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Re: HSC 2015 4U Marathon That wasn't what I did but that may work

Re: HSC 2015 4U Marathon \\ $Prove that any number of the form$ \ 4n + 2 \ $for positive integers$ \ n \ $cannot be written as the difference of two square numbers$

Re: HSC 2015 4U Marathon Try proving via contradiction

Re: HSC 2015 4U Marathon bump A translation of part (ii) is simply "show that no two members of the sequence F_n share a common factor"

Re: HSC 2015 4U Marathon \\ $i) Let$ \ z = x + A \ $for some constant$ \ A \ $so that$ \ (x+A)^3 + a(x+A)^2 + b(x+A) + c = 0 \\ $The coefficient of$ \ x^2 \ $is zero$ \\ \therefore 3x^2A + ax^2 = 0 \Rightarrow A = \frac{-a}{3} \\ \therefore \ z = x - \frac{a}{3} \ $is a sufficient...

Re: HSC 2015 3U Marathon Use the angle sum of a triangle being 180 We can see that in fact we know 2 angles, and these 2 angles are part of the same triangle, meaning .....

Re: HSC 2015 3U Marathon The first helpful thing to notice, is that the range of the cosine inverse function is between 0 and \pi. This means we can't do the old method of cancelling out the inverse cosine and the cosine because this would give us \pi + \alpha which since \alpha is acute...

Re: HSC 2015 4U Marathon \\ $Let$ \ F_n = 2^{2^n} + 1 \\ \\ $i) Show by mathematical induction that$ \ F_0F_1\dots F_{n-1} = F_n - 2 \ $for$ \ n \geq 1 \\ \\ $ii) Hence show that there is no$ \ F_i \ $and$ \ F_j \ $with$ \ i <j \ $such that there is an integer that divides into both$ \ F_i \...

Re: MX2 2015 Integration Marathon btw is there a non-strict lower bound for your question? It seems to me (at least on the positive integers) that the expression is monotone decreasing tending towards a limit, but then there is no non-strict lower bound

Re: HSC 2015 4U Marathon \\ $i) Show that if$ \ I_n = \int_0^{\pi /2} \sin^{2n}x \ dx \ $then$ \ \frac{I_{n}}{I_{n-1}} = \frac{2n-1}{2n} \\ $ii) Hence show that$ \ I_n = \frac{\pi}{2} \frac{1}{4^n} \binom{2n}{n} \\ $iii) Find$ \ \sum_{n=0}^{\infty} \frac{1}{16^n} \binom{2n}{n}

Re: HSC 2015 4U Marathon \\ $Consider the equation$ \ z^3 + az^2 + bz +c = 0 \ (*) \\ \\ $i) Find an appropriate substitution of the form$ \ z = x + A \ $that transforms the equation $ \ (*) \ $into one of the form$ \ x^3 + px + q = 0 \ (**) \\ \\ $ii) Find an appropriate substitution of the...

Re: HSC 2015 4U Marathon Probably 2

Re: MX2 2015 Integration Marathon \\ \ $Find in terms of$ \ n \ \int_0^{\pi} \frac{x\sin x}{(n-1)(n+1) + \sin^2x} \ dx

Re: HSC 2015 4U Marathon \\ $Prove that$ \ \frac{1}{6} n(n+1)(n+2) \ $is an integer for integers$ \ n \geq 1 Try to prove it without induction (try to generalize the above method of using evens/odds, or go for a purely algebraic approach)

Re: HSC 2015 4U Marathon These are very similar proofs just worded differently. For future reference a more rigorous way of writing it would be (I'm not sure how nitpicky they would be in the HSC, its good to be safe though): (2n+1)^2 - 1 = 4n(n+1) = 8 \left(\frac{n(n+1)}{2} \right) \\...

Re: HSC 2015 4U Marathon Also try to do so without induction (though in an exam situation they probably wouldn't force you to think of a non-inductive solution)

Re: HSC 2015 4U Marathon \\ $Show that if the polynomial$ \ x^3 + px + q = 0 \ $has exactly one real root then$ \ 4p^3 + 27q^2 > 0

Re: HSC 2015 4U Marathon Is that a typo? The number would be of the form (2n+1)^2 -1

Re: MX2 2015 Integration Marathon \int_0^1 \frac{\ln(x+1)}{\ln(2+x-x^2)} \ dx

Re: MX2 2015 Integration Marathon C_2 = e by approximating \int_1^n \ln x with n trapeziums each of width 1, I'll post a proper proof later and for the lower bound