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Re: HSC 2015 3U Marathon $Let $A,B,C$ be real numbers such that $\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+...+\frac{n}{2^n}=\frac{An+B}{2^n}+C$ holds for every positive integer $n$. Find the value of $A+B+C$.

Re: HSC 2015 3U Marathon Is it D? We have M M W W B B. If we consider the boys to be one group. we have M M W W BB, which can be arranged 5! ways. The boys can be arranged 2! ways. So we have 5!2!=240.

Re: HSC 2015 3U Marathon What do you mean 'reverse his steps'? Start from k>0?

re: HSC Physics Marathon Archive As a result of Einstein's Theory of Special Relativity, mass, time and length are variable quantities. Length contraction means the journey will be shorter in an astronaut's frame of reference Time dilation means the journey will be shorter in an astronaut's...

Re: HSC 2015 3U Marathon I'm not sure if your proof for n=k+1 is correct. You say you want to prove 4^{k+1}(k+4)>4^{k+2}, but you haven't really done so. I think what you should do is say RTP 4^{k+1}(k+4)>4^{k+2}, and then say "Consider the difference". Then simplify, show that the expression...

Re: HSC 2015 3U Marathon Hope this clears it up:

re: HSC Chemistry Marathon Archive I think you need to use all the given information. What I did was: 1. Find the volume (V1) of Barium hydroxide required to react completely with the Nitric acid 2. Find the amount of moles in 13.5 mL of 0.14M Barium hydroxide 3. Work out the concentration of...

Re: HSC 2015 3U Marathon Yep that's what I meant.

re: HSC Chemistry Marathon Archive I think you've forgotten to include the 22.35mL present due to the nitric acid when calculating the Total Volume.

Re: HSC 2015 3U Marathon Haha copied the question. Wasn't sure about the notation but no one pointed it out at the time. Would it just be f(x)?

Re: HSC 2015 3U Marathon Would you write it as a limit?

Re: HSC 2015 3U Marathon $1. Let $f(x)$ be a function such that $f(xy) = f(x)+f(y)$. Prove by induction that $f(a^n) = n \times f(a)$ for all $n\geq1$. $2. (a) Consider the circle $x^2+(y-k)^2=1$. For what value(s) of $k$ is the curve $y=x$ a tangent to the circle? \\ \indent (b) A ball of...

Re: HSC 2015 3U Marathon Correct answer is \frac{\pi}{2}. I think you're mistaking \tan ^{-1}(e^{-\infty}) for \tan ^{-1}(-e^{\infty}).

Re: HSC 2015 3U Marathon Here's a question Carrotsticks posted before:

Re: HSC 2015 3U Marathon You've made a mistake with this one. The others are all good.

Re: HSC 2015 3U Marathon Oh are they 2U? I'm just taking questions from old 3U marathons so idk.

Re: HSC 2015 3U Marathon Some more: $1. $\sum_{k=2}^n \frac{1}{\log_k u} $2. Evaluate $\int_{\frac{1}{2}}^{2} \frac{\ln x}{1+x^2} \, \mathrm{d}x $ using the substitution $u=\frac{1}{x} $3. $\lim_{c\to\infty}\int^c_{-c}\frac{\mathrm{d}x}{e^x+e^{-x}}

Re: HSC 2015 3U Marathon Nice. The solutions that I saw were (in white): 1. Use cosine double angle formula to change it to cot^2(x/2), then pythagorean identities. Not sure whether your method is easier or not though. 5. Change cot to cos/sin, simplify and then realise it is f'(x)/f(x).

Re: HSC 2015 3U Marathon Don't know what partial fractions are. For question 2, might be easier if you use the substitution u=1/x.

Re: HSC 2015 3U Marathon $1. $\int \frac{1+\cos{x}}{1-\cos{x}} \, \mathrm{d}x $2. $\int \frac{\cos{\frac{1}{x}}}{x^2} \, \mathrm{d}x $3. Evaluate $ \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{1}{3^{m+n}} $4. Find $\int \frac{1+2\sqrt{x}}{2\sqrt{x}(x+\sqrt{x})} \, \mathrm{d}x$ using the...