| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Linear equations in z and z* |
| Difficulty | Standard +0.3 Part (a) is a standard linear equation in w and w* requiring substitution of w = x + iy, equating real and imaginary parts, then solving simultaneous equations—routine A-level technique. Part (b) involves sketching standard loci (circle and half-line) and finding a minimum, which requires geometric visualization but uses familiar Further Maths content with straightforward execution. |
| 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 |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute and obtain a correct equation in \(x\) and \(y\) | B1 | |
| Use \(i^2 = -1\) and equate real and imaginary parts | M1 | |
| Obtain two correct equations in \(x\) and \(y\), e.g. \(x - y = 3\) and \(3x + y = 5\) | A1 | |
| Solve and obtain answer \(z = 2 - i\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show a point representing \(2 + 2i\) | B1 | |
| Show a circle with radius 1 and centre not at the origin | B1 FT | FT is on the point representing the centre |
| Show the correct half line from \(4i\) | B1 | |
| Shade the correct region | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out a complete method for finding the least value of \(\text{Im}\, z\) | M1 | |
| Obtain answer \(2 - \frac{1}{2}\sqrt{2}\), or exact equivalent | A1 |
## Question 8(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute and obtain a correct equation in $x$ and $y$ | B1 | |
| Use $i^2 = -1$ and equate real and imaginary parts | M1 | |
| Obtain two correct equations in $x$ and $y$, e.g. $x - y = 3$ and $3x + y = 5$ | A1 | |
| Solve and obtain answer $z = 2 - i$ | A1 | |
---
## Question 8(b)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show a point representing $2 + 2i$ | B1 | |
| Show a circle with radius 1 and centre not at the origin | B1 FT | FT is on the point representing the centre |
| Show the correct half line from $4i$ | B1 | |
| Shade the correct region | B1 | |
---
## Question 8(b)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out a complete method for finding the least value of $\text{Im}\, z$ | M1 | |
| Obtain answer $2 - \frac{1}{2}\sqrt{2}$, or exact equivalent | A1 | |8
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $( 1 + 2 \mathrm { i } ) w + \mathrm { i } w ^ { * } = 3 + 5 \mathrm { i }$. Give your answer 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 $z$ satisfying the inequalities $| z - 2 - 2 \mathrm { i } | \leqslant 1$ and $\arg ( z - 4 \mathrm { i } ) \geqslant - \frac { 1 } { 4 } \pi$.
\item Find the least value of $\operatorname { Im } z$ for points in this region, giving your answer in an exact form.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q8 [10]}}