Moderate -0.3 This question requires identifying two standard loci (perpendicular bisector and circle) and shading their intersection. While it involves multiple inequalities and geometric interpretation on the Argand diagram, the individual components are routine A-level techniques with no novel problem-solving required. The perpendicular bisector inequality and circle are both standard P3 content, making this slightly easier than average.
2 On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 1 + 2 i | \leqslant | z |\) and \(| z - 2 | \leqslant 1\).
FT centre not at the origin even if centre at \(1-2i\). Must clearly go through \((1,0)\) or \((3,0)\) (oe for FT mark).
Show the point representing \(1-2i\)
B1
Can be implied by correct perpendicular bisector
Show the perpendicular bisector of the line joining \(1-2i\) and the origin. Perpendicular to \(OP\) by eye and at midpoint of \(OP\) by eye sufficient. Must reach midpoint of \(OP\) and if extended will cut \(BE\).
B1 FT
FT on the position of \(1-2i\)
Shade the correct region. Dependent on all previous marks, except in case 3 below, and the perpendicular must cut axes between \(CF\) and \(BE\), but not actually through \(C\) or \(F\) and not through \(B\) or \(E\). Scale can be implied by dashes.
B1
1 Scale only on \(y\)-axis and \(2OA = OC\)
B1, B1FT, B1, B1FT, B1
2 Scale only on \(x\)-axis and \(2OB = OE\)
B1, B1FT, B1, B1FT, B1
3 No scale on either axis, but \(2OA = OC\) then \(2OB = OE\)
B0, B1FT, B0, B1FT, B1
5
## Question 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Show a circle centre $(2, 0)$ | B1 | |
| Show the relevant part of a circle with radius 1 | B1 FT | FT centre not at the origin even if centre at $1-2i$. Must clearly go through $(1,0)$ or $(3,0)$ (oe for FT mark). |
| Show the point representing $1-2i$ | B1 | Can be implied by correct perpendicular bisector |
| Show the perpendicular bisector of the line joining $1-2i$ and the origin. Perpendicular to $OP$ by eye and at midpoint of $OP$ by eye sufficient. Must reach midpoint of $OP$ and if extended will cut $BE$. | B1 FT | FT on the position of $1-2i$ |
| Shade the correct region. Dependent on all previous marks, except in case 3 below, and the perpendicular must cut axes between $CF$ and $BE$, but not actually through $C$ or $F$ and not through $B$ or $E$. Scale can be implied by dashes. | B1 | |
| 1 Scale only on $y$-axis and $2OA = OC$ | B1, B1FT, B1, B1FT, B1 | |
| 2 Scale only on $x$-axis and $2OB = OE$ | B1, B1FT, B1, B1FT, B1 | |
| 3 No scale on either axis, but $2OA = OC$ then $2OB = OE$ | B0, B1FT, B0, B1FT, B1 | |
| | **5** | |
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2 On an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $| z - 1 + 2 i | \leqslant | z |$ and $| z - 2 | \leqslant 1$.
\hfill \mbox{\textit{CAIE P3 2023 Q2 [5]}}