Moderate -0.3 This is a straightforward loci question requiring students to sketch a disc (circle with interior) centered at -1+i with radius 1, and a half-plane defined by arg(z-1) ≤ 3π/4, then shade their intersection. While it requires understanding of modulus and argument loci, these are standard A-level techniques with no problem-solving or novel insight needed—slightly easier than average due to being purely procedural.
2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z + 1 - i | \leqslant 1\) and \(\arg ( z - 1 ) \leqslant \frac { 3 } { 4 } \pi\).
Need some indication of scale or a correct label. Could just be mark(s) on the axes
Show a circle with radius 1 and centre not at the origin (or relevant part thereof)
B1
Show correct half line from 1 (or relevant part thereof)
B1
Shade the correct region on a correct diagram
B1
N.B. If they have very different scales on *their* 2 axes the diagram must match *their* scale - the 'circle' should be an ellipse. Allow freehand diagrams with clear correct intention.
## Question 2:
| Answer | Mark | Guidance |
|--------|------|----------|
| Show a circle with centre $-1 + i$ | B1 | Need some indication of scale or a correct label. Could just be mark(s) on the axes |
| Show a circle with radius 1 and centre not at the origin (or relevant part thereof) | B1 | |
| Show correct half line from 1 (or relevant part thereof) | B1 | |
| Shade the correct region on a correct diagram | B1 | N.B. If they have very different scales on *their* 2 axes the diagram must match *their* scale - the 'circle' should be an ellipse. Allow freehand diagrams with clear correct intention. |
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2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $| z + 1 - i | \leqslant 1$ and $\arg ( z - 1 ) \leqslant \frac { 3 } { 4 } \pi$.
\hfill \mbox{\textit{CAIE P3 2021 Q2 [4]}}