1. 1
   The complex numbers $z$ and $w$ are given by $z = 2\,\mathrm{cis}\left(\dfrac{\pi}{3}\right)$ and $w = 4\,\mathrm{cis}\left(\dfrac{\pi}{6}\right)$.

   (a)

   Find $zw$.

   [1 mark]

   • A$6\,\mathrm{cis}\left(\frac{\pi}{2}\right)$
   • B$8\,\mathrm{cis}\left(\frac{\pi}{2}\right)$
   • C$8\,\mathrm{cis}\left(\frac{\pi^2}{18}\right)$
   • D$8\,\mathrm{cis}\left(\frac{\pi}{6}\right)$

   (b)

   Find $\dfrac{z}{w}$ in Euler form.

   [1 mark]

   • A$2\mathrm{e}^{-\frac{\pi}{6}\mathrm{i}}$
   • B$-2\mathrm{e}^{\frac{\pi}{6}\mathrm{i}}$
   • C$\frac{1}{2}\mathrm{e}^{\frac{\pi}{6}\mathrm{i}}$
   • D$\frac{1}{2}\mathrm{e}^{\frac{\pi}{2}\mathrm{i}}$

   (c)

   Write $z$ in the form $a + b\mathrm{i}$, where $a, b \in \mathbb{R}$, giving exact values.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The complex number $u$ is given by $u = -2 - 2\mathrm{i}$. Arguments are taken in the interval $-\pi < \theta \le \pi$.

   (a)

   Write $u$ in modulus–argument form.

   [1 mark]

   • A$2\sqrt2\,\mathrm{cis}\left(-\frac{3\pi}{4}\right)$
   • B$2\sqrt2\,\mathrm{cis}\left(\frac{\pi}{4}\right)$
   • C$8\,\mathrm{cis}\left(-\frac{3\pi}{4}\right)$
   • D$2\sqrt2\,\mathrm{cis}\left(\frac{3\pi}{4}\right)$

   (b)

   Write $u^{*}$ in Euler form.

   [1 mark]

   • A$2\sqrt2\,\mathrm{e}^{-\frac{3\pi}{4}\mathrm{i}}$
   • B$-2\sqrt2\,\mathrm{e}^{\frac{3\pi}{4}\mathrm{i}}$
   • C$2\sqrt2\,\mathrm{e}^{\frac{\pi}{4}\mathrm{i}}$
   • D$2\sqrt2\,\mathrm{e}^{\frac{3\pi}{4}\mathrm{i}}$

   (c)

   Use your answer to part (a) to find $u^{2}$, giving your answer in the form $a + b\mathrm{i}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In an alternating-current circuit, the voltage $V$ volts, the current $I$ amps and the impedance $Z$ ohms are complex numbers related by $V = IZ$. A component has impedance $Z = 4\sqrt{3} + 4\mathrm{i}$ and carries a current $I = 5\mathrm{e}^{\frac{\pi}{4}\mathrm{i}}$. A calculator may be used in part (b).

   (a)

   Write $Z$ in the form $r\mathrm{e}^{\mathrm{i}\theta}$, where $r > 0$ and $-\pi < \theta \le \pi$.

   [3 marks]

   (b)

   Find $V$ in Euler form. Hence find $V$ in the form $a + b\mathrm{i}$, giving $a$ and $b$ correct to three significant figures, and state the angle by which the argument of the voltage exceeds the argument of the current.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $z = 1 + \mathrm{e}^{\mathrm{i}\theta}$ and $w = 1 - \mathrm{e}^{\mathrm{i}\theta}$, where $0 < \theta < \pi$.

   (a)

   (i) Show that $z = 2\cos\left(\dfrac{\theta}{2}\right)\mathrm{e}^{\frac{\mathrm{i}\theta}{2}}$.
   (ii) Hence write down $|z|$ and $\arg z$, justifying your value of $|z|$.
   (iii) Find $|z|$ and $\arg z$ when $\theta = \dfrac{2\pi}{3}$.
   (iv) Give a geometric reason, in the complex plane, why $\arg z = \dfrac{\theta}{2}$.

   [6 marks]

   (b)

   (i) Show that $w = -2\mathrm{i}\sin\left(\dfrac{\theta}{2}\right)\mathrm{e}^{\frac{\mathrm{i}\theta}{2}}$.
   (ii) Hence show that $\dfrac{z}{w}$ is purely imaginary, and interpret this result geometrically in the complex plane.

   [6 marks]

   Total for question 4: 12 marks

End of questions