VCE Specialist Maths practice questions

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1. Examination 2 · Section A · 1 mark · Complex numbers in polar form

   The principal argument of $-\sqrt{3} + i$ is
   Answer it, every step marked

2. Examination 1 · No technology · 5 marks · Complex numbers in polar form

   Let $z = 1 - \sqrt{3} i$ and $w = 2 i$.
   Answer it, every step marked

3. Examination 2 · Section B · 11 marks · Complex numbers in polar form

   The tip of a robot arm moves in a horizontal plane, modelled as the complex plane with the pivot of the arm at the origin $O$ and distances in metres. The tip starts at $z_0 = 2 \sqrt{3} + 2 i$. A turn command rotates the arm anticlockwise about $O$ through $\frac{\pi}{6}$ without changing its length, so after $n$ turn commands the tip is at $z_n = \left(\mathrm{cis}\left(\frac{\pi}{6}\right)\right)^{n} z_0$.
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4. Examination 2 · Section A · 1 mark · Complex numbers in polar form

   Let $z = 2 \, \mathrm{cis}\left(\frac{\pi}{3}\right)$ and $w = 4 \, \mathrm{cis}\left(-\frac{5 \pi}{6}\right)$. In polar form with the principal argument, $\dfrac{z}{w}$ is
   Answer it, every step marked

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