| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| Topic | Complex numbers 2 |
| Type | Direct nth roots: roots with geometric or algebraic follow-up |
| Difficulty | Standard +0.3 This is a standard complex numbers question requiring conversion to polar form, application of De Moivre's theorem for cube roots, and basic geometric calculation. While it involves multiple steps (converting 2+2i to polar form, finding three cube roots, sketching, and calculating area), these are all routine techniques for Further Maths students with no novel insight required. The exact multiple of π requirement and area calculation add slight complexity but remain within standard textbook exercises. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| \( | z^3 | = 2\sqrt{2}\) and \(\arg(z^3) = \frac{1}{4}\pi\) B1B1 |
**Question 7** [6+3 marks]
**(i)**
$|z^3| = 2\sqrt{2}$ and $\arg(z^3) = \frac{1}{4}\pi$ B1B1
$\Rightarrow z = \left(\sqrt{2},\ \frac{1}{12}\pi\right)$ cube-rooting modulus; arg $\div 3$ M1M1 (in at least the first case)
Other two roots: $\left(\sqrt{2},\ \frac{3}{4}\pi\right)$ and $\left(\sqrt{2},\ \frac{12}{12}\pi\right)$ A1A1
**[6]**
**(ii)**
Equilateral $\triangle$ with vertices in approx. correct places B1
Area $= 3 \times \frac{1}{2} \times \sqrt{2} \times \sqrt{2} \times \sin\left(\frac{2}{3}\pi\right) = \frac{3}{2}\sqrt{3}$ M1A1 (Give M1 for any correct area)
Accept **awrt** 2.60 (3 s.f.) from correct working
**[3]**7 (i) Find all values of $z$ for which $z ^ { 3 } = 2 + 2 \mathrm { i }$. Give your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $\theta$ is an exact multiple of $\pi$ in the interval $0 < \theta < 2 \pi$.\\
(ii) The vertices of a triangle in the Argand diagram correspond to the three roots of the equation $z ^ { 3 } = 2 + 2 \mathrm { i }$. Sketch the triangle and determine its area.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2016 Q7 [9]}}