19M.2.AHL.TZ1.H_6

pestleMathematicsAAHLPaper 219M· ahl-1-13-polar-and-euler-formsource ↗

Let $z = a + b{\text{i}}$ , $a$ , ${\text{b}} \in {\mathbb{R}^ + }$ and let ${\text{arg}}\,z = \theta$ .

Show the points represented by $z$ and $z - 2a$ on the following Argand diagram.

[1]

Find an expression in terms of θ for ${\text{arg}}\left( {z - 2a} \right)$ .

[1]

b.i.

Find an expression in terms of θ for ${\text{arg}}\left( {\frac{z}{{z - 2a}}} \right)$ .

[2]

b.ii.

Hence or otherwise find the value of θ for which ${\text{Re}}\left( {\frac{z}{{z - 2a}}} \right) = 0$ .

[3]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

Note: Award A1 for $z$ in first quadrant and $z - 2a$ its reflection in the $y$ -axis.

[1 mark]

$\pi - \theta$ (or any equivalent) A1

[1 mark]

b.i.

${\text{arg}}\left( {\frac{z}{{z - 2a}}} \right) = {\text{arg}}\left( z \right) - {\text{arg}}\left( {z - 2a} \right)$ (M1)

$= 2\theta - \pi$ (or any equivalent) A1

[2 marks]

b.ii.

METHOD 1

if ${\text{Re}}\left( {\frac{z}{{z - 2a}}} \right) = 0$ then $2\theta - \pi = \frac{{n\pi }}{2}$ , ( $n$ odd) (M1)

$- \pi < 2\theta - \pi < 0 \Rightarrow n = - 1$

$2\theta - \pi = - \frac{\pi }{2}$ (A1)

$\theta = \frac{\pi }{4}$ A1

METHOD 2

$\frac{{a + b{\text{i}}}}{{ - a + b{\text{i}}}} = \frac{{{b^2} - {a^2} - 2ab{\text{i}}}}{{{a^2} + {b^2}}}$ M1

${\text{Re}}\left( {\frac{z}{{z - 2a}}} \right) = 0 \Rightarrow {b^2} - {a^2} = 0$

$b = a$ A1

$\theta = \frac{\pi }{4}$ A1

Note: Accept any equivalent, eg $\theta = - \frac{{7\pi }}{4}$ .

[3 marks]

Examiners’ report

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b.i.

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b.ii.

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