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Re: HSC 2015 3U Marathon $(i) Use mathematical induction to show that $\ln(n!)>n$ for all positive integers $n \geq 6$. \\ \indent (ii) Hence show that $\frac{1}{n!} < \frac{1}{e^n}$ for all positive integers $n \geq 6$. \\ \indent (iii) Hence show that $\frac{1}{1!} + \frac{1}{2!} +...

Re: HSC 2015 3U Marathon $By definition $\cosh{x}=\frac{1}{2} (e^x + e^{-x})$ and $\sinh{x}=\frac{1}{2} (e^x - e^{-x})$. \\ \indent (i) Show that $2\sinh{x}\cosh{x}=\sinh{2x}$. \\ \indent (ii) Show that the equation $p\cosh{x} + q\sinh{x} = r$ can be written as $(p+q)e^{2x} - 2re^{x} + (p-q) =...

Re: HSC 2015 3U Marathon Yea that's why I included the link. Is that still not enough to use it?

Re: HSC 2015 3U Marathon $Consider the integral $J=\int^\pi_0\sin{x}~\mathrm{d}x$. \\ \indent (i) Determine the value of $J$. \\ \indent (ii) By dividing the interval from $x=0$ to $x=\pi$ into $n$ equal sub-intervals and using the mid-ordinate approximation, show that an approximate value of...

Re: HSC 2015 3U Marathon $(i)~~\,Prove that: $ \frac{\cos{\alpha}-\cos{(\alpha+2\beta)}}{2\sin{\beta}}=\sin{(\alpha+\beta})$. \\ \indent (ii) Hence use mathematical induction to show that $ \sin{\alpha}\,+\,\sin{3\alpha}\,+\,\sin{5\alpha}+...+\sin{\left [\left (2n-1 \right )\alpha \right...

$Draw a right-angled triangle in the second quadrant with $\sin{x}=k$. From this, you can find that $\cos{x}=-\sqrt{1-k^2}$. Now, $\tan{\left (\frac{\pi}{2}+x \right )}=-\cot{x}=\frac{-\cos{x}}{\sin{x}}=\frac{\sqrt{1-k^2}}{k}$. Edit: made a mistake.

Re: HSC 2015 3U Marathon $The equation $8x^3-36x^2+22x+21=0$ has roots which form an arithmetic progression. Find the roots.

Nah, only mc and options.

Not sure if they've been uploaded yet: http://imgur.com/a/ByfFx

If no one has uploaded MC and options by tonight, I will.

re: HSC Chemistry Marathon Archive It was in my trials too. The marking criteria (6-7 marks): - Describes ONE or TWO similarities between the chemical processes involved in the production of polyethylene and the named biopolymer - Describes at least TWO differences between the chemical...

re: HSC Chemistry Marathon Archive Not sure if this question has popped up before: Compare the chemical processes involved in the production of polyethylene and a named biopolymer (not cellulose) and evaluate the effect on society and the environment of both types of polymers. (7 marks)

re: HSC Chemistry Marathon Archive Methane reacts with a chlorine atom to form hydrogen chloride and a methyl radical. $Cl $ + $ CH$_4 \to $ HCl $ + $ CH$_3$\cdot . This prevents that chlorine atom from partaking in the destruction of the ozone. (Check page 254 of Conquering Chemistry)

I think it should be shaded. Why should the range matter? The inequality holds true for coordinates that are below the semicircle. If you were to graph y \leq x^2, you wouldn't exclude the area under the x-axis because the range of the parabola is y \geq 0. Someone correct me if I'm wrong.

http://community.boredofstudies.org/showthread.php?t=344092

re: HSC Physics Marathon Archive Yea it was in the Draft Writing Briefs posted by BOSTES (http://www.boardofstudies.nsw.edu.au/australian-curriculum/pdf_doc/physics-st6-draft-writing-brief-2015.pdf - page 24). I would have liked to try the Extension courses, particularly Extension Chemistry...

My bad, didn't notice it was missing. I've added it to my post.

Exam Paper

http://imgur.com/a/tA61F Question 11: http://imgur.com/ntz14kz

Re: HSC 2015 2U Marathon $Since the point $P(x_1,y_1)$ lies on $y=\sqrt{x-4}$, it can be expressed as $P(x_1,\sqrt{x_1-4})$. Using $m=\frac{y_2-y_1}{x_2-x_1}$, the gradient of $OP$ is $m=\frac{\sqrt{x_1-4}}{x_1}$. Since $OP$ is the tangent to $y=\sqrt{x-4}$ at $x=x_1$, you can differentiate...