:/ That sounds tricky haha. If you don't mind, could you walk me through finding the equation of the ray? e.g. if the question was like arg(z-1)=pi/4, arg(z+1)=pi/3 find arg(z)
Also, what are some of the really common complex number qs that one must now. From looking at past trials, finding the roots of unity and factorising over complex/real fields, finding square roots and then solving a quadratic are often the easy marks at the beginning of the test. Can you think of any others, especially to do with vectors (e.g. finding locus)
Thank you!
The method itself is actually not too bad. Take \( \arg (z-1) = \frac\pi4 \). The ray will just be one part of the line, through the point \( (1,0) \), with gradient \( m = \tan \frac\pi4 = 1 \). So using the point gradient form, you just have \( y = 1(x-1) \).
Having said that, whilst that's easy enough, if you did the same thing for \( \arg (z+1) = \frac\pi3 \) you would have \( y = \sqrt{3}(x + 1) \). And then you see the main problem - that \( \sqrt3\) is gonna be hella annoying. It's easy enough to go back to 2U here, and just do simultaneous equations to solve for the point of intersection. But would I expect a 4U student to want to deal with surds for this? Probably not.
Worse, they could move it away from -1 and 1 and give you a random complex number like \( -\sqrt{2} + i\). Or they could swap the angles into something which you'd have to do a bit of work to compute the exact values for.
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Quite a lot of the common ones are all included in my 4U notes book. Vector addition itself is uncommon altogether. For stuff like locus, there are many curves that you'd be expected to know how to draw. On top of that, it should be fairly easy to sketch regions in the Argand plane based off them.
A somewhat common culprit for the trials is to write down the minimum value of \( \arg z \), or \( |z| \), or the real part or whatever, given that you've
already sketched its locus. Sometimes they'll swap that out for the maximum value as well. These were covered in my trial lectures last year and you should look at the notes section for a few examples on those.
A lot of the more popular ones tend to overlap into the polynomials topic as well.