Question 8 - A-Level Maths

Exam Board	OCR
Module	Further Pure Core AS (Further Pure Core AS)
Year	2023
Session	June
Topic	Complex Numbers Arithmetic
Type	Linear equations in z and z*

Solve the equation \(\omega + 2 + 7 \mathrm { i } = 3 \omega ^ { * } - \mathrm { i }\).
Prove algebraically that, for non-zero \(z , z = - z ^ { * }\) if and only if \(z\) is purely imaginary.
The complex numbers \(z\) and \(z ^ { * }\) are represented on an Argand diagram by the points \(A\) and \(B\) respectively.
1. State, for any \(z\), the single transformation which transforms \(A\) to \(B\).
2. Use a geometric argument to prove that \(z = z ^ { * }\) if and only if \(z\) is purely real.