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Re: MX2 2015 Integration Marathon NEXT QUESTION $ Evaluate $ \int_0^{\pi}\frac{x}{2+\sin x}dx.

Re: HSC 2015 4U Marathon NEXT QUESTION Prove that all the roots of the equation z^n\cos n\alpha+z^{n-1}\cos (n-1)\alpha+\cdots+z\cos\alpha=1 $ where $ \alpha $ is real, lie outside the circle $ |z|=\frac12.

Re: HSC 2015 4U Marathon Alternatively, denoting the centre of the circle be C, point of contact be P, then OP=2\sqrt2, CP=\sqrt2, \sin\angle{COP}=\frac{CP}{OP}=\frac12, \angle{COP}=\frac{\pi}{6}, $ therefore, arg$z=\frac{\pi}{4}-\frac{\pi}{6}=\frac{\pi}{12}

yes exactly the same, i wrote that way is to easily see what are semi-major and semi-minor axes of the ellipse.

$ when $ |z|=2, x^2+y^2=|z|^2=4, $ sub. this into $ u $ and $ v, u=x-\frac{x}{4}=\frac{3}{4}x, v=y+\frac{y}{4}=\frac{5}{4}y. $ Now from these solve for $ x $ and $ y, x=\frac43u, y=\frac45v. $ therefore, $ \left(\frac43u\right)^2+\left(\frac45v\right)^2=4, $ so the locus is $...

$ sub. $ x=2at $ and $ y=at^2 $ into $ y=mx+b. $ this leads to a quadratic equation about $ t, $ which should have two roots $ t=p $ and $ t=q. $ finally product of roots concludes the proof. $

Re: HSC 2015 3U Marathon no, the questions says the person continues to walk after he arrives at the point C, which means the person stops at the point on AC produced, but not between AC

Re: HSC 2015 3U Marathon $ Yes, there was a typo, should be $ h=\frac{\sqrt{2}+\sqrt{10}}{4}d

Re: HSC 2015 2U Marathon This is pretty much what I would do.

Re: MX2 2015 Integration Marathon $ both approaches are good, i would sustitute $ x=\sqrt{3}\tan u, $ and substitute $ u=\frac{\pi}{2}-v $ to integrate $ \int_{\frac\pi6}^{\frac\pi3}\ln\tan udu

Re: How to do very well in HSC Mathematics exams when your results don't reflect effo doing plenty of practice questions helps to consolidate formulas, concepts and ideas. but you need to summarise all by yourself, like once you study a new formula or a new approach, do you hightlight it and...

Re: MX2 2015 Integration Marathon lol a 100% magic done by substitution. anyone can find a way to play this magic: \int_1^3\frac{\ln x}{3+x^2}dx

Re: HSC 2015 2U Marathon sorry but it is 8pi/21. u must have used plus sign instead of minus sign between two pieces?

Re: HSC 2015 4U Marathon NEXT QUESTION $ Show that $ x^2+y^2+xy=4 $ represents an ellipse by rewriting the equation into the form $ d_S=e\times d_D, $ where $ d_S $ denotes the distance of $ P(x,y) $ from some fixed point S and $ d_D $ denotes the distance of $ P(x,y) $ from some fixed...

Re: HSC 2015 4U Marathon excellent!

Re: HSC 2015 4U Marathon part one is a well done. part two was a lot confusing: make sure you understand the definition of the argument of a complex number Note: both part one and two can be solved from a geometric point of view

Re: HSC 2015 4U Marathon - Advanced Level I still don't understand, if g(x) = sqrt(1 + f(x)^2) and f(nx)=g(nx), isn't it that f(nx)^2=1/2 identically?

Re: HSC 2015 4U Marathon - Advanced Level $ It follows from the limiting sum formula that $ F(z)=\frac{A}{1-\frac{z}{\alpha}}+\frac{B}{1-\frac{z}{\beta}}=A\sum_{k=0}^{\infty}\left(\frac{1}{\alpha}\right)^kz^k+B\sum_{k=0}^{\infty}\left( \frac {1}{\beta} \right)^kz^k=\sum_{k=0}^{\infty}\left\{A...

Re: HSC 2015 4U Marathon - Advanced Level It is just the limiting sum formula for a GP with common ratio absolute value less than 1.

Re: HSC 2015 4U Marathon $ Let $ z=\cos\alpha+i\sin\alpha, $ then from DeMoivre's theorem, $ z+z^{-1}=2\cos\alpha, z^3+z^{-3}=2\cos3\alpha, z^5+z^{-5}=2\cos5\alpha. $ From the Binomial Theorem, $ (z+z^{-1})^5=(z^5+z^{-5})+5(z^3+z^{-3})+10(z+z^{-1}), $ which gives $...