| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Topic | Complex Numbers Argand & Loci |
| Type | De Moivre's theorem applications |
| Difficulty | Standard +0.8 This is a substantial multi-part question requiring modulus/argument calculation with exact values (including use of a given tan result), finding cube roots using De Moivre's theorem, converting between forms, geometric interpretation on Argand diagram, and a final part requiring insight about rotation/scaling to move a triangle. While systematic, it demands careful execution across multiple techniques and geometric understanding, placing it moderately above average difficulty. |
| 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 numbers4.02s Roots of unity: geometric applications |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \( | w | = \sqrt{(\sqrt{3}-1)^2+(\sqrt{3}+1)} = \sqrt{4-2\sqrt{3}+4+2\sqrt{3}} = \sqrt{8}\) or \(2\sqrt{2}\) M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sqrt[3]{ | w | }\); \(\frac{\arg(w)}{3}\) These method marks can be earned for just the first root M1M1 |
**(i)** $|w| = \sqrt{(\sqrt{3}-1)^2+(\sqrt{3}+1)} = \sqrt{4-2\sqrt{3}+4+2\sqrt{3}} = \sqrt{8}$ or $2\sqrt{2}$ **M1 A1**
$\arg(w) = \tan^{-1}\!\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\times\frac{\sqrt{3}+1}{\sqrt{3}+1}\right) = \tan^{-1}(2+\sqrt{3}) = \frac{5}{12}\pi$ **M1 A1**
**[4]**
**(ii)(a)** $z^3 = \left(2\sqrt{2},\,\frac{5}{12}\pi\right),\;\left(2\sqrt{2},\,\frac{29}{12}\pi\right),\;\left(2\sqrt{2},\,-\frac{19}{12}\pi\right)$ or $\frac{53}{12}\pi$
$\sqrt[3]{|w|}$; $\frac{\arg(w)}{3}$ These method marks can be earned for just the first root **M1M1**
$\Rightarrow z = \left(\sqrt{2},\,\frac{5}{36}\pi\right),\;\left(\sqrt{2},\,\frac{29}{36}\pi\right),\;\left(\sqrt{2},\,-\frac{19}{36}\pi\right)$ **A** marks for the 2nd & 3rd roots; $r\,\mathrm{e}^{\wedge}(\mathrm{i}\theta)$ forms equally acceptable **A1 A1**
**[4]**
**(b)** $z_1, z_2, z_3$ the roots of $z^3 - 0\cdot z^2 + 0\cdot z - w = 0$ **M1**
$\Rightarrow z_1z_2z_3 = w = (\sqrt{3}-1)+\mathrm{i}(\sqrt{3}+1)$ **A1**
**ALT.** Multiplying the 3 roots together in any form
**[2]**
**(c)** Three points in approx. correct places **M1**
All equally spaced around a circle, centre $O$, radius $\sqrt{2}$ (Explained that $\Delta_1$ equilateral) **M1 A1**
$l = 2\times\sqrt{2}\sin\!\left(\frac{1}{2}\times\frac{2}{3}\pi\right) = \sqrt{6}$ **M1**
or by the *Cosine Rule* **A1**
**[5]**
**(d)** $k = \exp\!\left\{-\mathrm{i}\cdot\frac{5}{36}\pi\right\}$ or $\exp\!\left\{-\mathrm{i}\cdot\frac{29}{36}\pi\right\}$ or $\exp\!\left\{\mathrm{i}\cdot\frac{19}{36}\pi\right\}$ **B1**
**[1]**11 The complex number $w = ( \sqrt { 3 } - 1 ) + \mathrm { i } ( \sqrt { 3 } + 1 )$.
\begin{enumerate}[label=(\roman*)]
\item Determine, showing full working, the exact values of $| w |$ and $\arg w$.\\[0pt]
[You may use the result that $\tan \left( \frac { 5 } { 12 } \pi \right) = 2 + \sqrt { 3 }$.]
\item (a) Find, in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, the three roots, $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$, of the equation $z ^ { 3 } = w$.\\
(b) Determine $z _ { 1 } z _ { 2 } z _ { 3 }$ in the form $a + \mathrm { i } b$.\\
(c) Mark the points representing $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$ on a sketch of the Argand diagram. Show that they form an equilateral triangle, $\Delta _ { 1 }$, and determine the side-length of $\Delta _ { 1 }$.\\
(d) The points representing $k z _ { 1 } , k z _ { 2 }$ and $k z _ { 3 }$ form $\Delta _ { 2 }$, an equilateral triangle which is congruent to $\Delta _ { 1 }$, and one of whose vertices lies on the positive real axis. Write down a suitable value for the complex constant $k$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2012 Q11 [11]}}