| Exam Board | WJEC |
|---|---|
| Module | Further Unit 4 (Further Unit 4) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Direct nth roots: roots with geometric or algebraic follow-up |
| Difficulty | Standard +0.3 This is a straightforward Further Maths complex numbers question requiring conversion to modulus-argument form (routine calculation: r=5, θ=arctan(4/3)), then finding cube roots using De Moivre's theorem and plotting them. The triangle recognition (equilateral) follows directly from the symmetry property of nth roots. While it involves multiple steps and Further Maths content, each component is standard bookwork with no novel insight required, making it slightly easier than average overall. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02d Exponential form: re^(i*theta)4.02f Convert between forms: cartesian and modulus-argument4.02k Argand diagrams: geometric interpretation4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| \( | z | = \sqrt{3^2 + 4^2} = 5\) |
| \(\arg(z) = \tan^{-1}\frac{4}{3} = 0.93\) | B1 | Condone degrees |
| \(\therefore z = 5e^{0.93i}\) | B1 | Radian form |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sqrt[5]{z} = \sqrt[5]{5e^{0.31i + \frac{2\pi ni}{3}}}\) | M1 | FT (a) |
| \(z_1 = \sqrt[5]{5}e^{0.31i} = 1.63 + 0.52i\) (1.63, 0.52) | A1 | A1 complex form for 1 root |
| \(z_2 = \sqrt[5]{5}e^{2.40i} = -1.26 + 1.15i\) (-1.26, 1.15) | m1 A1 | \(+\frac{2\pi ni}{3}\); A1 for all correct coordinates |
| \(z_3 = \sqrt[5]{5}e^{4.50i} = -0.36 - 1.67i\) (-0.36, -1.67) |
| Answer | Marks |
|---|---|
| Equilateral (triangle) | B1 |
## Part a)
$|z| = \sqrt{3^2 + 4^2} = 5$ | B1 |
$\arg(z) = \tan^{-1}\frac{4}{3} = 0.93$ | B1 | Condone degrees
$\therefore z = 5e^{0.93i}$ | B1 | Radian form
## Part b) i)
$\sqrt[5]{z} = \sqrt[5]{5e^{0.31i + \frac{2\pi ni}{3}}}$ | M1 | FT (a)
$z_1 = \sqrt[5]{5}e^{0.31i} = 1.63 + 0.52i$ (1.63, 0.52) | A1 | A1 complex form for 1 root
$z_2 = \sqrt[5]{5}e^{2.40i} = -1.26 + 1.15i$ (-1.26, 1.15) | m1 A1 | $+\frac{2\pi ni}{3}$; A1 for all correct coordinates
$z_3 = \sqrt[5]{5}e^{4.50i} = -0.36 - 1.67i$ (-0.36, -1.67) | |
## Part b) ii)
Equilateral (triangle) | B1 |A complex number is defined by $z = 3 + 4\mathrm{i}$.
\begin{enumerate}[label=(\alph*)]
\item Express $z$ in the form $z = re^{i\theta}$, where $-\pi \leqslant \theta \leqslant \pi$. [3]
\item \begin{enumerate}[label=(\roman*)]
\item Find the Cartesian coordinates of the vertices of the triangle formed by the cube roots of $z$ when plotted in an Argand diagram. Give your answers correct to two decimal places.
\item Write down the geometrical name of the triangle. [5]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 4 2019 Q1 [8]}}