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You can also easily find LCM and GCD on a calculator that doesn't have a specific button for it, like so: .

Couldn't they just as easily make (say) Maths the compulsory subject that ATAR is based on (if there is a need to have one common subject)? In other words, saying we need a common subject doesn't explain why English (or any particular subject) needs to be the compulsory one.

$\noindent It also works for non-integer but \textbf{constant} multiples of $x$, e.g.$ $$\begin{align*} \frac{d}{dx}(\pi x) &= \frac{d}{dx}(\underbrace{x + x + x + \cdots + x}_{\pi \text{ times}}) \\ &= \underbrace{ \frac{d}{dx}(x) + \frac{d}{dx}(x) + \frac{d}{dx}(x) + \cdots +...

$\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...

$\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...

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

$\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...

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.

$\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.$

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...

Depends what you said exactly I think.

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

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

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

$\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 $p'$.$

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.

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...

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...

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 $a_k \ge a_{k-1}+2$...

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