| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Challenging +1.8 This question requires students to interpret geometric loci in the complex plane (a circle and a ray), then determine when they intersect at exactly two points. It involves visualizing the geometry, finding the perpendicular distance from the circle's center to the ray using coordinate geometry, and solving an inequality involving surds. While the individual components are standard Further Maths topics, synthesizing them into a complete solution with exact values requires solid geometric insight and careful algebraic manipulation across multiple steps. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
Given that there are two distinct complex numbers $z$ that satisfy
$$\{z: |z - 3 - 5i| = 2r\} \cap \left\{z: \arg(z - 2) = \frac{3\pi}{4}\right\}$$
determine the exact range of values for the real constant $r$.
[7]
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q16 [7]}}