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Re: HSC 2015 3U Marathon well, another way is applying \binom{m}{l}=\binom{m-1}{l-1}+\binom{m-1}{l} (which is a well-known result in any textbook) again and again...

Re: HSC 2015 4U Marathon - Advanced Level 10\sqrt{3}

Re: HSC 2015 4U Marathon - Advanced Level next question f(x) $ is an even function defined for all real values of $ x. $ The graph of $ y=f(x) $ is symmetrical about the line $ x=1. $ For every $ 0\leq x_1, x_2\leq\frac{1}{2}, $ there holds $ f(x_1+x_2)=f(x_1)\cdot f(x_2). $ Let $...

Re: HSC 2015 4U Marathon next one $ Find the locus of all points through which two tangents drawn to the ellipse $ \frac{x^2}{16}+\frac{y^2}{9}=1 $ are perpendicular to each other. $

Re: HSC 2015 4U Marathon new question $ The equations of two ellipses are (i) $ 4x^2+9y^2=36 $, (ii) $ 2x^2+3y^2=30 $. The tangent to ellipse (i) at the point $ R(3\cos\theta,2\sin\theta) $ meets the ellipse (ii) at the points $ P $ and $ Q $. Show that the tangents at $ P $ and $ Q $ to...

Re: HSC 2015 4U Marathon Let the incircles of triangle ADB and ADC both touch AD at the point E. (given) Then DE=(AD+BD-AB)/2, DE=(AD+CD-AC)/2. (use "two tangents through an external point are equal" several times) Therefore, AB-BD=AC-CD. (manipulate simple algebra on the above equations) This...

Re: HSC 2015 3U Marathon i don't know what were you doing, but did you try quotient rule because y is a quotient of two functions

Re: HSC 2015 3U Marathon the question was asking you prove the derivative of sine is cosine, but you actually admit that the derivative of cosine is sine when you use l'hospital rule. this is obviously not allowed. a method inside the scope of high school is: using double angle 1-\cos...

there is no region bounded by these two parabolas, the correct way to refer the area is "enclosede by parabolas ... and ... and the x axis" the solution is easy, split the region into two parts: 0 to 1, and 1 to 2. integrate respectively and add up.

oh yes the second minus should be plus

thats good, but can you simplify a little bit, because you are going to solve y''=0

no. i meant your y''=?

what have you got for the second derivative?

case (i) three letters all different:7p3 case (ii) two alike one different: 2c1 times 6c1 times 3!/2! then sum up

Re: HSC 2015 4U Marathon - Advanced Level ok here is how i modify the question: $ Given that $ n $ is an even integer greater than $ 0. $ If $ \sum_{k=0}^{\frac{n}{2}}{{n}\choose{k}}\cos(n-2k)\theta=1 $ is true for some values of $ \theta, $ show that $ n=2 $ and find the values of $ \theta.

Re: HSC 2015 4U Marathon - Advanced Level this does not make sense to me, whatever your answer is, say n=2, the equation can't be true for random c.

Re: HSC 2015 4U Marathon - Advanced Level i can answer your question \sum_{k=0}^{\frac{n}{2}}{{n}\choose{k}}\cos(n-2k)\theta but im extremely confused about how theta and c are random

Re: HSC 2015 4U Marathon - Advanced Level find n in terms of c and theta then ?

Re: HSC 2015 4U Marathon - Advanced Level just clarify: if the last term is +c, what is c? and what is \theta? do u mean finding n given that equation is true for all \theta? or do u mean finding both n and \theta ?

Re: HSC 2015 4U Marathon new question $ Find the number of real roots of the equation $ x-2015\sin x=0.