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Re: MX2 Integration Marathon Here is my solution from request I = \int_1^{\sqrt{3}} (1+x^2)^{3/2} \ dx x=\tan u I_5 = \int_{\pi/4}^{\pi/3} \sec^5 u \ du = I_3 + \int_{\pi /4}^{\pi /3} \sec^3 u \tan^2 u \ du Do that integral by parts using: a = \tan u da = \sec^2 u...

Re: HSC 2013 4U Marathon $Find$ \prod_{m=1}^{n-1} \sin \frac{m\pi}{n}

Re: HSC 2013 4U Marathon $Prove$ x^x + y^y > x^y +y^x $For distinct positive reals$ \ \ x,y edit: my bad

Re: HSC 2013 4U Marathon 1) \ \ P'(x) = 5x^4 - 5c $Hence the stationary points are solutions to the equation$ \ \ \ x^4 = c $If$ \ \ c < 0 \ \ $then there are no stationary points$ $Considering the fact that$ \ \ P(0) = 1 > 0 \ \ \lim_{x \to \infty} P(x) =+\infty \ \ \lim_{x \to...

Re: HSC 2013 4U Marathon *please be a teacher* *please be a teacher*

Re: HSC 2013 4U Marathon 1/2 ? --- \sum_{k=1}^n \frac{1}{k(k+1)(k+2)}

Re: HSC 2013 4U Marathon If I understood the definition of cardinality and subsets correctly, we simply need to prove that \sum_{m \ \ even} \binom{n}{m} = \sum_{m \ \ odd} \binom{n}{m} Since C(n,m) is the number of ways to pick m elements from a set of n. 0 = (1-1)^n =...

Re: HSC 2013 4U Marathon I got the question from art of problem solving problem number 9.3.18, and it assumes differentiability, is there a way to oprove that it is differentiable using only the property? (can someone post a question please)

Re: HSC 2013 4U Marathon let a be the double root P'(a) = na^{n-1} + k = 0 \Rightarrow a^{n-1} = \frac{-k}{n} \ \ \fbox{1} P(a) = a^n + ka - s = 0 a(a^{n-1} + k) - s = a(k - \frac{k}{n}) - s = 0 a \frac{k}{n} = \frac{s}{n-1} a^{n-1} \left( \frac{k}{n} \right)^{n-1} = \left(...

Re: HSC 2013 4U Marathon Nice method, alternatively we can differentiate with respect to x (we aren't considering an x-y plane anyway) To get: yf'(yx) = f'(x) Then set x=1 to get f'(y) = 3/y then solve

Re: HSC 2013 4U Marathon 0< x, y< 1 $Prove that$ \frac{x}{1+y} + \frac{y}{1+x} \leq 1

Re: HSC 2013 4U Marathon That's true, I think my calculation is wrong, I am still intent on the grouping idea. I do agree that its also independent of n ----- Take our selected points: P1 P2 P3 ..... P(nk) The number of ways to split these points into k groups without moving them is...

Re: HSC 2013 2U Marathon yep welld one

Re: HSC 2013 2U Marathon what yep 91/2

Re: HSC 2013 2U Marathon $Evaluate$ \sin^2 1^{\circ} + \sin^2 2^{\circ} + \sin^2 3^{\circ} + \dots + \sin^2 88^{\circ} + \sin^2 89^{\circ} + \sin^2 90^{\circ}

Re: HSC 2013 2U Marathon dat fail holy crap

Re: HSC 2013 2U Marathon wait oh my god

Re: HSC 2013 2U Marathon I don't know what foreign mathematics you are going by But in this country when we use ..... we mean infinite series

Re: HSC 2013 2U Marathon yea lol there is no solution to the question you get ln(x) = -1, which is beyond even the 4U syllabus

Re: HSC 2013 4U Marathon To have pk points make k polygons, then there are p points per polygon, in order for there to be no polygons with no common area, then all points for each polygon must be 'grouped together', that is if I have A1 A2 A3 belonging to one polygon and B1 B2 B3 belonging to...