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Re: HSC 2015 4U Marathon \sqrt{17} ?

Re: HSC 2015 4U Marathon Probably people are looking forward to your method, I can't see other reasons why this one has been outstanding so long

yes. last-first+1=3n-n+1=2n+1

u=\sqrt{t+1}, t=u^2-1, dt=2udu, t=0, u=1; t=3, u=2, \int_0^3\frac{t}{\sqrt{t+1}}dt=\int_1^2\frac{u^2-1}{u}2udu=2\int_1^2(u^2-1)du=2\left(\frac{u^3}{3}-u\right)_1^2=\frac{8}{3}.

Re: HSC 2015 3U Marathon answer is 3.

the parts c(i)-(iii) are hints for (iv), ie., you follow the steps for finding total area to find total volume. to be more clear, you find the volume from 0 to 1, and show that the volumes from k to k+1 form a GP, and finally find the limiting sum to get the total volume.

did you guys notice that the diagram was wrong? if you based your solution on the sketch given, you would get incorrect answer.

it is asking how many different ways of five cards that consist of one pair and three of a kind (ie, 2 ace and 3 sixs, or 2 sevens and 3 kings, etc) for the pair, 13C1 times 4C2 for the three alike, 12C1, 4C3

Hint: $ 1. $ y^\prime=\frac{x}{x+1}=\frac{(x+1)-1}{\sqrt{x+1}}=\frac{x+1}{\sqrt{x+1}}-\frac{1}{\sqrt{x+1}}=(x+1)^{\frac12}-(x+1)^{-\frac12} $ 2. $ V=\pi\int_0^1y^2dx

two vectors are same if and only if all components are same. a=b, so z=1, 3y+2=x+y, 5x-3=2x+y, solving simutaneously we have x=4/5, y=-3/5, z=1, so the magnitude of 1.25xi+5yj+6zk=i-3j+6k is \sqrt{1^2+(-3)^2+6^2}=\sqrt{46}.

Re: HSC 2015 4U Marathon $ The line $ y=mx+b $ is tangential to the ellipse $ \frac{x^2}{9}+\frac{y^2}{4}=1. $ Show that $ b^2=9m^2+4. $ Hence find the minimum area bounded by the tangent and the coordinate axes. $

when u do area or volume, first thing is always sketch the graph. for this question, if you sketch the parabola, you will find what region is referred to.

Re: HSC 2015 4U Marathon sorry 3030 was a typo, i actually corrected into 4030

Re: HSC 2015 4U Marathon There are seven different coloured jars. Each jar contains ten identical marbles, each being the same colour as the jar it is in. A total of ten marbles is selected from these seven jars. (a) How many different selections are possible if at least one marble is selected...

Re: HSC 2015 4U Marathon a little change would make a good question: $ Simplify $ \sin{x}+\sin2x+\sin3x+\cdots+\sin{nx}. $ Hence show that $ \sin\frac{\pi}{2015}+\sin\frac{2\pi}{2015}+ \sin \frac{3\pi}{2015}+\cdots+\sin\frac{\2014\pi}{2015}= \cot\frac{\pi}{4030}.

Re: HSC 2015 4U Marathon NEXT QUESTION Each of a set of equal cubes has its six faces painted red, orange, yellow, green, blue and white respectively so that no two cubes are alike and every arrangement of colours is used. How many cubes are there in the set?

Re: HSC 2015 4U Marathon $ let $ P(x)=(x^2-a^2)Q(x)+Ax+B $ (the division transformation). sub. $ x=\pm a, P(a)=aA+B, P(-a)=-aA+B. $ solving simultaneous equations for $ A, B, A=\frac{1}{2a}[P(a)-P(-a)], B=\frac{1}{2}[P(a)+P(-a)]. $ when $ P(x)=x^n-a^n, A=\frac{1-(-1)^n}{2}a^{n-1}...

Re: MX2 2015 Integration Marathon \int\frac{dx}{x^3\sqrt{1+2x-x^2}}

Re: MX2 2015 Integration Marathon \int\frac{x^3}{\sqrt{1+2x-x^2}}dx

Re: MX2 2015 Integration Marathon \int\frac{x^2-1}{x^4+1}dx