A2 June 2022 Paper 1 Q3

OCR ACurrent spec11 marksComplex Numbers

3 In this question you must show detailed reasoning.

(a) Find the roots of the equation \(2z^2 - 2z + 5 = 0\). [2]

The loci \(\mathrm{C}_1\) and \(\mathrm{C}_2\) are given by \(|z| = |z - 2\mathrm{i}|\) and \(|z - 2| = \sqrt{5}\) respectively.

(b)

(i) Sketch on a single Argand diagram the loci \(\mathrm{C}_1\) and \(\mathrm{C}_2\), showing any intercepts with the imaginary axis. [3]

(ii) Indicate, by shading on your Argand diagram, the region
\(\{z : |z| \leqslant |z - 2\mathrm{i}|\} \cap \{z : |z - 2| \leqslant \sqrt{5}\}\). [1]

(c)

(i) Show that both of the roots of the equation \(2z^2 - 2z + 5 = 0\) satisfy \(|z - 2| \lt \sqrt{5}\). [2]

(ii) State, with a reason, which root of the equation \(2z^2 - 2z + 5 = 0\) satisfies \(|z| \lt |z - 2\mathrm{i}|\). [1]

(d) On the same Argand diagram as part (b), indicate the positions of the roots of the equation \(2z^2 - 2z + 5 = 0\). [2]

Mark scheme (a)

Scheme	Marks	AO
DR \(z = \dfrac{2 \pm \sqrt{-36}}{4} \Rightarrow z = \dfrac{1}{2} \pm \dfrac{3}{2}\mathrm{i}\) or \(2\left(z^2 - z\right) = -5 \Rightarrow 2\left(z^2 - z + \dfrac{1}{4}\right) = -\dfrac{9}{2}\) \(\left(z - \dfrac{1}{2}\right)^2 = -\dfrac{9}{4} \Rightarrow z - \dfrac{1}{2} = \pm\dfrac{3}{2}\mathrm{i} \Rightarrow z = \dfrac{1}{2} \pm \dfrac{3}{2}\mathrm{i}\)	M1 A1	1.1 1.1
	[2]

Notes

M1: Using correct quadratic formula with correct substitutions
Or completing the square (Root of negative number must be seen)

A1: Final answer

Argand diagram on a grid: horizontal line Im(z) = 1 and a circle centre 2 radius root 5, crossing the imaginary axis at 1 and -1 — Mark scheme (b)

Scheme	Marks	AO
	B1 B1 B1	1.1 1.1 2.2a
	[3]

Scheme	Marks	AO
(ii) Major segment (area below line and inside circle) shaded.	B1	1.1
	[1]

Mark scheme (c)

Scheme	Marks	AO
(i) DR \(\|z - 2\| = \left\|-\dfrac{3}{2} \pm \dfrac{3}{2}\mathrm{i}\right\| = \sqrt{\left(-\dfrac{3}{2}\right)^2 + \left(\pm\dfrac{3}{2}\right)^2}\)	M1	2.1
\(= \sqrt{\dfrac{9}{2}} \lt \left(\sqrt{\dfrac{10}{2}}\right) = \sqrt{5}\)	A1	2.2a
	[2]

Notes

M1: ft For calculating modulus of their \(|z - 2|\) for at least one of their values of \(z\).

A1: AG. Must see clear statement of inequality (eg as shown or by values). Both roots
If shown that \(|z - 2|^2 \lt 5\), must explain that \(|z - 2| \lt \sqrt{5}\) follows from \(|z - 2| \geqslant 0\).

Scheme	Marks	AO
(ii) \(\dfrac{1}{2} - \dfrac{3}{2}\mathrm{i}\) because \(\operatorname{Im}\left(\dfrac{1}{2} - \dfrac{3}{2}\mathrm{i}\right) = -\dfrac{3}{2} \lt 1\)	B1	2.2a
	[1]

Notes

(ii) B1: Or \(\left|\dfrac{1}{2} - \dfrac{3}{2}\mathrm{i}\right| = \sqrt{\dfrac{5}{2}} \lt \left|\dfrac{1}{2} - \dfrac{3}{2}\mathrm{i} - 2\mathrm{i}\right| = \dfrac{5\sqrt{2}}{2}\) ft their roots as long as their roots are conjugate pairs with imaginary part of magnitude greater than one.
Could be explained in terms of loci (ie “because (the point representing) \(\frac{1}{2} - \frac{3}{2}\mathrm{i}\) is closer to \(O\) than it is to (the point representing) \(2\mathrm{i}\)”) but must be consistent with their diagram after (d).
Or “below the perpendicular bisector” or “below axis”
isw after acceptable answer