1. 1
   The cubic equation $z^{3} + az^{2} + bz + c = 0$, where $a, b, c \in \mathbb{R}$, has a root $z = 2 - 3i$. It also has a real root $z = 1$.

   (a)

   Which of the following must also be a root of the equation?

   [1 mark]

   • A$3 - 2i$
   • B$-2 + 3i$
   • C$-2 - 3i$
   • D$2 + 3i$

   (b)

   Find the value of $a$.

   [1 mark]

   • A$5$
   • B$-5$
   • C$-4$
   • D$4$

   (c)

   Find the value of $b$ and the value of $c$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The complex number $w$ is given by $w = 1 + \sqrt{3}\,i$.

   (a)

   Find $w^{6}$.

   [1 mark]

   • A$64i$
   • B$-64$
   • C$64$
   • D$12$

   (b)

   Find the smallest positive integer $n$ for which $w^{n}$ is a negative real number.

   [1 mark]

   • A$3$
   • B$6$
   • C$2$
   • D$9$

   (c)

   Find $w^{-2}$, giving your answer in the form $p + qi$, where $p, q \in \mathbb{R}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the equation $z^{3} = 4\sqrt{2}\,(-1 + i)$, where $z \in \mathbb{C}$. The three solutions are represented in the complex plane by the points $A$, $B$ and $C$.

   (a)

   Solve the equation, giving your answers in the form $re^{i\theta}$, where $r > 0$ and $-\pi < \theta \le \pi$.

   [3 marks]

   (b)

   Find the exact area of triangle $ABC$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $z = \cos\theta + i\sin\theta$, where $\theta \in \mathbb{R}$.

   (a)

   Prove by mathematical induction that $z^{n} = \cos n\theta + i\sin n\theta$ for all $n \in \mathbb{Z}^{+}$.

   [6 marks]

   (b)

   By considering $z^{3}$, show that $\cos 3\theta = 4\cos^{3}\theta - 3\cos\theta$. Hence find the exact solutions of the equation $8x^{3} - 6x - 1 = 0$.

   [6 marks]

   Total for question 4: 12 marks

End of questions