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Re: HSC 2014 4U Marathon \\ $Using complex numbers, show that the angle subtended by a semi-circle is a right angle$

The monotheistic faiths claim to be a source of theological knowledge, religion tells you what is right or wrong, religion tells you that you are created by a creator, who is wholy perfect and wholy powerful. Religion lays out a path towards the divine, religion gives us purpose to fulfill

It's pretty obvious that He does exist

Re: HSC 2014 4U Marathon - Advanced Level Oh, right, so it would be: f(x) = \displaystyle \begin{cases}\frac{\ln x}{x-1} , \ x > 0 | x \neq 1 \\ 1, x=1 \end{cases}

Re: HSC 2014 4U Marathon - Advanced Level \\ \frac{ \text{d}((x-1)f(x)) }{\text{d}x} = \frac{1}{x} \\ \Rightarrow \ (x-1)f(x) = \int \frac{\text{d}x}{x} \\ \Rightarrow \ f(x) = \frac{\ln (Ax)}{x-1} \ $for some constant$ \ A > 0

Re: MX2 Integration Marathon \\ $Letting$ \ I_{n} = \int_0^{\frac{\pi}{2}} \frac{\sin((2n)x)}{\sin x} \ $d$x \ $where$ \ n \in \mathbb{N} \\ $Find in series form$ \ J_n = \int_0^{\frac{\pi}{4}} \tan^{2n}x \ $d$x \ $where$ \ n \in \mathbb{N} \\ $Hence prove$ \ \sum_{n=0}^{\infty}...

Thanks for taking the time to answer I don't really understand much of what you're saying though, at what level do you learn about this?

Re: HSC 2014 2U Marathon lol ty

So, just for fun I wanted to investigate the properties of this function: f(x) = \int_0^{\frac{\pi}{2}} \frac{\sin (xt)}{\sin t} \ $d$t , \ $and$ \ x \in \mathbb{R} Right now I know: \\ $Define$ \ I(N) = \int_0^{\frac{\pi}{2}} \frac{\sin ((2N)t)}{\sin t} \ $d$t , \ N \in \mathbb{N} \\...

Re: MX2 Integration Marathon \\ $Find$ \ \int_0^{\frac{\pi}{2}} \frac{\sin ((2n+1)x)}{\sin x} \ $d$x \ $where$ \ n \in \mathbb{N}

Re: HSC 2014 4U Marathon - Advanced Level (can someone post a good question please)

Re: HSC 2014 4U Marathon - Advanced Level Yep my bad

Re: HSC 2014 4U Marathon Yep pretty much what I was getting at

What makes a chaotic system chaotic? ex. Is there a specific reason *why* for example the limiting value of, x_{n+1} = rx_n(1+x_n) is chaotic at r > ~3.4?

Re: HSC 2014 2U Marathon TOPIC: Arithmetic and Geometric Series \\ $i) Show that$ \ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \\ $ii) Hence show that$ \ (k+1)^3 - k^3 = 3k^2 + 3k + 1 \\ $iii) Show that$ \ 1 + 2 + 3 + \dots + 100 = 5050 \ \ $(HINT: Arithmetic series)$ \\ $iv) Show that$ \ (2^3...

Re: MX2 Integration Marathon I would of substituted x = e^u, but pretty much the same length (still have to IBP twice)

That is an invalid proof, the triangles ADB and ABE are congruent if x = y, which is something you need to prove first If you can prove triangles ADB and ABE are congruent without using x = y (such as proving BD = AE), then you can conclude x = y which would be a valid proof.

Re: HSC 2013 3U Marathon Thread \\ $A sequence is defined as$ \\\\ a_0 = 1 \\ a_{n+1} = \frac{a_n}{a_n+1} \\ \\ $Prove by mathematical induction that$ \ \ a_n = \frac{1}{n+1} \ $for$ \ n \geq 0 \\ $EXTENSION (still within 3U knowledge), can you find another non-inductive method of finding$ \...

Re: HSC 2014 4U Marathon HINT: Try to prove the relationships for C_n(cos theta) instead of C_n(x)

Re: HSC 2014 4U Marathon That looks a little hand-wavy tbh (unless by "last bit" you mean those lines in your proof for part (i)) There is an easier way to do it though