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Re: HSC 2015 3U Marathon The answer I have is 1-\frac{\pi}{4}. Note that in this scenario, the value of k is \sqrt{2}. So in your area equation, the \sin^{-1}{\left (\frac{1}{\sqrt{2}}+\frac{k}{2} \right )} is outside the domain of -1 to 1 for inverse sine functions. Can you explain what you did?

Re: HSC 2015 3U Marathon If anyone is unsure what the scenario looks like, here you go.

Re: HSC 2015 3U Marathon Yep.

Re: HSC 2015 3U Marathon Question 1: \begin{align*}\textup{RTP }f(a^n)&=n\cdot f(a) \textup{ for }n\geq1\\ \textup{1. Show true for }n&=1 \\ \textup{LHS}&=f(a^1)=f(a) \\ \textup{RHS}&=1\times f(a)=f(a)=\textup{LHS} \\ \textup{Therefore, true for }n&=1. \end{align*} \begin{align*}\textup{2...

re: HSC Chemistry Marathon Archive Read through this.

Possible flowcharts/diagrams: - Titration method, or part of it - Production of a polymer from natural sources - Production of a biopolymer - Industrial production of ethanol from sugar cane - Identifying cations or anions - AAS - Water tests, e.g. TDS, BOD Can anyone think of more?

re: HSC Chemistry Marathon Archive Acid + Base -> Salt + Water. Or do you mean how do you know it is Ba(NO3)2 and not BaNO3 that is produced?

Yep I changed it. My bad, didnt consider that case: Case 5: 2 pairs of unique numbers (1122) 10*1*9*1 = 90 Total ways = 5860 + 90 = 5950 combinations.

Oh, well here's what I did: Case 1: 4 unique numbers (1234) 10*9*8*7 = 5040. 10 numbers for the first (0-9), 9 for the second (0-9 excluding the first number) etc. Case 2: 2 unique numbers (1123) 10*1*9*8 = 720 Case 3: 1 unique numbers (1112) 10*1*1*9 = 90 Case 4: 0 unique numbers (1111)...

Do you know the answer? I'm getting 5860.

Thanks!

Hey can someone do this question for me: $Find the general solution(s) of $\sin{x}+\cos{x}=\cos{2x}$.

No you don't need to. The conditions for a horizontal point of inflexion are: f'(x)=0 f''(x)=0 f'(x) is the same on either side Clearly f'(D)=0 as it is on the x-axis. Since D is also a stationary point, that means f''(D)=0. Now the y-axis in this graph is f'(x). f'(x) is greater than 0 on...

f'(x) > 0 on both sides of D. You're looking at the gradient of the first derivative, not the gradient of the function.

That's because this is a graph of the first derivative, not the function itself. The first derivative is positive on both sides of D.

Re: 2015 permutation X2 marathon Yep I think so.

Re: HSC 2015 3U Marathon $Let $\alpha=\frac{m\pi}{n}$ where $m$ and $n$ are positive relatively prime integer numbers. Find $m+n$ if $0 \leq \alpha \leq \frac{\pi}{2}$ and $\sin{\alpha}=\frac{1-\sqrt{2}}{\sqrt{6-3\sqrt{2}}-\sqrt{2+\sqrt{2}}}$.

Re: HSC 2015 3U Marathon $Solve for $x$: $2^{3x}-\frac{8}{2^{3x}}-6\left (2^x-\frac{1}{2^{x-1}} \right )=1

Re: 2015 permutation X2 marathon Sorry, that's not my intention. I figured I'd forget to post this one if I didn't now. I did attempt your question, but I have no idea where to proceed after finding the discriminant. Your hint of using integration makes me think it is a portion of an area...

Re: 2015 permutation X2 marathon Question I saw: In a chess tournament, n women and 2n men participated, and each one of them played only one game with everybody else. The ratio of the number of games won by women to the number of games won by men is 7:5. Find the number of men that...