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I'd be more inclined to do it by contrapositive \begin{aligned}\left|z+\frac{1}{z}\right|>2&\Rightarrow|w|>2\\ &\Rightarrow |w^2-3|\ge|w|^2-3>4-3\\ &\Rightarrow |w||w^2-3|>2\times1\\ &\Rightarrow|w^3-3w|>2\\ &\Rightarrow\left| z^3+\frac{1}{z^3}\right|>2\end{aligned} \text{Hence }\left|...

2 things are wrong. It should be \ge not > in the question. Also you are assuming |z|=1 which is incorrect. Anyway you can use triangle ineqality. |w^2-3|+|3|\ge|w^2-3+3| Therefore |w^2-3|+3\ge|w|^2 and so |w^2-3|\ge|w|^2-3 ------ A simple example showing why you need \ge is if...

i. So you proved it true for n=1 in which case you know that \frac{a^3}{b^3}+\frac{b^3}{a^3}=2 and hence (a^3-b^3)^2=0 so a^3=b^3 Likewise for induction hypothesis, if true for n=k then \frac{a^{3k}}{b^{3k}}+\frac{b^{3k}}{a^{3k}}=2 and hence (a^{3k}-b^{3k})^2=0 so a^{3k}=b^{3k}. Then for...

Is coplanarity in the syllabus? The standard way to do it is with scalar triple product.

Cambridge again? Maybe this

They seem to be from cambridge so here are the solutions.

There is another geometric explanation too involving the scalar triple product.

Here you go

Just add up the areas of the rectangles and divide by 2 (1+0.5+0.5+1/3)/2=1.2 correct to 1 decimal place. Exact value is ln3 which is more like 1.1

Here you go.

I don't think that will be a benefit. You should be able to swap things like that on the same calculator, just as it was on other calculators.

There is also a new video for 1-variable statistics at You can see how much easier it is to get all the statistics at once compared with the current models where you have to get each one separately

Exponential form was added. Technically it wasn't in the old syllabus, but many teachers still taught it anyway as an extra because it is quite useful. Even the old Coroneos textbook had it. But now you have to do it. For example in this year's HSC exam Question 11(c)(i) \text{Write the...

Here are the itute solutions for Standard 2: https://www.itute.com/wp-content/uploads/2022-NSW-ESA-Mathematics-Standard-2-Solutions.pdf and for what I believe to be the first time online, the solutions to the Standard 1 paper which came today...

itute solutions: https://www.itute.com/wp-content/uploads/2022-NSW-ESA-Mathematics-Advanced-Solutions.pdf

They have now made a new video for normal distributions. When they do the question b it shows how to do the inverse normal which the older calculators couldn't do.

As well as the 15 comments after the article, the Sydney Morning Herald has also published the following letter to the editor today: "As a maths teacher with more than 40 years’ experience I support the comments by Scott Lankshear with regard to the recent Mathematics Extension 1 paper (“HSC...

A few months before Coroneos died I rang him for what I thought was going to be a 2 minute conversation about a book. It quickly diverged to many other things and turned into a 5 hour conversation. One of the things discussed was the length of the 2 unit papers. Back in 1967 the entire paper was...

matrix education solutions: https://www.matrix.edu.au/2022-hsc-maths-extension-2-exam-paper-solutions/

Matrix Education solutions: https://www.matrix.edu.au/2022-hsc-maths-ext-1-exam-paper-solutions/