| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Loci of complex numbers |
| Difficulty | Standard +0.3 Part (a) is a standard complex number manipulation requiring division (multiply by conjugate), conversion to modulus-argument form, and calculator workâroutine A-level technique. Part (b) involves recognizing a circle locus and finding minimum distance from origin, which requires geometric insight but is a common exam pattern. Overall slightly easier than average due to straightforward application of standard methods. |
| Spec | 4.02d Exponential form: re^(i*theta)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02f Convert between forms: cartesian and modulus-argument4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| *EITHER*: Multiply numerator and denominator by \(1 + 2i\), or equivalent, or equate to \(x + iy\), obtain two equations in \(x\) and \(y\) and solve for \(x\) or for \(y\) | M1 | |
| Obtain quotient \(-\frac{4}{5} + \frac{7}{5}i\), or equivalent | A1 | |
| Use correct method to find either \(r\) or \(\theta\) | M1 | |
| Obtain \(r = 1.61\) | A1 | |
| Obtain \(\theta = 2.09\) | A1 | |
| *OR*: Find modulus or argument of \(2 + 3i\) or of \(1 - 2i\) | B1 | |
| Use correct method to find \(r\) | M1 | |
| Obtain \(r = 1.61\) | A1 | |
| Use correct method to find \(\theta\) | M1 | |
| Obtain \(\theta = 2.09\) | A1 | |
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show a circle with centre \(3 - 2i\) | B1 | |
| Show a circle with radius \(1\) | B1ft | Centre not at the origin |
| Carry out a correct method for finding the least value of \( | z | \) |
| Obtain answer \(\sqrt{13} - 1\) | A1 | |
| Total: 4 |
## Question 8(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| *EITHER*: Multiply numerator and denominator by $1 + 2i$, or equivalent, or equate to $x + iy$, obtain two equations in $x$ and $y$ and solve for $x$ or for $y$ | M1 | |
| Obtain quotient $-\frac{4}{5} + \frac{7}{5}i$, or equivalent | A1 | |
| Use correct method to find either $r$ or $\theta$ | M1 | |
| Obtain $r = 1.61$ | A1 | |
| Obtain $\theta = 2.09$ | A1 | |
| *OR*: Find modulus or argument of $2 + 3i$ or of $1 - 2i$ | B1 | |
| Use correct method to find $r$ | M1 | |
| Obtain $r = 1.61$ | A1 | |
| Use correct method to find $\theta$ | M1 | |
| Obtain $\theta = 2.09$ | A1 | |
| **Total: 5** | | |
## Question 8(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show a circle with centre $3 - 2i$ | B1 | |
| Show a circle with radius $1$ | B1ft | Centre not at the origin |
| Carry out a correct method for finding the least value of $|z|$ | M1 | |
| Obtain answer $\sqrt{13} - 1$ | A1 | |
| **Total: 4** | | |8
\begin{enumerate}[label=(\alph*)]
\item Showing all necessary working, express the complex number $\frac { 2 + 3 \mathrm { i } } { 1 - 2 \mathrm { i } }$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$. Give the values of $r$ and $\theta$ correct to 3 significant figures.
\item On an Argand diagram sketch the locus of points representing complex numbers $z$ satisfying the equation $| z - 3 + 2 i | = 1$. Find the least value of $| z |$ for points on this locus, giving your answer in an exact form.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q8 [9]}}