Question 3 - A-Level Maths

Exam Board	OCR
Module	Further Pure Core 1 (Further Pure Core 1)
Year	2022
Session	June
Topic	Complex Numbers Argand & Loci

3 In this question you must show detailed reasoning.

Find the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$. The loci $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$ are given by $| z | = | z - 2 \mathrm { i } |$ and $| z - 2 | = \sqrt { 5 }$ respectively.
1. Sketch on a single Argand diagram the loci $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$, showing any intercepts with the imaginary axis.
2. Indicate, by shading on your Argand diagram, the region $$\{ z : | z | \leqslant | z - 2 i | \} \cap \{ z : | z - 2 | \leqslant \sqrt { 5 } \} .$$
1. Show that both of the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$ satisfy $| z - 2 | < \sqrt { 5 }$.
2. State, with a reason, which root of the equation $2 z ^ { 2 } - 2 z + 5 = 0$ satisfies $| z | < | z - 2 i |$.
On the same Argand diagram as part (b), indicate the positions of the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$.