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I have never seen them asking you to derive these in the exams. It is highly unlikely they would ask you to derive them. The derivation of them is in the syllabus document in Section 5.7 of Page 40: http://www.boardofstudies.nsw.edu.au/syllabus_hsc/pdf_doc/maths23u_syl.pdf

Yeah. That's because we can't divide by 0, so we need to deal with this case separately. Welcome!

Remember to include the case of tan x = 0 (if we divide through by tan x in the equation, we are ignoring the possibility of tan x = 0). When tan x = 0, the equation is also satisfied, and this gives us the solutions you missed.

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread You need the absolute value sign to make sure answer is non-negative, since |z| is non-negative. (Draw a diagram.)

Oh yeah, whoops. Cheers!

Re: MX2 2016 Integration Marathon Ah right, of course. My bad. :)

Re: MX2 2016 Integration Marathon $\noindent I think it was a typo by omegadot. The final answer should clearly depend on $a$, so one of those terms in the bracket should probably have an $a$ in it.$

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $This is a pretty typical one. For the induction step, assume that $1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{1}{6}n(n+1)(2n+1)$, and use this to show that $1^2 + 2^2 + 3^2 + \cdots + n^2 + (n+1)^2 =...

Re: HSC 2015 4U Marathon - Advanced Level Yeah. I just posted it here to save people trouble of having to search it. Haha

Re: HSC 2015 4U Marathon - Advanced Level For anyone interested, here is the edited version:

Re: MX2 2016 Integration Marathon A lot of these integrals are harder than would appear in the HSC papers (if they did appear, they'd be broken down into sub-parts, making them easier to do), so you shouldn't stress too much. :)

Re: HSC 2016 4U Marathon - Advanced Level \textbf{ANOTHER QUESTION}$ $\noindent Let $A = \sin \theta _1 + \sin \theta _2 + \cdots + \sin \theta_n$, where $n$ is a fixed positive integer with $n\geqslant 3$, and each $\theta _i >0$ and $\theta_1 + \theta _2 + \cdots + \theta _n = 2\pi$. Show...

Re: MX2 2016 Integration Marathon $The reason this pattern doesn't work for odd exponents can be seen from looking at the general term in the binomial expansion. The expansion of $\left(\frac{2}{x} + x\right)^{2n+1}$ ($n \in \mathbb{N}$) has general term $\binom{2n+1}{k} 2^{2n+1-k} \cdot...

Re: MX2 2016 Integration Marathon Two possibilities: 1) He tried some small values and noticed a pattern 2) His intuition. :)

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread What's the question?

$Let the first term be $a$ and the common ratio be $r$. The second term is $ar$. So the sum of all terms except the first is $\frac{ar}{1-r}$ (limiting sum). This is given to be equal to $5a$. So we have $ $$\begin{align*} \frac{ar}{1-r} &= 5a \\ \Rightarrow \frac{r}{1-r}&= 5 \quad...

Re: HSC 2016 4U Marathon It's referring to the force up and down the plane, right (like in the direction of friction forces in these Q's that have friction)?

Re: MX2 2016 Integration Marathon $Nice. Here was my method:$ $Let $I=\int _a ^b \left(f(x) + f^{-1} (x) \right) \text{ d}x$, so $I = \int _a ^b f(x) \text{ d}x + \int _a ^b f^{-1} (x) \text{ d}x$. For the second integral here, let $u=f^{-1}(x)$, so $x = f(u)$, $\mathrm{d}x = f^\prime (u)...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $You can do it using a first derivative table of values, yes. This is a good idea when the second derivative is very messy to compute. In this question though (after realising the trick of considering the area $A$ as a function...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread It must be a typo, they meant minimised. It is clear by drawing a quick sketch that the area can be made arbitrarily large by making the line near-vertical or near-horizontal. (This shows why it is a good idea to quickly sketch...