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are beads of same colour identical ?

Re: HSC 2015 4U Marathon next question $ The hyperbola $ xy=c^2 $ meets the ellipse $ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $ at $ P\left(ct_1,\frac{c}{t_1}\right) $ and $ Q\left(ct_2,\frac{c}{t_2}\right) $ where $ t_1>t_2>0. $ Tangents to the hyperbola at $ P $ and $ Q $ meet in $ T, $ while...

Re: HSC 2015 4U Marathon when the speed is q, no side force, we have N_q\cos\phi=mg, N_q\sin\phi=mq^2/r which yields gr=\frac{q^2}{\tan\phi} ....(1) when the speed is p>q, the side force F is downwards, we have N_p\cos\phi-F\sin\phi=mg, N_p\sin\phi+F\cos\phi=mp^2/r which (by dividing...

Re: HSC 2015 4U Marathon adding up all five equations we obtain 6(x_1+x_2+x_3+x_4+x_5)=6(1+2+4+8+16) or x_1+x_2+x_3+x_4+x_5=31 then subtracting this equation from each of the original equations respectively

8\pi/3 the bigger solid minus the smaller solid

this is one such kind of question $ Find the volume obtained when the region bounded by the parabola $ y=2x-x^2 $ and the x axis is rotated about the y axis. $

of course one CAN use t=\tan x because \cos^2x is essentially \cos2x . don't tell me 3U students do not need to know \cos^2x=\frac{1+\cos2x}{2} :D

Re: HSC 2015 4U Marathon yes of course, because the semicircle and the smaller circle are also tangent(internally), the centres and the point of contact are colinear. isn't this a theorem in circle geometry?

Re: HSC 2015 4U Marathon $ let the bigger circle have radius $ x $ and the smaller circle have radius $ y. $ so the semicircle has radius $ 2x. $ Now we can establish the equation by Pythagoras $ (x+y)^2-(x-y)^2=(2x-y)^2-y^2. by finding the relationship between x and y, and using the given...

depends t=\tan x or t=\tan\frac{x}{2}

Re: HSC 2015 2U Marathon lol if this question appeared in 2U exam, the diagram would have been given. Ekman, you got it right.

$ let $ u=3^x. u^3-13u^2+39u-27=0, (u^3-27)-(13u^2-39u)=0, (u-3)(u^2+3u+9)-13u(u-3)=0, (u-3)(u^2+3u+9-13u)=0 (u-3)(u^2-10u+9)=0, (u-3)(u-1)(u-9)=0 u=3, 1, 9. $ so $ x=1,0,2.

Re: HSC 2015 2U Marathon why the plus 2 at the end?

Re: HSC 2015 2U Marathon correctly represented by the integration. but this one different

Re: HSC 2015 3U Marathon the integral was correct anyway, not sure about the answer, the evaluation of integral looks wrong

Re: HSC 2015 2U Marathon sorry but all tries are wrong. you do need to draw a diagram

Re: HSC 2015 2U Marathon hint:1. draw a diagram 2. with area no \pi

the corrected one was already there in the bottom in my last post

question 2, the first three terms shown were wrong. was this question from a book? trash it into a bin. 5+13+21+dots+(8n-3)=n(4n+1)

Re: HSC 2015 4U Marathon new question $ Find the limit $ \lim_{n\rightarrow\infty}\frac{2^{\frac{1}{n}}+2^{\frac{2}{n}}+2^{\frac{3}{n}}+\cdots+2^{\frac{n}{n}}}{n}.