A2 October 2021 Paper 1 Q12

OCR MEICurrent spec4 marksComplex Numbers

12 Fig. 12 shows a rhombus OACB in an Argand diagram. The points A and B represent the complex numbers \(z\) and \(w\) respectively.

Prove that \(\arg(z + w) = \frac{1}{2}(\arg z + \arg w)\). [4]

Scheme	Marks	AO
Let \(\angle\mathrm{DOA} = \alpha,\ \angle\mathrm{COA} = \beta\) \(z + w\) is represented by C \(\arg(z + w) = \alpha + \beta\)	B1	3.1a
\(\angle\mathrm{BOC} = \beta\) (diagonal of rhombus bisects \(\angle\mathrm{BOA}\))	M1	2.1
\(\arg z + \arg w = \alpha + (\alpha + 2\beta)\) \(= 2(\alpha + \beta)\)	M1	2.1
so \(\arg(z + w) = \frac{1}{2}(\arg z + \arg w)\)	A1	2.2a
	[4]