| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 Part (a) is routine complex division using conjugate multiplication. Part (b)(i) requires identifying standard loci (circle and perpendicular bisector) and shading their intersection—straightforward geometric interpretation. Part (b)(ii) involves finding the argument at a boundary point, requiring some coordinate geometry but following standard methods. Overall slightly easier than average due to predictable techniques. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| EITHER: Multiply numerator and denominator by \(1 - 4i\), or equivalent, and use \(i^2 = -1\) | M1 | |
| Simplify numerator to \(-17 - 17i\), or denominator to 17 | A1 | |
| Obtain final answer \(-1 - i\) | A1 | |
| OR: Using \(i^2 = -1\), obtain two equations in \(x\) and \(y\), and solve for \(x\) or \(y\) | M1 | |
| Obtain \(x = -1\) or \(y = -1\), or equivalent | A1 | |
| Obtain final answer \(-1 - i\) | A1 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Show a point representing \(2 + i\) in relatively correct position | B1 | |
| Show a circle with centre \(2 + i\) and radius 1 | B1\(\sqrt{}\) | |
| Show the perpendicular bisector of the line segment joining \(i\) and 2 | B1 | |
| Shade the correct region | B1 | 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply that the angle between the tangents from the origin to the circle is required | M1 | |
| Obtain answer 0.927 radians (or 53.1°) | A1 | 2 marks |
**(a)**
**EITHER:** Multiply numerator and denominator by $1 - 4i$, or equivalent, and use $i^2 = -1$ | M1 |
Simplify numerator to $-17 - 17i$, or denominator to 17 | A1 |
Obtain final answer $-1 - i$ | A1 |
**OR:** Using $i^2 = -1$, obtain two equations in $x$ and $y$, and solve for $x$ or $y$ | M1 |
Obtain $x = -1$ or $y = -1$, or equivalent | A1 |
Obtain final answer $-1 - i$ | A1 | 3 marks
**(b)(i)**
Show a point representing $2 + i$ in relatively correct position | B1 |
Show a circle with centre $2 + i$ and radius 1 | B1$\sqrt{}$ |
Show the perpendicular bisector of the line segment joining $i$ and 2 | B1 |
Shade the correct region | B1 | 4 marks
**(b)(ii)**
State or imply that the angle between the tangents from the origin to the circle is required | M1 |
Obtain answer 0.927 radians (or 53.1°) | A1 | 2 marks7
\begin{enumerate}[label=(\alph*)]
\item The complex number $\frac { 3 - 5 \mathrm { i } } { 1 + 4 \mathrm { i } }$ is denoted by $u$. Showing your working, express $u$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\item \begin{enumerate}[label=(\roman*)]
\item On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities $| z - 2 - \mathrm { i } | \leqslant 1$ and $| z - \mathrm { i } | \leqslant | z - 2 |$.
\item Calculate the maximum value of $\arg z$ for points lying in the shaded region.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2014 Q7 [9]}}