| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Equations with z/z* or zz* terms |
| Difficulty | Standard +0.3 Part (a) requires manipulation of z/z* which is a standard technique (multiply by z*, use |z|²) but needs careful algebraic handling. Part (b) involves routine loci sketching and finding an intersection point with basic geometry. The question tests standard P3/Further Pure content without requiring novel insight—slightly easier than average due to straightforward methods throughout. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e 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 horizontal equation in \(x\) and \(y\) in any form | B1 | \(zz^*+iz-2z^*=0 \Rightarrow x^2+y^2+ix-y-2x+2iy=0\). Allow if still includes brackets and/or \(i^2\) |
| Use \(i^2=-1\) and equate real and imaginary parts to zero OE | *M1 | For their horizontal equation |
| Obtain two correct equations e.g. \(x^2+y^2-y-2x=0\) and \(x+2y=0\) | A1 | Allow \(ix+2iy=0\) |
| Solve for \(x\) or for \(y\) | DM1 | |
| Obtain answer \(\frac{6}{5}-\frac{3}{5}i\) and no other | A1 | OE, condone \(\frac{1}{5}(6-3i)\) |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show a circle with centre \(2i\) and radius \(2\) | B1 | |
| Show horizontal line \(y=3\) – in first and second quadrant | B1 | SC: For clearly labelled axes not in the conventional directions, allow B1 for a fully 'correct' diagram |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out a complete method for finding the argument (not by measuring the sketch) | M1 | \(\left(z=\sqrt{3}+3i\right)\). Must show working if using \(1.7\) in place of \(\sqrt{3}\) |
| Obtain answer \(\frac{1}{3}\pi\) (or \(60°\)) | A1 | SC: Allow B2 for \(60°\) with no working |
| Total | 2 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute and obtain a correct horizontal equation in $x$ and $y$ in any form | B1 | $zz^*+iz-2z^*=0 \Rightarrow x^2+y^2+ix-y-2x+2iy=0$. Allow if still includes brackets and/or $i^2$ |
| Use $i^2=-1$ and equate real and imaginary parts to zero OE | *M1 | For their horizontal equation |
| Obtain two correct equations e.g. $x^2+y^2-y-2x=0$ and $x+2y=0$ | A1 | Allow $ix+2iy=0$ |
| Solve for $x$ or for $y$ | DM1 | |
| Obtain answer $\frac{6}{5}-\frac{3}{5}i$ and no other | A1 | OE, condone $\frac{1}{5}(6-3i)$ |
| **Total** | **5** | |
---
## Question 7(b)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show a circle with centre $2i$ and radius $2$ | B1 | |
| Show horizontal line $y=3$ – in first and second quadrant | B1 | SC: For clearly labelled axes not in the conventional directions, allow B1 for a fully 'correct' diagram |
| **Total** | **2** | |
---
## Question 7(b)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out a complete method for finding the argument (not by measuring the sketch) | M1 | $\left(z=\sqrt{3}+3i\right)$. Must show working if using $1.7$ in place of $\sqrt{3}$ |
| Obtain answer $\frac{1}{3}\pi$ (or $60°$) | A1 | SC: Allow B2 for $60°$ with no working |
| **Total** | **2** | |
---7
\begin{enumerate}[label=(\alph*)]
\item Find the complex number $z$ satisfying the equation
$$z + \frac { \mathrm { i } z } { z ^ { * } } - 2 = 0$$
where $z ^ { * }$ denotes the complex conjugate of $z$. Give your answer in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\item \begin{enumerate}[label=(\roman*)]
\item On a single Argand diagram sketch the loci given by the equations $| z - 2 \mathrm { i } | = 2$ and $\operatorname { Im } z = 3$, where $\operatorname { Im } z$ denotes the imaginary part of $z$.
\item In the first quadrant the two loci intersect at the point $P$. Find the exact argument of the complex number represented by $P$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2019 Q7 [9]}}