Search results

Re: MX2 Integration Marathon A cool question my friend made $Prove$ \int_0^1 (e^{x^2} + e^{-x^2} ) \ dx > 2

Re: HSC 2013 4U Marathon You can, but it isn't necessary, however do whatever method you can find.

Re: HSC 2013 4U Marathon $A polynomial$ \ \ P(x) \ \ $has roots$ \ \ a_1, a_2, a_3, \dots , a_n $Find the polynomial with roots$ \ \ \pm a_1, \pm a_2, \pm a_3 , \dots , \pm a_n

Re: HSC 2013 4U Marathon $Prove for real$ \ x,y x^4+y^4+8 \geq 8xy

put the terms like a right angle triangle and add the columns: 1 x+x x^2+x^2 x^3+x^3+x^3 x^4+x^4+x^4+x^4 . . . $First column$ + $Second column$ + $Third column$ + ... \frac{1}{1-x} + \frac{x}{1-x} + \frac{x^2}{1-x} + \dots \frac{1}{1-x} \left( 1+ x + x^2+ x^3+ \dots \right)...

Re: HSC 2013 4U Marathon If by small angles you mean the limit \lim_{a\to 0} \frac{\sin (a)}{a} = 1 then yes you need to use this limit

Re: HSC 2013 4U Marathon $Carefully prove for positive real$ \ \ x \ \ $and integer$ \ n \lim_{n \to \infty} \frac{x^n}{n!} = 0

<bump>

It basically means that you are expect to be able to prove that if you rotate the curve x^2-y^2 = a^2 45 degrees you get the hyperbola xy= 1/2 a^2 So referring to different axis means to rotate it basically. The common proof of this is to take the parametric equations of the hyperbola (asec(t)...

To be honest it wasn't that bad, I thought I'd get tired with extension 2 english due in the break of the 2 sets of trials, but it all turned out fine. I should of maybe spent more time on Chemistry but its fine I guess. The second set of trials for maths was harder than the first, english was...

Re: HSC 2013 4U Marathon $i) Prove$ \ \ \frac{\sin x}{x} = \cos \left( \frac{x}{2} \right ) \cdot \cos \left( \frac{x}{4} \right) \cdot \cos \left( \frac{x}{8} \right ) \cdot \cos \left( \frac{x}{16} \right ) \cdot \dots $ii) Hence prove that$ \pi = 2\cdot \frac{2}{\sqrt{2}} \cdot...

Re: HSC 2013 4U Marathon Yeah I did have a quicker method than that. Post an outline of your solution if you want anyway, it doesn't need to be detailed.

Re: HSC 2013 4U Marathon Yes that actually works, quite clever. (assuming you mean m when you said (1-x)^m / (1-x) = (1-x)^(m-1)) My way was: Equating co-efficient of x^n in: x^n (1-x)^m + x^{n-1} (1-x)^m + \dots + (1-x)^m = \frac{1-x^{n+1}}{1-x} (1-x)^{m} = (1-x)^{m-1} - x^{n+1} (1-x)^{m-1}

Re: HSC 2013 4U Marathon Nice solution, I think my one might be yours 'in disguise' since it uses a similar concept My one was: First conclude that at least 1 of a,b,c needs to be > 1 if it satisfies abc=1. And at least 1 of a,b,c < 1. This leaves 2 cases, 2 of them > 1, and 1 of them < 1 or...

Re: HSC 2013 2U Marathon \frac{2^n - 1}{2^n} I'll leave it to you to explain why it is. i.e. I've turned the question into a 'Show that'

Re: HSC 2013 2U Marathon If n = say 10, then your probability becomes greater than 1 :P Just imagine n to be any other regular number, do it for 5 numbers first, then try to do it for any number n

Re: HSC 2013 2U Marathon you're answer should be dependent on 'n' =)

Re: HSC 2013 2U Marathon $A standard fair die is rolled$ \ n \ $times, the product of the scores is calculated$ $Show that the probability that this product is even is$ \frac{2^n-1}{2^n}

I tried to look at my notes where I 'solved' this question since I don't remember it, it turns out that in my 1am stupor I made a mistake in squaring (a/b + b/c + c/a)^2. Which led me to a cool 'solution' which had a substitution. But yeah its wrong lol, sorry about that.....

We have students who are trying to understand relativity yet they cannot even for the life of them calculate rate of motion in resisted medium due to the limitations of this course I'm not saying Chemistry is great, but rather they are on similar levels of rote learning and such. The syllabus is...