| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Optimization of modulus on loci |
| Difficulty | Standard +0.3 This is a straightforward locus question requiring recognition that the equation represents a circle with center (-3, 2) and radius 2, then finding the minimum distance from the origin using the formula |center| - radius. It involves standard techniques with minimal problem-solving demand, making it slightly easier than average. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show a circle with centre \(-3 + 2i\) | B1 | Allow curved figure with centre in roughly correct position. Accept marks or numbers on axes. B0B1 available for axes the wrong way round (and M1 A1 in part (b)) |
| Show a circle with radius 2 | B1 FT | FT centre not at the origin. Allow 'near miss' on \(x\) axis. Different scales on axes require an ellipse for B1 B1. Scales on axes and any label of radius must be consistent for B1 B1. Correct circle shaded scores B1 B0 |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out a correct method for finding the least value of \( | z | \) |
| Obtain answer \(\sqrt{13} - 2\) or \(\sqrt{17 - 4\sqrt{13}}\) | A1 | Or exact equivalent e.g. \(\sqrt{17 - \frac{26}{3}\sqrt{\frac{36}{13}}}\). Correct solution only. Allow A1 if exact answer seen and then decimal given |
| Total: 2 |
## Question 3(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show a circle with centre $-3 + 2i$ | B1 | Allow curved figure with centre in roughly correct position. Accept marks or numbers on axes. B0B1 available for axes the wrong way round (and M1 A1 in part (b)) |
| Show a circle with radius 2 | B1 FT | FT centre not at the origin. Allow 'near miss' on $x$ axis. Different scales on axes require an ellipse for B1 B1. Scales on axes and any label of radius must be consistent for B1 B1. Correct circle shaded scores B1 B0 |
| **Total: 2** | | |
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## Question 3(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out a correct method for finding the least value of $|z|$ | M1 | e.g. distance of centre from origin $-$ radius or find point of intersection of circle and $3y = -2x$ and use Pythagoras. If they subtract the wrong way round M0. If diagram is a reflection or rotation of correct diagram, M1 A1 available (requires equivalent work). Any other circle M0 |
| Obtain answer $\sqrt{13} - 2$ or $\sqrt{17 - 4\sqrt{13}}$ | A1 | Or exact equivalent e.g. $\sqrt{17 - \frac{26}{3}\sqrt{\frac{36}{13}}}$. Correct solution only. Allow A1 if exact answer seen and then decimal given |
| **Total: 2** | | |
---3
\begin{enumerate}[label=(\alph*)]
\item On an Argand diagram, sketch the locus of points representing complex numbers $z$ satisfying $| z + 3 - 2 \mathrm { i } | = 2$.
\item 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 2023 Q3 [4]}}