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Re: MX2 2015 Integration Marathon \\ $Find$ \ \int_0^1 x^r (\ln x)^n \ dx \\ \\ $Where$ \ r > 0 \ $and$ \ n \ $is a non-negative integer$ \\ \\ $You may assume that$ \ \lim_{x \to 0^+} x^r (\ln x)^n = 0

Re: MX2 2015 Integration Marathon \\ \lim_{\alpha \to \infty} \int_{-\alpha}^{\alpha} \frac{dx}{(x^2+a^2)(x^2+b^2)}

Re: HSC 2015 4U Marathon The way I gave the question made it HSC worthy, the first part guides you through, just use the ordinary logic you know from Volumes.

Re: HSC 2015 4U Marathon \\ $Consider the graphs$ \ y = x^n \ $and$ \ y = x \ $where$ \ n \ $is a positive integer greater than 1$ \\ \\ $i) Consider the point$ \ (t,t^n) \ $which lies on$ \ y = x^n \ $and$ \ 0 \leq t \leq 1 $, find a function of$ \ t \ $that represents the distance from this...

Re: HSC 2015 4U Marathon \\ x_n^2 = 9x_{n-1}^2 + 4x_{n-2}^2 - 12x_{n-1}x_{n-2} \ (*) \\ x_{n-1} = 3x_{n-2} - 2x_{n-3} \Rightarrow 2x_{n-3} = 3x_{n-2} - x_{n-1} \\ \Rightarrow 4x_{n-3}^2 = 9x_{n-2}^2 - 6x_{n-2}x_{n-1} + x_{n-1}^2 \\ \Rightarrow 6x_{n-1}x_{n-2} = x_{n-1}^2 + 9x_{n-2}^2 -...

Re: HSC 2015 4U Marathon \\ $i) Assume that$ \ n! + 1 = kP \ $where$ \ k \ $is some integer between$ \ n \ $and$ \ 2 \ $inclusive$ \\ \\ $So,$ \ 1 = kP - n! = k(P-M) \ $where$ \ M \ $is an integer because$ \ n! \ $contains$ \ k \ $as a factor$ \\ \\ $However this is a clear contradiction...

Re: HSC 2015 4U Marathon yes, but the equation must be only in terms of y_n, and the terms must be linear (so no sqrt(y_n), y_n^2 etc.)

Re: HSC 2015 4U Marathon \\ $Let the recurrence relation for the sequence$ \ x_n \ $be$ \ x_n = 3x_{n-1} - 2x_{n-2} , \ (n \geq 2) \ $with$ \ x_0 = 0 \ $and$ \ x_1 = 1 \\ \\ $Find a recurrence relation for the sequence$ \ y_n = x_n^2 \ $involving only linear forms of$ \ y_n, y_{n-1}, $etc.$

Re: HSC 2015 4U Marathon A prime number is a positive integer greater than 1 that is divisible by only 1 and itself. i) Show that (n! + 1) is not divisible by any number from 2 to n inclusive ii) Hence show that the number of primes is infinite

Re: HSC 2015 4U Marathon That is sufficient to prove that if the radius and the one of the axes of the ellipse were equal, then they are tangent to each other. However the question is asking, given that they are tangent, that therefore the radius and one of the axes are equal. So while the...

Re: HSC 2015 4U Marathon \\ $Prove that the only way an ellipse and a circle with the same centre, can be tangent to each other at their intersection, is if the radius of the circle is the same in length as the semi-minor or semi-major axis of the ellipse$

Re: HSC 2015 4U Marathon - Advanced Level \\ $Prove that if$ \ \alpha, \beta , \gamma\ $are roots of a monic cubic polynomial, then these roots are real if and only if$ \ (\alpha - \beta)^2(\beta - \gamma)^2(\gamma - \alpha)^2 > 0

Re: MX2 2015 Integration Marathon Yes you got it

Re: MX2 2015 Integration Marathon The expression you get is in terms of m, and n, but the 'm' is a constant. The recurrence function should only be single variable

Re: MX2 2015 Integration Marathon \\ $Let$ \ I_n = \int_0^1 x^m (1-x)^n \ dx \\ \\ $Find a recurrence relation, where the relation is itself$ \\ $only in terms of$ \ n \ $(so no$ \ I(m,n))

The Gold/EXP Zombie Pigman Grinder is complete I followed the following structure and video almost precisely: At the start he shows how the pigman farm is used. To get to the farm, one has to enter the portal (the one that leads to the nether Fortress), behind the portal there is a Diorite...

Re: MX2 2015 Integration Marathon \\ $i) Explain why$ \ \int_0^a f(x) \ dx = \int_0^a f(a-x) \ dx \\ \\ $ii) Hence find$ \ \int_0^{\pi/4} \ln (1 + \tan x) \ dx

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I have decided that I cannot be bothered to turn this Skeleton Dungeon into a grinder, so if anyone wants to have a go at it, go right ahead, I will just continue with the Zombie Pigman Farm project