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  1. I

    How do you know when to use u-substitution for integration and what u is? (example included)

    $\noindent $\int f(\sin x)\, d(\sin x)$ essentially just means $\int f(u)\, du$ where $u=\sin x$. If you don't like using $d(\sin x)$, just write it using a $u$-substitution, integrate with respect to $u$, and then write the answer in terms of $x$.$ $\noindent For example, to find $\int e^{\sin...
  2. I

    Further Vectors help

    $\noindent \textbf{Hints:} From the dot product, you should be able to find $\left\|\mathbf{b}\right\|$ (remember the formula $\boxed{\mathbf{a}\cdot\mathbf{b}=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\cos\theta}$). You also know / can work out the angle $\mathbf{a}$ makes to the...
  3. I

    Further Vectors help

    You should probably show us the steps in your working, in order for us to be able to answer that.
  4. I

    Further Vectors help

    $\noindent \textbf{Hint:} Recall that we have $(\mathbf{u}\cdot\mathbf{v})^2 = \left\| \mathbf{u}\right\|^2\left\| \mathbf{v}\right\|^2\cos^2 \theta$, where $\theta$ is the angle between the two vectors. You should get a quadratic equation in $a$ from this, which you should know how to solve...
  5. I

    HSC 2019 NSW School Ranking

    The ranking really should be based on ATAR; maybe the Sydney Morning Herald doesn't get access to the ATAR scores, so doesn't rank based on that.
  6. I

    Do they allow you to use “reversing the step” in the HSC? (Nature of proof)

    $\noindent You could in theory add the $\color{blue}\Leftrightarrow$ (or even $\color{blue}\Leftarrow$) symbol before each line after the first line to make the proof valid. However, I'm not sure if the HSC markers would accept it.$
  7. I

    pigeon hole principle help!

    You can show that if there are 7 stamps placed, then there must be a row of 3 stamps as follows: Suppose 7 stamps are placed, then consider the "blank" squares (squares that don't have a stamp in them). There are 2 blank squares (because there are 7 stamps placed in 9 squares). Since there are...
  8. I

    Mathematics Extension 1 Exam Predictions/Thoughts

    Depends what you said exactly I think.
  9. I

    Mathematics Extension 1 Exam Predictions/Thoughts

    $\noindent Note that it is not generally true that if $\cos A = \cos B$, then $\cos \left(\frac{\pi }{2}-A\right) = \cos\left(\frac{\pi}{2}-B\right)$.$
  10. I

    My solutions to the 2019 Mathematics Extension 2 Paper

    $\noinent Do you mean whether you needed to express your answer in radians? I would be very surprised if they penalised you for expressing the answer in degrees.$
  11. I

    Maths Extension 2 predictions/thoughts?

    I would be very surprised if they penalised you for expressing the answer in degrees.
  12. I

    Help on Past Paper questions

    $\noindent As long as you can show that $\alpha_{k}$ is a root for all $k =1,\ldots, m$ and can explain why the $\alpha_{k}$ are all different (i.e. if $k\neq j$, then $\alpha_{k} \neq \alpha_{j}$), then you are done, no need to use $p'$.$
  13. I

    Help on Past Paper questions

    Q7(b)(i) is a special case of the rational root theorem. For a proof of this, you can see https://en.wikipedia.org/wiki/Rational_root_theorem#Proofs.
  14. I

    BoS Maths Trials 2019

    Why is it obviously 42C9? $\noindent This answer follows from noting that we need to insert at least one 0 between each 1. If there are $K$ $1$'s and $N-K$ $0$'s, there are $K-1$ gaps between $1$'s, so after putting one $0$ in each gap, there are now $N-K-(K-1)=N-2K+1$ $0$'s left to place...
  15. I

    BoS Maths Trials 2019

    I got 42C9 using the method I posted above, so I assumed there was a simpler method. For the equivalence between the method your teacher used and the original question, it is as follows: The bit string of length 50 corresponds to your choice of whether you pick each number from 1-50 or not. A...
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    BoS Maths Trials 2019

    There's probably a simpler way to do it, but here's a sketch for one way. $\noindent In general we can ask the same question but with $K$ integers chosen from $\{1,\ldots,N\}$.$ $\noindent Assume the integers are $a_1 < a_2 < \cdots < a_K$ ($K=9$ in your example). We need $a_k \ge a_{k-1}+2$...
  17. I

    What course in which uni do most Ruse graduates go to?

    Is that for a specific university, or altogether? And is the limit only for James Ruse, or do all schools have such a limit?
  18. I

    Wrong Intended Answer in HSC 2011

    For convenience, here's the link to the paper: https://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2011exams/pdf_doc/2011-hsc-exam-mathematics-ext2.pdf. Here are the Sample Answers...
  19. I

    HSC 2018-2019 MX2 Marathon

    For the chord in a circle problem, see also https://en.wikipedia.org/wiki/Bertrand_paradox_(probability).
  20. I

    Solving trigonometric functions Multiple choice HSC

    $\noindent The solutions to $\tan x + 2 = 0$ for $x$ between $0^{\circ}$ and $360^{\circ}$ are indeed $180^{\circ} + \tan^{-1}(-2)$ and $360^{\circ} + \tan^{-1}(-2)$.$ $\noindent To see this, note that if we don't restrict $x$, then $\tan x + 2 = 0 \iff \tan x = -2$ has a solution $x =...