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Re: HSC 2013 4U Marathon Here is a neat question I made, its a little hand-holdy I guess. $i) Consider the polynomial$ \ \ P(z) = z^m -1 $The roots of the polynomial is$ \ \ z_1, z_2, z_3, \dots , z_m $Find these roots$ \ \ \ \fbox{1} $ii) Let$ \ \ S_k = z_1^k + z_2^k + z_3^k + \dots +...

The statement is true for n=1 If the statement is true for n=k, then it is true for n=k+1 Therefore it is true for all integers n => 1 by Mathematical Induction

Re: HSC 2013 3U Marathon Thread Yep nice work. Also I should probably say x > 0 for that question.

Re: HSC 2013 4U Marathon $For some positive real values$ \ \ \{x_k \} $Prove that$ \frac{1}{n} \sum_{k=1}^{n} x_k \geq \left(\frac{1}{n} \sum_{k=1}^n x_k^{-1} \right)^{-1} $State the condition for equality$ (don't feel the obligation to answer the questions I'm posting in order, answer...

Re: HSC 2013 3U Marathon Thread $Prove that$ x^x \geq \left( \frac{1}{e} \right ) ^{\frac{1}{e}}

Re: HSC 2013 4U Marathon If you don't mind me expanding on this :P $ii) Using part (i), prove that$ e = \sum_{k=0}^{n} \frac{1}{k!} + \frac{(-1)^{n+1}}{n!}e \int_0^1 t^n e^{-t} \ dt \ \ \ \fbox{3} $Let$ J_n = \int_0^{1} t^n e^{-t} \ dt $iii) Show that$ 0 < J_n < \frac{1}{n+1} \ \ \...

Re: HSC 2013 4U Marathon Great question Consider the integral: I_k = \int_0^{x} \frac{t^{k-1}}{(k-1)!} f^{(k)} (x-t) \ dt Integrating by parts: u = f^{(k)} (x-t) du = -f^{(k+1)}(x-t) \ dt dv= \frac{t^{k-1}}{(k-1)!} dt v = \frac{t^k}{k!} I_{k} = \frac{t^k}{k!}...

Ok some space cleared, you can send now.

I regret wasting my time in Physics at all which is why I finally dropped it. I learnt more Physics from youtube than I did from this horrible HSC course. Main reason I dropped it is due to marks though :P

This is a little out of syllabus, but it will help you understand BCS theory I think. What are known as electron-cooper pairs are formed because as the electrons move through the lattice, the positive charges shift slighty towards the path of the electrons since positives are attracted to it...

Re: HSC 2013 4U Marathon $Consider for some positive real values$ \ \ a, \ \ b, \ \ k $Define the sequence$ \ \ \{S_n \} S_n = \frac{a}{kn+b} + \frac{a}{kn+2b} + \frac{a}{kn+3b} + \dots + \frac{a}{n(k+b)} $i) Prove that$ 0 < S_n < \frac{a}{k} \ \ \ \fbox{2} $ii) Prove that$...

Ah yep I see that, clever, thanks. \frac{(53)!}{50! \timess 3!} right? Because there are 53 elements to arrange, 3 dividers 50 tokens, and they are not distinct.

Thank you. I see what you mean by simplifying the problem down. For example, if we split 12 people into 3 groups of 4, the ways to do so is: \binom{12}{4} \binom{8}{4} \binom{4}{4} right? (I think) But what if 2 people cannot be in the same group? Would it be, first putting those people in...

So I have come to the realisation that the only thing weighing me down at all is Permutations and Combinations. And I want to perfect my ability to do the harder problems in Perms/Combs I find that for the harder problems, I usually end up not seeing something, end up double counting or not...

Re: HSC 2013 4U Marathon Ah ok I see. So, we can establish a fairly accurate bound for e: e < \frac{87}{32} using the inequality you made above. Then I guess we could use: \int_0^1 \frac{x^4(1-x)^4}{1+x^2} = \frac{22}{7} - \pi Since that function is always positive then: \pi <...

Re: HSC 2013 4U Marathon Interesting, is there a direct way of proving this without computing pi/4 whatsoever? I can find upper bounds for pi and e using primitive methods but they are too weak, and I end up getting something lower than 1/3 :/ I was able to get e < 3 using trapezium and...

Re: HSC 2013 4U Marathon a+b \geq 2\sqrt{ab} \therefore e^{1/e} + e^{1/\pi} \geq 2e^{\frac{1}{2e + \frac{1}{2\pi}}} Now, lets establish some approximations for e and pi. $We will assume that$ e = \sum_{k=0}^{\infty}\frac{1}{k!} Since this sum is increasing, therefore: e >...

Re: HSC 2013 4U Marathon Holy crap hahahaha you're right Wow I can't believe I went through all that and not notice that simple transformation...

Re: HSC 2013 4U Marathon Great question! Here is my solution: On the cartesian plane, sketch the curve y=ln(x). Then draw upper rectangles from x=k to x=1. We establish the inequality: \ln(k) + \ln(k-1) + \dots + \ln(1) > \int_1^k \ln x \dx = k\ln k - k + 1 \ln \frac{k!}{k^k} > 1- k...

Re: HSC 2013 4U Marathon Can someone please post a challenging inequality to prove? (preferably the ones involving a, b, c, d etc)