Search results

Re: HSC 2013 3U Marathon Thread $A parabola is a plane curve, whereby it is a locus of the points that are equidistant from a fixed line, the directrix, and a fixed focus$ $By considering these elements of a parabola and some variable point, rotate the parabola$ \ \ x^2=4ay \ \ $about the...

Re: HSC 2013 3U Marathon Thread $Simplify$ \binom{n}{0} \binom{n}{2} + \binom{n}{2} \binom{n}{4} + \binom{n}{4}\binom{n}{6} + \dots + \binom{n}{n-2} \binom{n}{n}

Re: HSC 2013 3U Marathon Thread For those who haven't done it yet, think integration. ===== $Prove that$ \ \ \tan^{-1}(x) + \tan^{-1} \left(\frac{1}{x} \right ) = \frac{\pi}{2}

Re: HSC 2013 4U Marathon P(x) \ $is a polynomial of a degree at least 2 such that$ \ \ P'(a) = 0 \ $. Show that when$ \ P(x) \ $is divided by$ \ (x-a)^2 \ $the remainder is$ \ \ P(a)

Re: MX2 Integration Marathon \int \sqrt{a+\sqrt{b+\sqrt{x}}} \ dx

Re: HSC 2013 4U Marathon Yep. I particularly like this question, it came from an old HSC paper: $If we can consider the Earth to be a sphere of uniform shape and density, then the gravitational acceleration towards the centre of the Earth above the Earth's surface is proportional to$ \ \...

Re: HSC 2013 4U Marathon The upper bound is correct, but d can be negative. Integration marathon

Re: HSC 2013 4U Marathon A whole day and no one answered it, its up to anyone by then. An alternative approach is simply letting it be f(x), finding f'(x) factorising x out and solving the quadratic. ====== $The lengths of a triangle are the first three terms in an arithmetic sequence with...

I don't want to make another thread for these small questions, so if a mod could please change the name into 'Small questions' Another one is: Why in the hydration of ethylene to produce ethanol, a hydrogen donor is needed? I can't see why this would be so, but its importance is mentioned...

I'm not very experienced with modulo and congruency, but using the 4 properties that I do know: a \equiv a \mod m $if$ \ a \equiv b \mod m \ \ b \equiv a \mod m \ \ \fbox{1} $if$ a \equiv c \mod m \ $and$ \ b \equiv d \mod m $then$ \ \ a+b \equiv c+ d \mod m \ \ , ab \equiv cd \mod m...

Oh ok, so its just common sense really :s

It is said that the fact that Glucose polymerizes and alternates the glucose monomers so that the Cellulose is a polymer of alternating sides of glucose, apparently because of this it makes the chain straight? Why is this so?

re: HSC Chemistry Marathon Archive $Explain the chemistry behind the addition polymerisation of polyethylene$

Re: HSC 2013 4U Marathon Using De'Moivers Theorem: \frac{\sin(2n+1)x}{\sin^{2n+1}x} = \binom{2n+1}{1} \cot^{2n} x - \binom{2n+1}{3} \cot^{2n-2}x + \dots + (-1)^n \sin(2n+1)x = 0 x= \frac{k\pi}{2n+1} z= \cot^2 x \binom{2n+1}{1} z^n - \binom{2n+1}{3} z^{n-1} + \dots + (-1)^{n} = 0...

I'm not sure about last year, I imagine there would be some considering its difficulty, but iirc people on this forum say that the last person to get 100% Raw was someone in 1995 or something.

Re: HSC 2013 3U Marathon Thread $Prove that for any triangle, the line joining the midpoints of 2 sides of the triangle, is parallel to the third side. And is half its length$

(1): \frac{a-a}{m} = 0 = n \ \ n \in \mathbb{Z} (2): \frac{a-b}{m} = n \ \ n \in \mathbb{Z} \ \ \ \fbox{1} -1 \times \fbox{1} \Rightarrow \frac{b-a}{m} = -n = k \ \ k \in \mathbb{Z} \therefore \ b \equiv a \ \mod{m} \ \ $if$ \ \ a \equiv b \ \mod{m} n, k, r are all integers (3)...

Re: MX2 Integration Marathon $It is known that any integral of the form$ \ \ \int \frac{P(x)}{Q(x)} \ dx $For some polynomials$ \ \ P \ \ $and$ \ \ Q $The integral can be found$ $Prove that any integral of the form$ \ \ \int_0^{\pi } x f(\sin x) \ dx \ \ $can be evaluated$ $For some...

Re: MX2 Integration Marathon Well done The idea is correct, but there is a mistake when you substitute cos(k) back in, it is not: \cos (k) = 1 - \frac{u^2}{(a-b)} But rather the square root of that expression. An easier substitution is: t^2 = \frac{a-x}{x-b} Then re-arranging: x =...

Re: HSC 2013 3U Marathon Thread (Note I'm giving numbers here because the expression you get if it were all pro numerals would be very big and ugly) ==== Some engineers are in a laboratory in space, they are testing a new gravity simulation machine on a projectile, and assessing the effects...