Search results

Re: HSC 2016 4U Marathon $\textbf{NEW QUESTION}$\\\\$The complex numbers w and z with respective arguments $\phi$ and $\theta$ are two distinct points on the Argand diagram where $\phi < \theta$. The area of the triangle formed by the origin and these two points is A.$\\\\$Show that...

Re: HSC 2016 4U Marathon \\$Let $z=r(\cos\theta + i\sin\theta)\\arg(r^3\cos 3\theta+ r^3 i\sin 3\theta)=\frac{\pi}{3}\\3\theta = \frac{\pi}{3}\\\theta = \frac{\pi}{9}, \frac{7\pi}{9}, \frac{-5\pi}{9}\\$Three rays originating from the origin, travelling in the three arguments above. The centre...

It still doesn't really make sense in my head as to why we divide by 5 instead of 6. When I imagine these problems I usually think of the amount of arrangements of people in a line and then think of how many ways these arrangements can be represented on a circle. Because each "straight line...

In how many ways can the host and hostess and 4 visitors be arranged around a round table if the host and hostess sit next to each other? My line of thinking: \\$Consider host and hostess as one group. There are 5 groups, and within the host and hostess, they can be arranged 2! ways.$\\5...

If someone could do a worked solution for this, I'd appreciate it. $If $|z|=r$ and $arg\: z=\theta$, show that $\frac{z}{z^2+r^2}$ is real and give its value.

Re: HSC 2016 4U Marathon Yeah you're right, nvm.

Re: HSC 2016 4U Marathon What?

Re: HSC 2016 4U Marathon arg(z) refers to the angle created by the line from the origin to the complex number z and the positive real axis on the Argand diagram. arg(z)\neq arg(z)+2\pi as adding 2\pi to one side fundamentally changes the value. Take for example...

Re: HSC 2016 4U Marathon I'm 90% sure this is wrong. If you could tell me what's wrong about it, that would be great. I can't even picture this locus in my head. It's a weird one. arg(z)-arg(z+i)=\frac{5\pi}{4}\\\tan^{-1}(\frac{y}{x})-\tan^{-1}(\frac{y+1}{x}) =...

My teacher said they won't really change the exams that much at all. I don't think it is anything to fear to worry about.

Nine people are to be seated at a thin rectangular table with 4 people on one side and five on the other. How many different ways can these nine people be seated if: Mary and John must sit on the same side Mary and John must sit on opposing sides

Re: HSC 2016 4U Marathon Why does replacing z with root z yield a quadratic with roots of \alpha^2 and \beta^2?

Re: HSC 2016 4U Marathon NEXT QUESTION for anyone in the 2016 cohort: $Use de Moivre's theorem to express $\cos{6\theta}$ as a polynomial in terms of \cos{\theta}. HINT IF NEEEDED (highlight): You will need to use the binomial expansion for a binomial of 6th degree.

Re: HSC 2016 4U Marathon Thank you for posting a question which is within the scope of this year's cohort. I enjoyed doing this one. $i) By polynomial division: $z^5-1=(z-1)(z^4+z^3+z^2+z+1)\\$The roots of $z^5-1$ are: $z=1, cis\frac{2\pi}{5}, cis\frac{-2\pi}{5}, cis\frac{4\pi}{5}...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread This is an imaginary number question, why is it in the year 11 3U thread? Anyway, this is how I'd work it out, and yes it does use the factor theorem...

Like on the same spot?

1. Both the ice skate and the satellite are undergoing a form of circular motion. This circular motion entails tangential momentum but also some form of a centripetal force causing a centripetal acceleration. This acceleration does not effect the magnitude of the object's velocity, but rather...

Re: HSC Chemistry Marathon 2016 Cobalt-60 is used in medicine to irradiate cancerous cells in radiotherapy. Cobalt-60 has a half life of about 5 years which means it can last for a substantial period of time before needing replacement delivering the dosages required. Furthermore, cobalt-60...

I'm having trouble with a question from a past paper, please help. A rocket is fired vertically but begins to turn towards the tangent of Earth's surface. At a height of 260m the rocket runs out of fuel and is travelling at a 65 degree angle to the horizontal and follows a parabolic...

Re: HSC 2016 MX2 Combinatorics Marathon Yeah, I changed it.