4 (a) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z − 4 − 3i| ≤ 2 and Rez ≤ 3. (b) Find the greatest value of arg z for points in this region.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. B1 Show a circle with centre 4+3i. Accept a curved shape with correct point roughly in the middle.
2. B1 Show a circle with radius 2 and centre not at the origin. The shape should be consistent with their scales
3. B1 Show correct vertical line. Enough to meet correct circle twice or complete line for any other circle.
4. B1 Shade the correct region on a correct diagram Any other shading must be accompanied by words to explain which region is required
5. M1 Carry out a complete method for finding the greatest value of arg(z) e.g tan⁻¹(3/4)+sin⁻¹(2/√25) (0.6435+0.4115)
6. A1 Obtain answer 1.06 (accept 1.055 or 1.056) radians or 60.45° (accept 60.4° or 60.5°)