The shaded region in the Argand diagram, bounded by a line and a circle, represents the complex numbers z satisfying Rez < 2 and |z-(3+i)|≤2. The point P shown on the diagram is one of the points of intersection of the line and the circle. [Figure 3.1]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

1. [1] (a) Complete method to obtain equation in y only. E.g. (2-3)² + (y−1)² = 4. Or consider triangle and state ±(√3-1).
2. [1] (a) Obtain 2+i(1-√3). Or exact equivalent in the form x + iy. Allow 2-i(√3-1). Must not be coordinates, and not x, y stated separately.
3. [1] (b) State one relevant angle, e.g. tan⁻¹(1/3) or sin⁻¹(2/√10).
4. [1] (b) Complete method to obtain the required angle. (E.g., by adding angles such as tan⁻¹(1/3) + sin⁻¹(2/√10)).
5. [1] (b) Obtain 1.01 radians or 57.7°. Accept AWRT.
6. [1] (b) Alternative Method: If y=mx is a tangent to the circle, form (x-3)²+(mx-1)² = 4 leading to a quadratic in x, then use the discriminant. This yields m = (3+2√6)/5. Must be choosing the positive root.
7. [1] (b) Alternative Method: Substitute the value of m into an appropriate expression to find the angle (e.g., θ = π/2 - sin⁻¹((9-√6)/5)).
8. [1] (b) Alternative Method: Obtain 1.01 radians or 57.7°. Accept AWRT.