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9709 P31 - Nov 2023 - Q2
1972
On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(|z - 2i| \leq |z + 2 - i|\) and \(0 \leq \arg(z + 1) \leq \frac{1}{4}\pi\).
Solution
1. Identify the points \(2i\) and \(-2+i\) on the Argand diagram.
2. The inequality \(|z - 2i| \leq |z + 2 - i|\) represents the region on or above the perpendicular bisector of the line segment joining \(2i\) and \(-2+i\).
3. The perpendicular bisector can be found by calculating the midpoint of \(2i\) and \(-2+i\), which is \(0 + \frac{1}{2}i\), and drawing a line perpendicular to the segment.
4. The inequality \(0 \leq \arg(z + 1) \leq \frac{1}{4}\pi\) represents the region within the angle formed by the positive real axis and the line with gradient 1 passing through \(-1, 0\).
5. Shade the region that satisfies both conditions: above the perpendicular bisector and within the specified angle.
