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Re: HSC 2013 2U Marathon \int \frac{dx}{x\ln x}

Re: HSC 2013 3U Marathon Thread Something like that, nice work.

Re: HSC 2013 4U Marathon This is a little easier: $Prove$ \sum_{k=0}^{n} \binom{n}{k} \cos(k \theta) = 2^n \cos^n \left(\frac{\theta}{2} \right ) \cos \left(\frac{n\theta}{2} \right) \ \ \ \ \fbox{3}

Yep I remember he posted a thread for his channel some time ago, feel free to bump it if you can find it.

Re: HSC 2013 4U Marathon $Prove that$ \prod_{k=1}^{n} \cos \left(\frac{\pi(2k-1)}{4n} \right ) = \frac{\sqrt{2}}{2^n} \ \ \ \ \ \fbox{6} (please inform me if there is a mistake)

Yeah that's true, it seems to be the rules to follow when we are trying to symbolise what the author of an equation is trying to mean. And because we write from left to right, I guess that's why we only take multiplication and division from left to right?

Yes because BODMAS for multiplication and division should always go from left to right. But the real question is. Why? How can you 'prove' BODMAS?

http://www.youtube.com/user/Letslearnscience I just wanted to give this great guy a 'bump' for his great videos on Chemistry, Physics and Biology as a sort of thank you. He has comprehensive videos on specific syllabus points. It really helped me in Chemistry short answer questions, and...

Yes you have the tools, you aren't supposed to know HOW to do everything, some questions should challenge people in their critical thinking skills. It makes absolute sense. There maybe should be some questions as free marks to 'warm' people up. IQ tests are skewed since you can easily...

Example of a 'critical thinking question' (albeit quite simple) $Use trapezoidal rule to estimate$ \ \ ln 2 \ \ $to 3 decimal places$ If you rote learn and don't know how to critical think, you think 'oh derp we didnt learn this at school, I guess I'll just have to skip it because I don't...

There are 6 ways to put the dice face down into the box, either the number 1, 2, 3, 4, 5 or 6 touches the bottom. For each time we do this, there are 4 different orientations that can be done via rotating. Therefore, 6*4=24

Its asking for unique cubes, so lets say we have our cube where red is top and white is bottom. Another arrangement (according to your solution) is white on top red on bottom, but they are the same cube. First lets arrange the top and bottom faces of the cube. We pick 2 colours from the...

Re: HSC 2013 3U Marathon Thread Yep nice work. ==== $Find without integrating$ \int_b^a \sqrt{a^2-x^2} \ dx a>b>0

Re: HSC 2013 3U Marathon Thread $i) Show that$ \ \ (n+1)! - n! = n \cdot n! $ii) Hence prove that$ \ \ \sum_{k=1}^{n} k \cdot k! = (n+1)! - 1 $iii) Similarly, find an expression for$ \sum_{k=1}^{n} k! (k^2 + k + 1)

Re: HSC 2013 4U Marathon Yep you have lol. ====== $Let$ \ \ H_n = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} $i) Prove that$ H_n = \int_0^1 \frac{x^n - 1}{x - 1} \ dx $ii) Hence prove that$ H_n = \sum_{k=1}^{n} \frac{(-1)^{k-1}}{k} \binom{n}{k}

Ah yes, I meant that there should also be some sort of wind I guess, not resistance. And if thats the case, say wind is 5m/s in positive z, then \dot{z} = 5 right?

Ok then, I set myself the problem: 'A projectile is fired with velocity V at an angle theta to the x-axis. Air resistance is proportional to the velocity of the projectile. Find the distance of the point of projection to the point of landing.' So, the only forces for the y-component would be...

But why can't z in terms of t, x in terms of t and y in terms of t not be asked? We aren't dealing with multi-variable functions if we just consider the parametric forms and deal with that only? My motivation for doing 3D is because I want to model air resistance not only parallel to the x and y...

Oh ok, do you know why 3D Projectiles aren't in the syllabus? From what I am experimenting with, it doesn't to get more complex than simply deriving \ddot{z} along with \ddot{x} and \ddot{y} Or is there something more deeper involved?

Is is possible for the BOS to ask non-uniform circular motion? I can derive the equation of acceleration quite easy and I was wondering if they hold the ability to ask it since I haven't really seen any questions with non-uniform motion. Also is 3D projectile motion in the syllabus? I...