A region, a boundary and membership

Choose your method, connect the parts and explain your conclusions. These original questions include written reasoning and sketches as well as exact answer checks.

Suggested marks guide how much working to show. The site does not automatically award examination marks for proofs, diagrams or methods. Use scaffolded fluency practice when you need a hint first.

A-level core. Course filters use selected objective mappings; none of these tasks establishes complete paper coverage.

Before you start

Complex loci. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

S = {z : |z − 3| ≤ 3, Im(z) ≥ 0}.

Argand axes: horizontal real part, vertical imaginary part.

Part a

Sketch and shade S on an Argand diagram, identifying its boundaries.

Part b

Does z = 3 + 3i belong to S?

Part c

Explain how the diagram changes if ≤ is replaced by <.

Model solution and review criteria

Part a

Shade the closed upper half-disc centred at (3,0) with radius 3. Include its semicircular arc and real-axis diameter from 0 to 6.

Use a circle centred on the real axis.
Shade only its upper half.
Include both boundaries because the inequalities are non-strict.

Part b

Its modulus relative to the centre is exactly 3, and its imaginary part is positive.

Check both conditions.

Part c

Exclude the circular arc. Points of the diameter strictly inside the disc still satisfy Im(z) ≥ 0 and remain included.

Remove only the affected circle boundary.
Keep the admissible diameter interior.

Compare your own reasoning. No automatic examination marks are awarded.