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Re: HSC 2013 2U Marathon Yea correct ---- (this one is different I assure you) $Solve$ 1+ \cos x + \cos^2 x + \cos^3 x + \dots = \cot x \csc x

Its a topic called partial fractions in 4U This lol they would never ask a straight out series in 3U, let alone give it without some guidance in 4U

Re: HSC 2013 2U Marathon Yea well done ------- $Find the solutions to the equation$ 1+ x + x^2 + \dots = 5x

LOOKS

w0t

Re: HSC 2013 2U Marathon $Find the domain and range of$ f(x) = \sqrt{1-\sqrt{x-2}}

Well its a very simple fraction split so technically its a 4U question but yea lol

It is asking for what values of n is there, that there exists a term in the expansion of (2x^3-1/x)^n, that is constant and NOT zero (The reason why it says non-zero is because every expansion has a zero constant term since A+0 = A)

T_n = \frac{1}{(2n-1)(2n+1)} = \frac{1}{2}\left(\frac{1}{2n-1} - \frac{1}{2n+1} \right) \sum_{k=1}^{n} T_k = \frac{1}{2} \sum_{k=1}^n \frac{1}{2k-1} - \frac{1}{2k+1} Now, try expanding that sum out, it doesn't have a common difference, but see what happens when you write that series out in...

Re: HSC 2013 2U Marathon f(x) = C\ln(x) ---- To find a proper functional solution we do: \frac{d}{dA} f(AB) = \frac{d}{dA} (f(A) + f(B)) B f'(AB) = f'(A) Subbing in A=1 f'(B) = \frac{f'(1)}{B} Integrating f(B) = C \ln B

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Re: HSC 2013 2U Marathon $i) Show that$ 1 + (\cos \theta + \sin \theta) + (\cos^2 \theta + \sin^2 \theta) + (\cos^3 \theta + \sin^3 \theta) + \dots = \frac{1-\sin \theta \cos \theta}{(1-\cos \theta)(1-\sin \theta)} $ii) Show that this sum is always positive$

Re: HSC 2013 4U Marathon First move the complex numbers and make z_3 the origin z_1 \cdot \left( \cos \frac{\pi}{3} + i\sin \frac{\pi}{3} \right) = z_2 $Through manipulation we find$ z_1^2 + z_2^2 = z_1 z_2 z_1 \rightarrow z_1 - z_3 z_2 \rightarrow z_2 - z_3 This corresponds to an...

Re: MX2 Integration Marathon \int_1^e \ln^{n-1}x \left( \ln x + n) \ dx n \geq 1

Re: HSC 2013 2U Marathon Yes I'm quite furious that you managed to answer that ---- $i) Show that the values of$ \ \ a \ \ $that satisfy$ \int_{-a}^a e^{x} - 1 \ dx $Satisfy the equation$ e^{a} - e^{-a} = 2a $ii) Show that$ \ \ a=0 \ \ $is the only solution$

Re: HSC 2013 4U Marathon Are you sure the last line of the question isn't a typo? Any 3 points are cyclic

Re: HSC 2013 4U Marathon Yep that is correct, I would think that you'd need to mention: \lim_{n \to \infty} \frac{1}{\prod_{k=0}^n (1+x^{2^k})}} = 0 There are some cases where the sum would go to infinity and there would be mass cancellation (telescoping), but you still have a...

Re: HSC 2013 2U Marathon I've got secret insider information into the contents of the exams next week and I'm trying to sabotage you all

Re: HSC 2013 3U Marathon Thread $Prove by mathematical induction that the sum of interior angles of an$ \ n \ $sided polygon is$ 180(n-2) \ \ $in degrees measure$

Re: HSC 2013 2U Marathon $Give an example of a function$ \ \ f(x) \ \ $such that$ f(AB) = f(A) + f(B) \ \ $for positive values of$ \ A,B