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Re: HSC 2015 4U Marathon NEXT QUESTION $ Find the equation(s) of the tangent(s) drawn from the origin to the curve $ y=3-\frac{1}{x^2}.

y=1+\frac{9}{x^2-9}, $ the denominator $ x^2-9\geq-9, $ hence the fraction $ \frac{9}{x^2-9}\leq-1 $ (when $ x^2-9<0 $) or $ \frac{9}{x^2-9}>0 $ (when $ x^2-9>0 $). therefore the range is $ y\leq0 $ or $ y>1. but you can actually first sketch the graph using asymptotes and turning points, then...

Re: HSC 2015 4U Marathon NEXT QUESTION $ Find the values of $ k $ for which the equation $ 4x^3+4x^2+x+k=0 $ has more than one real root. $

Re: HSC 2015 3U Marathon incorrect. your calculation is true for the volume of solid generated by rotating that region about x axis, but not y axis.

Re: HSC 2015 3U Marathon NEXT QUESTION $ The region enclosed between $ y=2x-x^2 $ and the $ x $ axis is rotated about the $ y $ axis. Find the volume of the solid formed. $

Re: HSC 2015 2U Marathon To my best understanding, it doesn't matter who tries or solves the question. What matters is everyboday lets the ball rolling, ie, when one question is surely solved, post a new question! The students still have chance to try the question that is already solved because...

Re: HSC 2015 2U Marathon ok then this is a new question $ Evaluate $ \int_{-1}^1\ln\left(x+\sqrt{x^2+1}\right)dx $ using properties of integral. $

Re: HSC 2015 2U Marathon $ It is a GP with $ a=k^r, 0< $ common ration $=k^{\frac1k}<1. $ So its limiting sum $=\frac{k^r}{1-k^{\frac1k}}.

Re: HSC 2015 2U Marathon $ do u mean $ k^r +k^\frac{rk+1}{k} +k^\frac{rk+2}{k}+\cdots

Re: HSC 2015 4U Marathon \cot^2\frac{\pi}{12}+\cot^2\frac{5\pi}{12}=14...

Re: HSC 2015 4U Marathon \sum\alpha=\alpha+\beta+\gamma+\delta=p\Rightarrow\alpha+\beta=\gamma+\delta=\frac{p}{2} \sum\alpha\beta=(\alpha+\beta)(\gamma+\delta) + \alpha \beta+\gamma \delta=q\Rightarrow\frac{p}{2}\cdot\frac{p}{2} + \alpha \beta + \gamma\delta=q\Rightarrow\alpha\beta +...

Re: HSC 2015 3U Marathon k{{n}\choose{k}}=\frac{k\cdot n!}{k!(n-k)!}=\frac{n\cdot(n-1)!}{(k-1)!(n-k)!}=n{{n-1}\choose{k-1}} \therefore\quad...

Re: HSC 2015 4U Marathon Hint: Use the triangle inequality about complex numbers and solve an inequation to get |z|<= something.

Re: HSC 2015 4U Marathon NEXT QUESTION $ Find the greatest value of $ |z| $ if the complex number $ z $ satisfies $ \left|z-\frac{4}{z}\right|=8.

Re: HSC 2015 4U Marathon I knew that, just want more people get involved. here is my go: z^2=a^2(\cos2\theta+i\sin2\theta), z^2+a^2=a^2[(1+\cos2\theta)+i\sin2\theta]=a^2(2\cos^2\theta+2i\sin\theta\cos\theta)=2a^2 \cos \theta (cos\theta+i\sin\theta), \therefore\quad\frac{z}{z^2+a^2}=\frac{a{\rm...

Re: HSC 2015 4U Marathon i think it is because you input "enter" in the "tex" environment. Here I post my question: $ The complex number $ z $ satisfies $ |z-1-5i|=|z+2-3i|. $ Find the minimum value of $ |z|.

Re: HSC 2015 3U Marathon but only binomial expansion is needed to solve this problem. I've had one method in mind. P.S. if you can prove it is true for 0<=p<=1, that actually means it is true for all real values of p. Because both LHS and RHS are polynomials of p of degree less than or equal...

Re: HSC 2015 3U Marathon NEXT QUESTION $ For what values of positive constant $ a, $ the coefficient of $ x^{13} $ in the expansion of $ (3+ax)^{20} $ is the greatest.

Re: HSC 2015 4U Marathon what about b=1,c=-2; or b=-2,c=4; or b=pi, c=-4/(pi+1), etc.:smile:

and even more unusual, the interest is computed monthly while the investment is made weekly. Note that 1month=30/7weeks, which is not a whole numbher.