17M.1.AHL.TZ1.H_2

pestleMathematicsAIHLPaper 117M· ahl-1-13-polar-and-euler-formsource ↗

Consider the complex numbers ${z_1} = 1 + \sqrt 3 {\text{i, }}{z_2} = 1 + {\text{i}}$ and $w = \frac{{{z_1}}}{{{z_2}}}$ .

By expressing ${z_1}$ and ${z_2}$ in modulus-argument form write down the modulus of $w$ ;

[3]

a.i.

By expressing ${z_1}$ and ${z_2}$ in modulus-argument form write down the argument of $w$ .

[1]

a.ii.

Find the smallest positive integer value of $n$ , such that ${w^n}$ is a real number.

[2]

Markscheme / solution

${z_1} = 2{\text{cis}}\left( {\frac{\pi }{3}} \right)$ and ${z_2} = \sqrt 2 {\text{cis}}\left( {\frac{\pi }{4}} \right)$ A1A1

Note: Award A1A0 for correct moduli and arguments found, but not written in mod-arg form.

$\left| w \right| = \sqrt 2$ A1

[3 marks]

a.i.

${z_1} = 2{\text{cis}}\left( {\frac{\pi }{3}} \right)$ and ${z_2} = \sqrt 2 {\text{cis}}\left( {\frac{\pi }{4}} \right)$ A1A1

Note: Award A1A0 for correct moduli and arguments found, but not written in mod-arg form.

$\arg w = \frac{\pi }{{12}}$ A1

Notes: Allow FT from incorrect answers for ${z_1}$ and ${z_2}$ in modulus-argument form.

[1 mark]

a.ii.

EITHER

$\sin \left( {\frac{{\pi n}}{{12}}} \right) = 0$ (M1)

$\arg ({w^n}) = \pi$ (M1)

$\frac{{n\pi }}{{12}} = \pi$

THEN

$\therefore n = 12$ A1

[2 marks]

Examiners’ report

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a.i.

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a.ii.

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