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Re: HSC 2015 4U Marathon $ Using properties of Complex Numbers and standard Geometry, the locus is the major arc of a circle with radius $\frac{4}{\sqrt3}$ and centre $(0, \frac{2}{\sqrt3})$. Area of the entire circle is therefore $\frac{16\pi}{3}$. \\ NOTE that the points $(\pm2,0)$ are not...

Re: HSC 2015 4U Marathon NEW QUESTION $ Show that the remainder when the polynomial $P(x)$ is divided by $x^2-a^2$ is $\frac{1}{2a}[P(a)-P(-a)]x+\frac{1}{2}[P(a)+P(-a)]$. \\ Find the remainder when $P(x)=x^n-a^n$ is divided by $x^2-a^2$ for non-negative values of $n$.

Re: HSC 2015 4U Marathon Yeah sorry, the RHS should be 'n' not '1', just fixed it, thanks Frank.

Re: HSC 2015 4U Marathon $ Using squares of the sum and products of roots we can find expressions for the coefficients of our new polynomial to get $x^6+2x^4-3x^2-9$. $ NEXT QUESTION $ If $\alpha$ is a non-real $n$th root of 1, show that $(1-\alpha)(1-\alpha^2)...(1-\alpha^{n-1})=n$ $.

Re: HSC 2015 4U Marathon $ We can see that $z=0$ is a root to the polynomial, and since the coefficients of $P(z)$ are real, then by the Complex conjugate root theorem, either two cases are possible for the remaining $5$ roots: \\ 1. Two pairs of complex conjugate roots + 1 real root \\ 2...

Have you tried the UNSW Subject Reviews thread? There is lots of info there and you may come across some reviews for the subjects you listed. Should give you a brief idea on what each course is like in terms of difficulty, lecturers, tutors, content and assessments etc.

Re: HSC 2015 4U Marathon $ Hmm, I am not able to see how the lower bound is $-\frac{\pi}{4}$ since the line $y=-x$ is not tangent to the locus I got to be as $(x-\frac{3}{2})^2+(y-1)^2=\frac{9}{4}$. Upper bound for $\arg(z)$ should be $\frac{\pi}{2}.$ Please correct me if I am wrong on...

Re: UNSW Chit Chat Thread 2015. Yep, all part of the course pack, it will save your life :)

Re: UNSW Chit Chat Thread 2015. Both Kress and Dick are decent lecturers, but Kress' lecture notes are better imo. I don't know which lecturer will be running which stream (Alg or Calc) but last year Kress ran the Calculus stream, and Dick ran the Calc stream for MATH1251 in SEM2 so I don't...

Re: HSC 2015 4U Marathon Yep, the rest falls in place. :)

Re: HSC 2015 4U Marathon NEXT QUESTION: $ By considering $z=\cos\theta+i\sin\theta$, show that \\ $\sin\theta-\frac{\sin3\theta}{3^2}+\frac{\sin5\theta}{3^4}-\frac{\sin7\theta}{3^6}+...=\frac{72\sin\theta}{82+18\cos2\theta}

Re: HSC 2015 4U Marathon $ Consider the polynomial $P(x)=x^3+x+12$, then $P'(x)=3x^2+1>0$ for all real $x$, thus the function is always increasing along the reals and so there only exists one real root to $P(x)=0$. Since the coefficients of $P(x)$ are real, then the other two roots are...

Re: HSC 2015 4U Marathon $ Need Frank to confirm this, but I got the equation of the tangent to be $y=12x-9$. Idea is that you need to realise that since there exists two distinct points at which the tangent touches the curve, there exists two double roots for the equation $f(x)-mx-b=0$ where...

Re: HSC 2015 4U Marathon $ If we want to solve the polynomial equation $z^{5}-\(1+\sqrt{2}i)^3=0$, then our roots will satisfy the equation $(z-\alpha)(z-\beta)(z-\gamma)(z-\delta)(z-\epsilon)=0$, where $\alpha, \beta, \gamma, \delta, \epsilon$ are the fifth roots of $\(1+\sqrt{2}i)^3$. From...

Re: HSC 2015 4U Marathon NEW QUESTION: $ Show that if for any complex number $z$ with $|z|=1$, then Re\left(\frac{1-z}{1+z} \right)=0

Re: HSC 2015 4U Marathon Yep, pretty much the right idea, but I think you mixed up some plus signs with minus signs. Answer in simplest form is \! $ $z=\frac{3}{2}\left[\sqrt{2+\sqrt2}+i\sqrt{2-\sqrt2} \right]$ $ .

Re: HSC 2015 4U Marathon NEW QUESTION: $ Find the Cartesian form of the complex number whose modulus and argument is $3$ and $\frac{\pi}{8}$ respectively.

Re: HSC 2015 3U Marathon Alternatively, you could try substituting $ $x=\tan\frac{\theta}{2}$ $ and use the half-angle formula to make the differentiation easier to do. Should yield the same answer though :)

Re: HSC 2015 4U Marathon Don't have time to completely do it now, but set $ $x=\sqrt{3}\cos\theta$ $ and use the given identity to find the solutions to the equation.

Re: HSC 2015 4U Marathon $ $ y=\pm{2x}$ $