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I put it into wolframalpha and got \sqrt{e}

Files for Extension 2: https://www.mediafire.com/file/bna01lzg3wb4gmr/hsc-feedback-day-2023-files-e2.zip/file (about 82 MB) A somewhat larger one is for all courses: https://www.mediafire.com/file/vr8rzp5deiq2v05/hsc-feedback-day-2023-files.zip/file (about 339 MB)

You can do it like this: (a) ^{12}C_3=220 (b) \textstyle\frac{^{12}C_3\times{}^9C_3}{2}=9240 (c)(i) \textstyle\frac{12\times({}^9C_3\times1+{}^8C_3\times2+{}^7C_3\times3+{}^6C_3\times4+{}^5C_3\times5+{}^4C_3\times6+{}^3C_3\times7)}{2}=2772 (c)(ii) 9240-2772=6468

They made a new video for regression:

Here are 2 methods. Method 1 - trig substitution \text{One way is to let }x=\cos^2\theta\text{ so that }\cos\theta=\sqrt x,\ \sin\theta=\sqrt{1-x}, \theta=\sin^{-1}\sqrt{1-x}\text{ and }dx=-2\cos\theta\sin\theta d\theta. \begin{aligned}\text{Then...

There are other ways to write the answer like -\sqrt{x-x^2}+\cos^{-1}\sqrt{1-x}+C, etc., but whichever way you write it, it should have an inverse trig function.

\text{How about }-\sqrt{x-x^2}-\sin^{-1}\sqrt{1-x}+C \text{If you differentiate this you will get }\sqrt\frac{x}{1-x} \text{If you differentiate your answer, you don't.}

It is from New Senior Mathematics Two Unit Course for Years 11 & 12 by Fitzpatrick. (1st ed, 1984) Ex 8d Q10 page 191 However it has been reused in the second and 3rd editions too. In the more up-to-date 3rd edition it is Ex 2.8 Q10 on page 47 (slightly reworded, but essentially the same).

I don't think it is outside the syllabus if you prove this first (pretty easy to do): \text{Prove that for all non-negative real numbers }a,b,\frac{a+b}{2}\le\sqrt{\frac{a^2+b^2}{2}}\text{ with equality }\iff a=b

Some dude (can't remember who) came here before and tried to use the AM-GM inequality to do it but it didn't work. They kind of had a good idea but had the inequality the wrong way around. However the AM-QM inequality does work (instead of the AM-GM one)...

The corrections to incorrect Cambridge solutions to the Extension 1 Year 12 textbook are now at http://www.angelfire.com/ab7/fourunit/Cambridge-y12-e1-sol-cor.pdf and together with the Year 11 corrections at http://www.angelfire.com/ab7/fourunit/Cambridge-y11-e1-sol-cor.pdf this further reduces...

You can also do it with calculus. f(\theta)=|e^{i\theta}-2|+|e^{i\theta}+2|\text{ can be simplified to }\sqrt{5-4\cos\theta}+\sqrt{5+4\cos\theta} \text{Then }f'(\theta)=\frac{2\sin\theta}{\sqrt{5-4\cos\theta}}-\frac{2\sin\theta}{\sqrt{5+4\cos\theta}} \text{Maximum value occurs when...

I think the most up-to-date Cambridge Extension 1 solutions are best sourced from the links I previously posted: https://www.cambridge.edu.au/education/titles/CambridgeMATHS-Stage-6-Mathematics-Extension-1-Year-11-Online-Teaching-Suite/ and...

Here are the corrections to incorrect Cambridge solutions to the Extension 1 Year 11 textbook: http://www.angelfire.com/ab7/fourunit/Cambridge-y11-e1-sol-cor.pdf This now leaves the following somewhat smaller list: Missing Year 11: 1D Q8e, 8f, 9e, 9f, 10e, 10f 1G Q6g, 6h, 7 2C Q6j, 6k, 6l...

Update to Yr12 Cambridge Extension 1 solutions. Added 1I Q9a, 9b, 9c, 10a, 10b: "Answers are contained in the question." - acceptable for these parts but not 9d or 10c 3G Q8e 3H Q10e, 11e 4B Q18, 19 9B Q14bii: solution is there but is incorrectly labelled 14bi 9C Q13e, 13f Chapter 9 review...

This rather complicated post is to update the list of missing Cambridge Year 11 Extension 1 solutions. As complicated as it is, the end result is to shorten the list and hence make it easier to complete the solutions. I will make a similar post later for Cambridge Yr 12 Extension 1. It is a...

Here is a more visual explanation

I think we will be doing this: |\overrightarrow{a}|=\sqrt{5^2+1^2}=\sqrt{26},\overrightarrow{b}=x\overrightarrow{i}+y\overrightarrow{j},|\overrightarrow{b}|=\sqrt{x^2+y^2} \overrightarrow{a}\cdot\overrightarrow{b}=5x+y=12 \cos58^\circ=\frac{\overrightarrow{a}\cdot{\overrightarrow...

Most of these Extension 2 ones have now been added except for 3E Q3e and 6G Q17. But they are now available at https://www.angelfire.com/ab7/fourunit/Cambridge-e2-3EQ3e-6GQ17.pdf and that completes the solutions for Cambridge Extension 2. I will update the lists for Extension 1 soon.