| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Geometric relationships on Argand diagram |
| Difficulty | Standard +0.8 Part (a) is a standard simultaneous equations problem with complex numbers requiring algebraic manipulation. Part (b) requires geometric insight on the Argand diagram—constructing a point C such that BC is perpendicular to AB with specific length ratio, then translating this geometric relationship into complex number operations (likely rotation and scaling). The combination of algebraic technique, geometric visualization, and the no-calculator constraint with exact surds makes this moderately challenging but still within standard Further Maths territory. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.02l Geometrical effects: conjugate, addition, subtraction |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Solve for \(z\) or for \(w\) | M1 | |
| Use \(i^2 = -1\) | M1 | |
| Obtain \(w = \frac{i}{2-i}\) or \(z = \frac{2+i}{2-i}\) | A1 | |
| Multiply numerator and denominator by the conjugate of the denominator | M1 | |
| Obtain \(w = -\frac{1}{5} + \frac{2}{5}i\) | A1 | |
| Obtain \(z = \frac{3}{5} + \frac{4}{5}i\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| EITHER: Find \(\pm\left[2 + (2-2\sqrt{3})i\right]\) | B1 | |
| Multiply by \(2i\) (or \(-2i\)) | M1* | |
| Add result to \(v\) | DM1 | |
| Obtain answer \(4\sqrt{3} - 1 + 6i\) | A1 | |
| OR: State \(\frac{z-v}{v-u} = ki\), or equivalent | M1 | |
| State \(k = 2\) | A1 | |
| Substitute and solve for \(z\) even if \(i\) omitted | M1 | |
| Obtain answer \(4\sqrt{3} - 1 + 6i\) | A1 |
## Question 11(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Solve for $z$ or for $w$ | M1 | |
| Use $i^2 = -1$ | M1 | |
| Obtain $w = \frac{i}{2-i}$ or $z = \frac{2+i}{2-i}$ | A1 | |
| Multiply numerator and denominator by the conjugate of the denominator | M1 | |
| Obtain $w = -\frac{1}{5} + \frac{2}{5}i$ | A1 | |
| Obtain $z = \frac{3}{5} + \frac{4}{5}i$ | A1 | |
**Total: 6**
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## Question 11(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| **EITHER:** Find $\pm\left[2 + (2-2\sqrt{3})i\right]$ | B1 | |
| Multiply by $2i$ (or $-2i$) | M1* | |
| Add result to $v$ | DM1 | |
| Obtain answer $4\sqrt{3} - 1 + 6i$ | A1 | |
| **OR:** State $\frac{z-v}{v-u} = ki$, or equivalent | M1 | |
| State $k = 2$ | A1 | |
| Substitute and solve for $z$ even if $i$ omitted | M1 | |
| Obtain answer $4\sqrt{3} - 1 + 6i$ | A1 | |
**Total: 4**11 Throughout this question the use of a calculator is not permitted.
\begin{enumerate}[label=(\alph*)]
\item The complex numbers $z$ and $w$ satisfy the equations
$$z + ( 1 + \mathrm { i } ) w = \mathrm { i } \quad \text { and } \quad ( 1 - \mathrm { i } ) z + \mathrm { i } w = 1$$
Solve the equations for $z$ and $w$, giving your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\item The complex numbers $u$ and $v$ are given by $u = 1 + ( 2 \sqrt { 3 } ) \mathrm { i }$ and $v = 3 + 2 \mathrm { i }$. In an Argand diagram, $u$ and $v$ are represented by the points $A$ and $B$. A third point $C$ lies in the first quadrant and is such that $B C = 2 A B$ and angle $A B C = 90 ^ { \circ }$. Find the complex number $z$ represented by $C$, giving your answer in the form $x + \mathrm { i } y$, where $x$ and $y$ are real and exact.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2017 Q11 [10]}}