| Exam Board | WJEC |
|---|---|
| Module | Further Unit 1 (Further Unit 1) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Modulus-argument form conversion |
| Difficulty | Moderate -0.8 This is a straightforward application of modulus-argument form conversion and geometric transformations. Part (a) requires converting z from polar to Cartesian form (routine) and recognizing that a 90° clockwise rotation multiplies by -i. Part (b) is simple division. All steps are standard techniques with no problem-solving insight required, making it easier than average. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02m Geometrical effects: multiplication and division |
| Answer | Marks | Guidance |
|---|---|---|
| \(z = 6\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right) = 3 + 3\sqrt{3}i\), \(w = 6\left(\cos\frac{-\pi}{6} + i\sin\frac{-\pi}{6}\right) = 3\sqrt{3} - 3i\) | M1A1, A1 | M1 for either \(z\) or \(w\); or \(\frac{11\pi}{6}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{z}{w} = \frac{3 + 3\sqrt{3}i}{3\sqrt{3} - 3i} = \frac{(3 + 3\sqrt{3}i)(3\sqrt{3} + 3i)}{(3\sqrt{3} - 3i)(3\sqrt{3} + 3i)} = \frac{9\sqrt{3} + 9i + 27i - 9\sqrt{3}}{27 + 9} = \frac{z}{w} = i\) | M1, A1A1, A1 | FT (a); A1 numerator; A1 denominator; dep on all previous marks awarded |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left | \frac{z}{w}\right | = \frac{6}{6} = 1\), \(\arg\left(\frac{z}{w}\right) = \frac{\pi}{3} - \frac{-\pi}{6} = \frac{\pi}{2}\), \(\frac{z}{w} = 1 \times \left(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\right)\), \(\frac{z}{w} = i\) |
**Part (a):**
$z = 6\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right) = 3 + 3\sqrt{3}i$, $w = 6\left(\cos\frac{-\pi}{6} + i\sin\frac{-\pi}{6}\right) = 3\sqrt{3} - 3i$ | M1A1, A1 | M1 for either $z$ or $w$; or $\frac{11\pi}{6}$
**Part (b):**
**Method 1:**
$\frac{z}{w} = \frac{3 + 3\sqrt{3}i}{3\sqrt{3} - 3i} = \frac{(3 + 3\sqrt{3}i)(3\sqrt{3} + 3i)}{(3\sqrt{3} - 3i)(3\sqrt{3} + 3i)} = \frac{9\sqrt{3} + 9i + 27i - 9\sqrt{3}}{27 + 9} = \frac{z}{w} = i$ | M1, A1A1, A1 | FT (a); A1 numerator; A1 denominator; dep on all previous marks awarded
**Method 2:**
$\left|\frac{z}{w}\right| = \frac{6}{6} = 1$, $\arg\left(\frac{z}{w}\right) = \frac{\pi}{3} - \frac{-\pi}{6} = \frac{\pi}{2}$, $\frac{z}{w} = 1 \times \left(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\right)$, $\frac{z}{w} = i$ | (B1), (B1), (M1), (A1) | FT (a)
---3. The complex numbers $z$ and $w$ are represented by the points $Z$ and $W$ in an Argand diagram.
The complex number $z$ is such that $| z | = 6$ and $\arg z = \frac { \pi } { 3 }$.\\
The point $W$ is a $90 ^ { \circ }$ clockwise rotation, about the origin, of the point $Z$ in the Argand diagram.
\begin{enumerate}[label=(\alph*)]
\item Express $z$ and $w$ in the form $x + \mathrm { i } y$.
\item Find the complex number $\frac { z } { w }$.
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 1 2019 Q3 [7]}}