n-1 \ \ $and$ \ \ a_k = \{0,1,2,3,4,5,6,7,8,9\} \\ (a_k \neq 0 \ $and if$...

Re: HSC 2014 4U Marathon \\ $For non-zero$ \ A,B,C \ $and$ \ A+B = C \ $if$ \ Ae^{ax} + Be^{bx} = Ce^{cx} \ \ (*) \ \ \ $for all$ \ x \\ \\ $i) Prove that$ \ a=b=c \ $is necessarily true$ \\ \\ $ii) If we drop the condition$ \ A+B = C \ $is there any$ \ a,b,c \ $so that the identity (*) holds...

Yep, I was merely pointing out that at her level the operation can not be justified

It is mathematically incorrect to do so if the sigma refers to an infinite sum, i.e.: \frac{d}{dx}\sum_{k=0}^{\infty} f_k(x) \neq \sum_{k=0}^{\infty} f_k'(x) Whereas: \frac{d}{dx} \sum_{k=0}^n f_k(x) = \sum_{k=0}^{n} f_k'(x) This is because firstly: \frac{d}{dx} (f(x) + g(x)) =...

Re: HSC 2014 4U Marathon - Advanced Level \\ $Consider the constant of$ \\ \\ (1+x)^{4n}\left(1 + \frac{1}{x} \right)^{4n} + (1+x)^{4n} \left(1 - \frac{1}{x} \right)^{4n} = \frac{(1+x)^{8n}}{x^{4n}} + \frac{(x^2-1)^{4n}}{x^{4n}} \\ \\ \sum_{k=0}^{4n} \binom{4n}{k}^2 +...

Re: MX2 Integration Marathon \\ \int_0^{\pi/12} \frac{dx}{(\sin x + \cos x)^4}

Re: MX2 Integration Marathon Yep well done, something like that, using geometric series *coughCSSA2011cough*

Harder 3U Topics Mostly because of a greater variety, and more imaginative questions

Re: MX2 Integration Marathon \\ \int_0^{\infty} |e^{-x} \sin x| \ dx

Re: MX2 Integration Marathon \int_0^{\pi /2} \frac{1}{1+\sin^2x} \ dx

Proselytism is essential in the 2 fastest growing religions in the world, Christianity and Islam As is evident in the Gospel of Matthew: "Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost: Teaching them to observe all things...

Re: MX2 Integration Marathon Yea his solution was pretty much exact same as mine As for your solution to this, it is quite long and you will need to evaluate \tan \frac{\pi}{8} Here is my solution: \int_0^{\pi/4} \frac{\tan x }{1 + \sin x} \ dx = \int_0^{\pi}{4} \frac{\tan x (1-\sin...

Re: MX2 Integration Marathon That will probably not work, this is beyond HSC difficulty though (but within the scope of the syllabus) The answer is n\frac{\pi}{2} , it uses recursion in the solution, I can post my one on request ----------------- \int_0^{\pi /4}\frac{\tan x}{1 + \sin x} \ dx

Re: HSC 2014 4U Marathon The way I did the upper bound was: f(x) = 1 - \frac{x^2}{\pi^2} - \cos x \ \Rightarrow \ f'(x) = - \frac{2x}{\pi} + \sin x \\ $Since the function$ \ g(x) = \sin x \ $is concave, then the straight line through$ \ (0, g(0)), \left(\frac{\pi}{2} , g\left(\frac{\pi}{2}...

Re: HSC 2014 4U Marathon - Advanced Level Yes well done, my solution was a little more direct: \\ $If$ \ F \ $is the primitive of$ \ f \ $which exists since$ \ f \ $is differentiable and hence continuous$ \\ \\ \therefore \ F(x+y) - F(y) = f(y) \sin x + F(x) \cos y \\ \frac{d}{dx} (F(x+y)...

Re: HSC 2014 4U Marathon - Advanced Level \\ $Suppose the function$ \ f \ $which is differentiable on$ \ \mathbb{R} \ $is such that$ \\ \\ \int_{y}^{x+y} f(t) \ dt = \int_0^x f(y) \cos t + f(t) \cos y \ dt \\ $Find$ \ f(x+y) \ $in terms of$ \ f(x) \ $and$ \ f(y) \\ \\ $Hence find$ \ f(x)