1. 1
   The complex numbers $z$ and $w$ are given by $z = 3 - 4\mathrm{i}$ and $w = 1 + 2\mathrm{i}$.

   (a)

   Find $zw$.

   [1 mark]

   • A$3 - 8\mathrm{i}$
   • B$-5 + 2\mathrm{i}$
   • C$11 - 2\mathrm{i}$
   • D$11 + 2\mathrm{i}$

   (b)

   Find $|z|$.

   [1 mark]

   • A$5$
   • B$25$
   • C$7$
   • D$\sqrt{7}$

   (c)

   Find $\dfrac{z}{w}$, giving your answer in the form $a + b\mathrm{i}$, where $a, b \in \mathbb{R}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The complex number $z = a + b\mathrm{i}$, where $a, b \in \mathbb{R}$, satisfies the equation $z + 2z^{*} = 9 - 4\mathrm{i}$, where $z^{*}$ is the complex conjugate of $z$.

   (a)

   Find $z$.

   [1 mark]

   • A$3 - 4\mathrm{i}$
   • B$3 + 4\mathrm{i}$
   • C$3 - \frac{4}{3}\mathrm{i}$
   • D$9 + 4\mathrm{i}$

   (b)

   Which statement correctly describes the point representing $z^{*}$ in the complex plane?

   [1 mark]

   • AIt is the point $(-3, 4)$, in the second quadrant
   • BIt is the point $(3, 4)$, in the first quadrant
   • CIt is the point $(3, -4)$, in the fourth quadrant
   • DIt is the point $(-3, -4)$, in the third quadrant

   (c)

   Find $z^{2}$, and write down its real part and its imaginary part.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The complex number $z$ is given by $z = 5 + 12\mathrm{i}$. A calculator may be used in part (a) only.

   (a)

   Find the modulus of $z$ and the argument of $z$, giving the argument in radians correct to three significant figures.

   [3 marks]

   (b)

   Find the two complex numbers $w$ such that $w^{2} = z$, giving your answers in the form $a + b\mathrm{i}$, where $a, b \in \mathbb{R}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In the complex plane, the points $P$ and $Q$ represent the complex numbers $p = 1 + 2\mathrm{i}$ and $q = 2 - \mathrm{i}$ respectively, and $O$ is the origin.

   (a)

   (i) Show that $|p| = |q|$.
   (ii) Find $\dfrac{p}{q}$ in the form $a + b\mathrm{i}$.
   (iii) Find $|p - q|$, and hence show that triangle $OPQ$ has a right angle at $O$.

   [6 marks]

   (b)

   The point $T$ represents the complex number $t = x + y\mathrm{i}$, where $x, y \in \mathbb{R}$. Given that $t\,t^{*} + 2t = 3 + 2p$, find the two possible values of $t$, giving your answers in exact form.

   [6 marks]

   Total for question 4: 12 marks

End of questions