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Re: HSC 2015 4U Marathon Right, a more rigorous way of showing it would be to do this: \\ \sum_{k=0}^{n-1} |p - x_k|^2 = \sum_{k=0}^{n-1} (|p|^2 + |x_k|^2 - (p\bar{x_k} + \bar{p} x_k)) = \sum_{k=0}^{n-1} 2 - \sum_{k=0}^{n-1}(p\bar{x_k} + \bar{p} x_k) = 2n - p\sum_{k=0}^{n-1}\bar{x_k} -...

Re: HSC 2015 4U Marathon Yep that is correct, your explanation for why those extra terms cancel out is a little vague, but that's the general idea

Re: HSC 2015 4U Marathon \\ $Show that the curves$ \ x^2-y^2 = a^2 \ $and$ \ xy =c^2 \ $(for constants$ \ a,c $), intersect at right angles$

Re: HSC 2015 4U Marathon - Advanced Level Right, my bad

Re: HSC 2015 4U Marathon - Advanced Level \\ $Determine the integer solutions to$ \ x^2 - y^2 = p \ $for primes$ \ p \ $and positive integers$ \ x,y

Re: HSC 2015 4U Marathon - Advanced Level \\ $Show that$ \ y^2 = x^2 + x + 1 \ $has no integer solutions$

Re: HSC 2015 4U Marathon - Advanced Level Why is this a condition for k tails being the most probable outcome?

Re: HSC 2015 3U Marathon \\ $Prove through mathematical induction that$ \ n! \geq 2^n \ $for integers$ \ n \geq 4

Re: HSC 2015 4U Marathon - Advanced Level \\ $A biased coin is tossed$ \ n \ $times, and the probability that the coin lands tails is$ \ p $. The most probable outcome of the coin tosses is that tails comes up$ \ k \ $times. Find all possible values of$ \ p.

Re: HSC 2015 4U Marathon \\ $a) Show that for complex$ \ u,v \ $then$ \ |u+v|^2 = |u|^2 + |v|^2 + uv \left(\frac{\bar{u}}{u} + \frac{\bar{v}}{v} \right) \\ $b) Let the points$ \ X_0, X_1, X_2, \dots, X_{n-1} \ $represent the roots of$ \ z^n = 1 \ $on the Argand plane. Let$ \ P \ $represent...

Re: HSC 2015 4U Marathon Yes well done, for the other users who attempted this question (your attempt was deleted in a site crash yesterday), the key part I was looking for is what I bolded

Re: HSC 2015 4U Marathon \\ $Out of the set of shapes$ \ \{ $equilateral triangle$, \ $square$, \dots, \ $regular$ \ n$-gon$ \ , \dots \ $circle$ \} \\ \\ $All of which have identical perimeter, show that the circle has the largest area$

Re: HSC 2015 4U Marathon \\ $The number of real roots is equal to the number of non-negative roots multiplied by 2 and take away 1 (to avoid double counting zero)$ \\ 641 \pi < 2015 < 642\pi \\ $Therefore, there are no roots to the equation$ \ \frac{x}{2015} = \sin x \ $for$ \ x \geq...

Re: HSC 2015 4U Marathon \\ $i) Expand$ \ (4+i)(5+3i) \\ $ii) Hence show that$ \ \tan^{-1} \left( \frac{1}{4} \right) + \tan^{-1} \left(\frac{3}{5} \right) = \frac{\pi}{4}

Re: HSC 2015 4U Marathon \\ $Finding the area of the rhombus is the same as finding the four times the area of the triangle of the first quadrant$ \\\\ $The triangle is represented by the equation$ \ y = k - x\tan \beta \\ \\ $Where$ \ k \ $is a constant such that the line is tangent to the...

Re: HSC 2015 4U Marathon - Advanced Level I misinterpreted "beat", my bad

Re: HSC 2015 4U Marathon - Advanced Level \\ $Let the scores$ \ A \ $and$ \ B \ $be denoted by scores$ \ 1 \ $and$ \ 0 \ $respectively. So someone with$ \ 3 \ A $'s and$ \ 5 \ B $'s, means that they have a score of$ \ 1 + 1 + 1 + 5\cdot 0 = 3 \\ \\ $Therefore, for a student to beat another...

I didn't say that you clown You're as dumb as the muslims who said the first statement (so pretty dumb)

“It was execution style, a bullet in every head,” Abu-Salha said Wednesday morning. “This was not a dispute over a parking space; this was a hate crime.This man had picked on my daughter and her husband a couple of times before, and he talked with them with his gun in his belt. And they were...

Re: HSC 2015 4U Marathon \\ $Rigorously explain why polynomials with real co-efficients with an odd degree have at least one real root$