| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Argument calculations and identities |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on complex numbers requiring standard techniques: sketching points on an Argand diagram, division of complex numbers, and using argument properties. Part (iii) is slightly elegant but follows directly from recognizing that arg(u/u*) = 2arg(u). All parts are routine applications of core complex number concepts with no novel problem-solving required. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.02l Geometrical effects: conjugate, addition, subtraction |
| Answer | Marks | Guidance |
|---|---|---|
| Show \(u\) and \(u*\) in relatively correct positions | B1 | |
| Show \(u + u*\) in relatively correct position | B1♦ | |
| State or imply that \(OACB\) is a parallelogram | B1♦ | |
| State or imply that \(OACB\) has a pair of adjacent equal sides [The statement that \(OACB\) is a rhombus, or equivalent, earns B2♦.] | B1♦ | 4 |
| Answer | Marks |
|---|---|
| Multiply numerator and denominator of \(\frac{u}{u*}\) by \(2 + i\) | M1 |
| Simplify numerator to \(3 + 4i\) or denominator to \(5\) | A1♦ |
| Obtain answer \(\frac{3}{5} + \frac{4}{5}i\), or equivalent | A1♦ |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain two equations in \(u\) and \(y\), and solve for \(x\) or for \(y\) | M1 | |
| Obtain \(x = \frac{2}{5}\) or \(y = \frac{3}{5}\) | A1♦ | |
| Obtain answer \(\frac{3}{5} + \frac{4}{5}i\) | A1♦ | 3 |
| Answer | Marks |
|---|---|
| State or imply \(\arg\left(\frac{u}{u*}\right) = 2\arg u\) | M1 |
| Justify the given statement correctly | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Use \(\tan 2A\) formula with \(\tan A = \frac{1}{2}\) | M1 | |
| Justify the given statement correctly | A1 | 2 |
| [The f.t. is on \(-2 + i\) as complex conjugate.] |
**(i)**
| Show $u$ and $u*$ in relatively correct positions | B1 |
| Show $u + u*$ in relatively correct position | B1♦ |
| State or imply that $OACB$ is a parallelogram | B1♦ |
| State or imply that $OACB$ has a pair of adjacent equal sides [The statement that $OACB$ is a rhombus, or equivalent, earns B2♦.] | B1♦ | 4 |
**(ii) EITHER:**
| Multiply numerator and denominator of $\frac{u}{u*}$ by $2 + i$ | M1 |
| Simplify numerator to $3 + 4i$ or denominator to $5$ | A1♦ |
| Obtain answer $\frac{3}{5} + \frac{4}{5}i$, or equivalent | A1♦ |
**OR:**
| Obtain two equations in $u$ and $y$, and solve for $x$ or for $y$ | M1 |
| Obtain $x = \frac{2}{5}$ or $y = \frac{3}{5}$ | A1♦ |
| Obtain answer $\frac{3}{5} + \frac{4}{5}i$ | A1♦ | 3 |
**(iii) EITHER:**
| State or imply $\arg\left(\frac{u}{u*}\right) = 2\arg u$ | M1 |
| Justify the given statement correctly | A1 |
**OR:**
| Use $\tan 2A$ formula with $\tan A = \frac{1}{2}$ | M1 |
| Justify the given statement correctly | A1 | 2 |
[The f.t. is on $-2 + i$ as complex conjugate.] |The complex number $2 + \mathrm{i}$ is denoted by $u$. Its complex conjugate is denoted by $u^*$.
\begin{enumerate}[label=(\roman*)]
\item Show, on a sketch of an Argand diagram with origin $O$, the points $A$, $B$ and $C$ representing the complex numbers $u$, $u^*$ and $u + u^*$ respectively. Describe in geometrical terms the relationship between the four points $O$, $A$, $B$ and $C$. [4]
\item Express $\frac{u}{u^*}$ in the form $x + \mathrm{i}y$, where $x$ and $y$ are real. [3]
\item By considering the argument of $\frac{u}{u^*}$, or otherwise, prove that
$$\tan^{-1}\left(\frac{4}{3}\right) = 2\tan^{-1}\left(\frac{1}{2}\right).$$ [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2006 Q7 [9]}}