| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2003 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Geometric relationships on Argand diagram |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question testing basic complex number operations (squaring, reciprocal, modulus) and a simple similarity proof. All parts are routine applications of standard techniques with no novel insight required, though the similarity proof adds slight challenge beyond pure computation. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z) |
| Answer | Marks | Guidance |
|---|---|---|
| \(z^2 = (3 - 3i)(3 - 3i) = -18i\) | M1 A1 | (2) |
| (b) \(\frac{1}{z} = \frac{(3+3i)}{(3-3i)(3+3i)} = \frac{3+3i}{18} = \frac{1+i}{6}\) | M1 A1 | (2) |
| (c) \( | z | = \sqrt{(9+9)} = \sqrt{18} = 3\sqrt{2}\) |
| \( | z | = 18\) |
| \(\left | \frac{1}{z}\right | = \sqrt{\frac{1}{18}} = \frac{1}{3\sqrt{2}} = \frac{\sqrt{2}}{6}\) |
| (d) Two correct | B1 | (2) |
| Four correct | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(OB = 18, \quad OA = \frac{3\sqrt{2}}{\sqrt{2}/6} = 18\) | M1 A1 | |
| \(\angle AOB = \angle COD = 45 \therefore\) similar | B1 | (3) |
| (11 marks) |
$z^2 = (3 - 3i)(3 - 3i) = -18i$ | M1 A1 | (2) |
**(b)** $\frac{1}{z} = \frac{(3+3i)}{(3-3i)(3+3i)} = \frac{3+3i}{18} = \frac{1+i}{6}$ | M1 A1 | (2) |
**(c)** $|z| = \sqrt{(9+9)} = \sqrt{18} = 3\sqrt{2}$ | |
$|z| = 18$ | two correct | M1 |
$\left|\frac{1}{z}\right| = \sqrt{\frac{1}{18}} = \frac{1}{3\sqrt{2}} = \frac{\sqrt{2}}{6}$ | all three correct | A1 | (2) |
**(d)** Two correct | B1 | (2) |
Four correct | B1 |
| |
$OB = 18, \quad OA = \frac{3\sqrt{2}}{\sqrt{2}/6} = 18$ | M1 A1 |
$\angle AOB = \angle COD = 45 \therefore$ similar | B1 | (3) |
| | (11 marks) |
---13. Given that $z = 3 - 3 i$ express, in the form $a + i b$, where $a$ and $b$ are real numbers,
\begin{enumerate}[label=(\alph*)]
\item $z ^ { 2 }$,\\
(2)
\item $\frac { 1 } { z }$.\\
(2)
\item Find the exact value of each of $| z | , \left| z ^ { 2 } \right|$ and $\left| \frac { 1 } { z } \right|$.\\
(2)
The complex numbers $z , z ^ { 2 }$ and $\frac { 1 } { z }$ are represented by the points $A , B$ and $C$ respectively on an Argand diagram. The real number 1 is represented by the point $D$, and $O$ is the origin.
\item Show the points $A , B , C$ and $D$ on an Argand diagram.
\item Prove that $\triangle O A B$ is similar to $\triangle O C D$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2003 Q13 [11]}}