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Re: MX2 2015 Integration Marathon I = \int_1^3 \frac{\ln x}{3+x^2} \ dx u = \frac{x}{\sqrt{3}} , \ I_1 = \int_{1 / \sqrt{3}}^{\sqrt{3}} \frac{\frac{1}{2} \ln 3 + \ln u}{3 + 3u^2} \cdot \sqrt{3} \ du = \frac{1}{\sqrt{3}}\int_{1 / \sqrt{3}}^{\sqrt{3}} \frac{\ln \sqrt{3} + \ln u}{1 + u^2} \ du...

Re: MX2 2015 Integration Marathon Do you mean a substitution that does it all in one go? The only thing I can think of at the moment is: u = \frac{x}{\sqrt{3}} \Rightarrow \ I_1 u = \frac{\sqrt{3}}{x} \Rightarrow \ I_2 2I = I_1 + I_2 = \int \dots Where the dots are a tan inverse...

Let me show you the absurdity of your statements --------------------------------- 1. You reject the existence of God since you find no definite proof for His existence 2. Therefore, you want definite proof of something before you believe in it (from 1, if you reject 2 and affirm 1...

Re: HSC 2015 4U Marathon - Advanced Level \\ $A monic cubic has roots$ \ \alpha, \beta, \gamma \ $show that all 3 roots are real if and only if$ \ (\alpha - \beta)^2(\beta - \gamma)^2(\alpha - \gamma)^2 > 0

Re: HSC 2015 4U Marathon \\ $Describe the locus of$ \ z \ $in the Argand plane if$ \ \frac{1}{z} + \frac{1}{\bar{z}} = 1

Re: HSC 2015 4U Marathon - Advanced Level Nice! Here is my solution: \\ $The function$ \ f(x) = p(x) - p(-x) \ $has$ \ n \ $roots, at$ \ x=1,2,3, \dots, n \\ \\ $Since$ \ f \ $is the sum of polynomials it is itself a sum of polynomials, the leading term of$ \ p(x) \ $is identical to the...

Re: HSC 2015 4U Marathon That's almost how I would do it: After showing (\alpha + \beta - \gamma)(\alpha + \gamma - \beta)(\beta + \gamma - \alpha) = 8(1- \alpha)(1-\beta)(1-\gamma) p(x) = x^3 - 2x^2 + 3x - 1 = (x - \alpha)(x- \beta)(x-\gamma) \therefore \ p(1) = 1 =...

Re: HSC 2015 4U Marathon - Advanced Level \\ $Let,$ \ n \ $be a positive even integer, and let$ \ p(x) \ $be a polynomial of $ \ n$-th degree, such that, $ \ p(k) = p(-k) \ $, for $ \ k=1,2,3, \dots, n \ \\ $Prove that there exists a polynomial,$ \ q(x) \ $such that,$ \ p(x) = q(x^2)

Re: HSC 2015 4U Marathon - Advanced Level Done

Re: HSC 2015 4U Marathon - Advanced Level What about for example a polynomial with roots \sqrt{2} and \sqrt{8} ? Just knowing about the product of roots alone is not enough to conclude that each irrational pair needs its conjugate

Re: HSC 2015 4U Marathon - Advanced Level \\ $A polynomial$ \ p(x) \ $has integer coefficients,$ \ p(x) \ $also has a root of the form$ \\ p + \sqrt{q} \ $where$ \ p, q \ $are rational and$ \ q \ $is not a square of a rational$ \\ \\ $Prove that$ \ p(x) \ $also has a root of the form$ \ p -\sqrt{q}

Re: HSC 2015 4U Marathon \\ $The polynomial$ \ x^3- 2x^2 + 3x - 1 \ $has roots$ \ \alpha, \beta, \gamma \\ \\ $Find the value of$ \ (\alpha + \beta - \gamma)(\alpha + \gamma - \beta)(\beta + \gamma - \alpha)

Re: HSC 2015 4U Marathon - Advanced Level In HSC exams I would think that you'd need to justify the use of the "extended product rule", so how would you do it?

Re: HSC 2015 2U Marathon TOPIC: Maxima and Minima \\ $I have a constant amount of fencing that I want to turn into a rectangular paddock, and I want to maximise the area of the paddock. Show that no matter how much (constant) material I have, I should always make a square shaped paddock$

Re: HSC 2015 4U Marathon - Advanced Level \\ $Let$ \ p(x) \ $be a polynomial with degree$ \ n \ $with real roots$ \ \alpha_1, \alpha_2, \dots \alpha_n \\ \\ $Prove that:$ \ p'(x) = \frac{p(x)}{x - \alpha_1} + \frac{p(x)}{x - \alpha_2} + \dots + \frac{p(x)}{x - \alpha_n}

Re: HSC 2015 4U Marathon - Advanced Level Yep that's the right idea A more rigorous way of doing so would be "if E has no double roots, then each root is a single root, and so then there is some value 'c' So that E(c) < 0, but E(x) -> +inf as x grows large and E(x) -> +inf as x becomes...

Re: HSC 2015 2U Marathon For those who haven't started series yet, and are still doing differentiation and integration TOPIC: Integration \\ $The area enclosed between the curves$ \ y = x \ $and$ \ y=x^3 \ $is rotated about the$ \ x \ $axis, find the volume of the figure generated$

Re: HSC 2015 4U Marathon - Advanced Level \\ $NEW QUESTION - required knowledge: Polynomials and Complex Numbers$ \\$Let the polynomial$ \ E(x) = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \dots + \frac{x^n}{n!} \\ \\ $i) Prove that$ \ E(x) \ $has no double root$ \\ \\ $ii) Prove that if$ \ n \ $is...

Re: HSC 2015 4U Marathon - Advanced Level Here is my solution: \\ $Take the proposition that$ \ a_n \ $converges$ \\ \\ $1. If$ \ a_n \ $converges, then$ \ \lim_{n \to \infty} (a_{n+1} - a_n) = 0 \\ \\ $2. If$ \ \lim_{n \to \infty}(a_{n+1} - a_n) = 0 \ $then$ \ \left(\lim_{n \to \infty}...

who is that? and probably not