FP2 June 2013 Q2

EdexcelOld spec6 marksComplex Numbers

2. \[z = 5\sqrt{3} - 5\mathrm{i}\]

Find

(a) \(|z|\), (1)

(b) \(\arg(z)\), in terms of \(\pi\). (2)

\[w = 2\left(\cos\frac{\pi}{4} + \mathrm{i}\sin\frac{\pi}{4}\right)\]

Find

(d) \(\arg\left(\dfrac{w}{z}\right)\), in terms of \(\pi\). (2)

Scheme	Marks
\(z = 5\sqrt{3} - 5\mathrm{i} = r(\cos\theta + \mathrm{i}\sin\theta)\)
\(r = \sqrt{\left(5^2 \times 3 + 5^2\right)} = 10\)	B1
	(1)

Mark scheme (b)

Scheme	Marks
\(\arg z = \arctan\left(-\dfrac{5}{5\sqrt{3}}\right) = -\dfrac{\pi}{6}\) \(\left(\text{or } -\dfrac{\pi}{6} \pm 2n\pi\right)\)	M1A1
	(2)

Notes

M1 for \(\arg z = \arctan\left(\pm\dfrac{5}{5\sqrt{3}}\right)\), \(\tan(\arg z) = \pm\dfrac{5}{5\sqrt{3}}\), \(\arg z = \arctan\left(\pm\dfrac{5\sqrt{3}}{5}\right)\) or \(\tan(\arg z) = \pm\dfrac{5\sqrt{3}}{5}\) OR use their \(|z|\) with sin or cos used correctly

A1 for \(= -\dfrac{\pi}{6}\) \(\left(\text{or } -\dfrac{\pi}{6} \pm 2n\pi\right)\) (must be 4th quadrant)

Scheme	Marks
\(\left\|\dfrac{w}{z}\right\| = \dfrac{2}{10} = \dfrac{1}{5}\) or 0.2	B1
	(1)

Mark scheme (d)

Scheme	Marks
\(\arg\left(\dfrac{w}{z}\right) = \dfrac{\pi}{4} - \left(-\dfrac{\pi}{6}\right),\ = \dfrac{5\pi}{12}\) \(\left(\text{or } \dfrac{5\pi}{12} \pm 2n\pi\right)\)	M1,A1
	(2)
	(6 marks)

Notes

M1 for \(\arg\left(\dfrac{w}{z}\right) = \dfrac{\pi}{4} - \arg z\) using their \(\arg z\)

A1 for \(\dfrac{5\pi}{12}\) \(\left(\text{or } \dfrac{5\pi}{12} \pm 2n\pi\right)\)

Alternative for (d)

Scheme	Marks
Find \(\dfrac{w}{z} = \dfrac{\left(\sqrt{6} - \sqrt{2}\right) + \left(\sqrt{6} + \sqrt{2}\right)\mathrm{i}}{20}\)
\(\tan\left(\arg\dfrac{w}{z}\right) = \dfrac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\)	M1 from their \(\dfrac{w}{z}\)
\(\arg\left(\dfrac{w}{z}\right) = \dfrac{5\pi}{12}\)	A1 cao

Work for (c) and (d) may be seen together – give B and A marks only if modulus and argument are clearly identified
ie \(\dfrac{1}{5}\left(\cos\dfrac{5\pi}{12} + \mathrm{i}\sin\dfrac{5\pi}{12}\right)\) alone scores B0M1A0