Graphs Outline

Once again, here is a form for the Graphs topic from the syllabus.
You will need a PDF viewer such as Adobe Reader to be able to use the form.

Arnold & Arnold, Exercise 1.7 Solutions

Here are the solutions for Exercise 1.7 of Cambridge Mathematics 4 Unit textbook by Graham and Denise Arnold.
Remember that once you know the graph of $\displaystyle y=\sqrt{f(x)}$, then it is simply a matter of reflecting the square root curve in the $x$-axis to find the curve $\displaystyle y^2=f(x)$.

The reason for this is that if $\displaystyle y^2=f(x)$, then $\displaystyle y=\pm\sqrt{f(x)}$

Generalised Product Rule

When differentiating some products, the two and three unit courses generally use binomials in the product.  Occasionally, a product will arise that has three components; how can this be done?

Fortunately, there is an easy way to do this:
Consider $y=uvw$, where $u=u(x), v=v(x)$ and $w=w(x)$.  The derivative is then
$\frac{\mathrm{d}}{\mathrm{d}x}(uvw)=\frac{\mathrm{d}u}{\mathrm{d}x}vw+u\frac{\mathrm{d}v}{\mathrm{d}x}w+uv\frac{\mathrm{d}w}{\mathrm{d}x}$

The product can also be extended to any number of terms in the product.  It looks scary, but never let notation scare you in maths!

It should also be noted that doesn't appear in the syllabus documents - it just may make your life that little bit easier!

Implicit Differentiation

Here is a worksheet, with solutions, on implicit differentiation. At the heart, it is an application of the Chain Rule. All you really need to remember is that you differentiate $y$ as if it were an $x$, but just put $\frac{\mathrm{d}y}{\mathrm{d}x}$ after it.

Complex Numbers Checklist

Yes! You can find the Extension II syllabus on the Board of Studies website!
Yes! It has been hyperlinked at the end of the notes appearing elsewhere in this blog!

 No! These previous versions do not allow you to electronically check the points off, and then print the result!

However, using the ever-versatile $\LaTeX$, the syllabus is now new and improved! The version here does exactly what the spiel above says! As usual, you will need to use a PDF viewer that allows JavaScript, such as Adobe Reader.

Complex Numbers Assignment I

This is an assignment that was used in past years - it will be good revision for you!

Past HSC Questions on regions of the complex plane

Traditionally, some of these straightforward questions would appear in question 2 of the paper.  It would appear that with the introduction of multiple choice questions, they may start to show up there (at least there was one in 2013).

1995-2006 Past HSC Complex Numbers Questions

As promised, here are some questions from past HSC papers.  I can't take credit for putting this together (I found it on the internet several years ago), nor can I vouch for the reliability of the answers.  I am in the process of making some worked solutions, and these should be available shortly.

Complex Numbers on the Khan Academy

I presume that you have all heard of the Khan Academy; if not, it is a free resource that has screencasts of a myriad of subjects. Although it is from the US, and some of the material discussed may not be in our course, it is still an excellent resource to use. The complex numbers topics are found here.

Regions of the Complex Plane

As we move into the final part of the unit on complex numbers, we need to consider the different regions of the complex plane.
These sort of questions traditionally appear early in the paper - in the old regime, it would have been around question 2, and with the new papers, they seem to be appearing in multiple choice questions.  However, you should still know how to work these out from the given complex numbers.

Complex Numbers Multiple Choice Test

Embedded here is a multiple choice test on complex numbers.  It may look like an ordinary PDF, but it is interactive - you need to only click on the correct choice and it will give instant feedback.  (Another example of the beauty of using $\LaTeX$ for typesetting!)
Let me know if there are any problems or wrong answers!
You will need to use Adobe Reader to run it.  I use Sumatra, but this won't activate the interactivity.

1997 HSC - Higher Roots of Complex Numbers

Continuing with our study of roots of complex numbers, this particular question asked for the cube roots of a complex number.  Although the examiner's comments discuss several of the methods attempted by students to solve this one, personally, I think that using polar form (mod-arg)  is by far the easiest and most elegant.  However, as I have said on many occasions, don't let my mathematical "prejudices" influence you!  If you prefer one of the other methods, and can perform the computations correctly, you are at complete liberty to use it!

2009 HSC - Square Roots of Complex Numbers

In the 2009 HSC, a question was asked requiring students to find the square roots of a complex number, and to then use this result in solving a quadratic equation.
Remember, for the quadratic, because the coefficients are not real, then the roots do not occur as complex conjugate pairs.

Roots of Complex Numbers

Being able to find square and higher roots of complex numbers is an important part of the HSC course.
Here are some notes that may help you.