AS June 2025 Paper 1 Q1

EdexcelCurrent spec6 marksComplex Numbers

\[z = \sqrt{3} - 3\mathrm{i}\]

(a) Write \(z\) in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\) where \(-\pi \lt \theta \leqslant \pi\) (2)

(b) Show and label on a single Argand diagram

(i) the point \(P\) representing \(z\)

(ii) the point \(Q\) representing \(\mathrm{i}z\)

(2)

Mark scheme (a)

Scheme	Marks	AO
\(z = \sqrt{3} - 3\mathrm{i}\) so \(r = \sqrt{\left(\sqrt{3}\right)^2 + (\pm 3)^2}\) or \(\sqrt{12}\) or \(2\sqrt{3}\) or awrt 3.46 \(\theta = -\dfrac{\pi}{3}\) or \(-60\) or awrt \(-1.05\)	B1	1.1b
\(\{z=\}\ \sqrt{12}\left(\cos\left(-\dfrac{\pi}{3}\right) + \mathrm{i}\sin\left(-\dfrac{\pi}{3}\right)\right)\) or \(2\sqrt{3}\left(\cos\left(-\dfrac{\pi}{3}\right) + \mathrm{i}\sin\left(-\dfrac{\pi}{3}\right)\right)\) or \(\{z=\}\ \sqrt{12}\left(\cos\left(\dfrac{\pi}{3}\right) - \mathrm{i}\sin\left(\dfrac{\pi}{3}\right)\right)\) or \(\{z=\}\ 2\sqrt{3}\left(\cos\left(\dfrac{\pi}{3}\right) - \mathrm{i}\sin\left(\dfrac{\pi}{3}\right)\right)\)	B1	1.1b
	(2)

B1: Correct modulus or correct argument may be correct to 3 s.f for this mark

B1: Fully correct must be exact values for this mark, angle must be in radians

Mark scheme (c)

Scheme	Marks	AO
Rotation	B1	1.1a
\(\dfrac{\pi}{2}\) or 90° (anticlockwise) about the origin or \(\dfrac{3\pi}{2}\) or 270° clockwise about the origin	B1	1.1b
	(2)
	(6 marks)

(c) Note must be a single transformation, there is no follow through

B1: Deduces rotation

B1: Correct angle (radians or degrees), direction and centre

Alternatives must have scored M1A1 in (b)

Reflection	Matrix	Translation	Marks	AO
			B1	1.1a
\(\arg z = -\dfrac{\pi}{12}\) or \(y = \left(\sqrt{3} - 2\right)x\)	\(\begin{pmatrix}\mathrm{i} & 0\\ 0 & \mathrm{i}\end{pmatrix}\) or \(\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}\)	\(\begin{pmatrix}3 - \sqrt{3}\\ \sqrt{3} + 3\end{pmatrix}\)	B1	1.1b
			(2)

Alternative: must be from a correct (b)

B1: Deduces reflection or matrix or translation

B1: Correct line or matrix or vector