AS June 2022 Paper 1 Q5

OCR MEICurrent spec5 marksComplex Numbers

5 An Argand diagram is shown below. The circle has centre at the point representing \(1 + 3\mathrm{i}\), and the half line intersects the circle at the origin and at the point representing \(4 + 4\mathrm{i}\).

Argand diagram: a circle with centre 1 + 3i passing through the origin O, and a half line from O at 45 degrees meeting the circle again at 4 + 4i; the segment of the circle below the half line (between the half line and the arc, crossing just below the real axis) is shaded

State the two conditions that define the set of complex numbers represented by points in the shaded segment, including its boundaries. [5]

Mark scheme

Scheme	Marks	AO
\(\|z - 1 - 3\mathrm{i}\| \leqslant \sqrt{10}\)	M1 B1 A1	2.5 1.1 1.1
\(\arg(z) \leqslant \pi/4\)	M1 A1	1.1 1.1
	[5]

Notes

M1: (1st) circle of form \(|z - a| = b\) used

B1: radius \(\sqrt{10}\) soi

A1: (1st) all correct (must be \(\leqslant\))

M1: (2nd) half line \(\arg(z) = a\) used

A1: (2nd) all correct, condone \(\arg(z) \leqslant 45^\circ\)
Accept alternatives, e.g. \(\mathrm{Re}(z) \geqslant \mathrm{Im}(z)\), or \(|z - 1| \leqslant |z - \mathrm{i}|\)