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Re: MX2 2015 Integration Marathon \\ $See$ \ f(z) = \frac{2 - z^2}{1 + z + z^2 + z^3} \\ \\ = \frac{(2-z^2)(1-z)}{1 - z^4} = \frac{1}{2} \cdot \frac{1}{1+z} - \frac{3}{2} \cdot \frac{z}{1+z^2} + \frac{3}{2} \cdot \frac{1}{1+z^2} \\ I = \int { \frac { 1+\sin { x } +\sin ^{ 2 }{ x } +\sin ^{...

Re: HSC 2015 4U Marathon \\ $i) Prove that for$ \ x \ $sufficiently close to$ \ 0 \ $that$ \ \cos x < \frac{\sin x}{x} < 1 \\ $ii) Hence show that$ \ \lim_{x \to 0} \frac{\sin x}{x} = 1

Re: HSC 2015 4U Marathon \\ $i)$ \ \left( \sum_{i=1}^n x_i \right)^2 = \left(\sum_{i=1}^n x_i \right) \left(\sum_{i=1}^n x_i \right) \\ = x_1 \left(\sum_{i=1}^n x_i \right) + x_2 \left(\sum_{i=1}^n x_i \right) + \dots + x_n \left(\sum_{i=1}^n x_i \right) \\ = \sum_{k=1}^n x_k...

Re: HSC 2015 4U Marathon \\ $Let$ \ x_1,x_2, \dots, x_n \ $be non-negative real numbers$ \\ \\ $i) Explain why$ \ \left( \sum_{i = 1}^n x_i \right)^2 = \sum_{k=1}^{n} \sum_{j=1}^n x_j x_k \\ \\ $ii) Hence or otherwise prove$ \ n \sum_{i =1}^n x_i^2 \geq \left(\sum_{i=1}^n x_i \right)^2

Re: MX2 2015 Integration Marathon \\ \int_0^{\pi /2} \frac{dx}{a^2\sin^2x + b^2\cos^2x}

Re: HSC 2015 4U Marathon \\ $Show that for real$ \ x \ $and positive integer$ \ n \ $that$ \ e^x \geq \left(1 + \frac{x}{n} \right)^n (edit: this inequality only works for x > - n if n is even)

Re: HSC 2015 4U Marathon \\ $Let the polynomial$ \ P(x) \ $be a monic cubic polynomial with non-negative real roots$ \ a,b,c \\ $i) Find$ \ P(a) + P(b) + P(c) \ $in terms of$ \ a,b,c \\ $ii) Prove for non-negative reals$ \ x,y,z \ $that$ \ x^2 + y^2 + z^2 \geq xy + xz + zx \\ $iii) Hence...

Re: HSC 2015 4U Marathon - Advanced Level Wouldn't it matter how close the satellites are orbiting this planet? Or am I just misunderstanding the question

Re: HSC 2015 4U Marathon - Advanced Level Yep well done!

Re: HSC 2015 4U Marathon - Advanced Level Not too difficult (but better suited for this thread than the other): \\ $Let there be a sequence of positive integers$ \ a_1, a_2, a_3, \dots \ $such that$ \\ a_{n+1} = a_n + a_n^2 \ $and some$ \ a_1 \\ $Define another sequence of positive...

Re: HSC 2015 4U Marathon I've proven it for when the polynomial's domain is restricted among the reals \\ $Let$ \ P \ $be a complex polynomial among the reals with complex co-efficients$ \\ c_j = r_j + is_j \ $for$ \ j= 0, 1, 2, \dots, n \ $and$ \ r_j, s_j \in \mathbb{R} \\ $So,$ \...

You can mention the fact that Islam as a living tradition, (at least we are talking orthodox Sunni Islam here, and in a way Shia Islam, however the latter is not relevant to Sayyid Qutb), in that in Islam there are religious authorities, the class of ulema who are seen as the learned theologians...

Re: HSC 2015 4U Marathon \\ 1 - x + x^2 - x^3 + \dots + (-1)^{2n-1}x^{2n-1} = \frac{1-x^{2n}}{1+x} \\ 1 - x + x^2 - \dots + (-1)^{2n-2} x^{2n-2} = \frac{1 + x^{2n-1}}{1+x} \\ (1): \ \ \int_0^z 1 - x + x^2 - x^3 + \dots + (-1)^{2n-1}x^{2n-1} \ \mathrm{d}x = \int_0^z\frac{1}{1+x} \...

Re: HSC 2015 4U Marathon \\ $A solid has a circle of radius 1 as its base in a plane$ \ p \ $and every cross-section cut of the solid, perpendicular to the plane$ \ p $, is an equilateral triangle$ \\ \\ $Find the volume of the solid$

Re: MX2 2015 Integration Marathon Yea whatever it might be

Re: MX2 2015 Integration Marathon Integrate by parts, letting u = \sin \ln x to get I = \frac{x^2}{2} \sin \ln x - \frac{1}{2} \int x \cos \ln x \ \mathrm{d}x Integrate by parts again, letting u = \cos \ln x I = \frac{x^2}{2} \sin \ln x - \frac{1}{2} \left(\frac{x^2}{2} \cos \ln x -...

Do you guys want to continue with the server? There has been little to no activity in the past couple weeks

Re: HSC 2015 4U Marathon \\ $Consider the sequence of polynomials defined by$ \\ \\ 1) \ C_0(x) = 1 \\ 2) \ C_1(x) = x \\ 3) \ C_n(x) = 2xC_{n-1}(x) - C_{n-2}(x) , \ n \geq 2 \\ $i) Show that$ \ C_n(\cos \theta) = \cos n\theta \\ $ii) Let$ \ f_n(x,t) = t^n C_n(x) $. Show that$ \ f_n(x,t) =...

Re: HSC 2015 4U Marathon Just an exercise in mathematical/logical argument and proof: \\ $You may assume that the addition of any 2 positive integers is a positive integer$ \\ $i) Prove that the subtraction between any 2 positive integers is not always a positive integer$ \\ $ii) Prove...

Re: MX2 2015 Integration Marathon Something like a (-1)^{n-1} or (-1)^n at the front but other than that, that looks correct