SPM.1.AHL.TZ0.15

pestleMathematicsAIHLPaper 1SPM· ahl-1-13-polar-and-euler-formsource ↗

Let $w = a{{\text{e}}^{\frac{\pi }{4}{\text{i}}}}$ , where $a \in {\mathbb{R}^ + }$ .

for $a$ = 2,

find the values of ${w^2}$ , ${w^3}$ , and ${w^4}$ .

[2]

a.i.

draw $w$ , ${w^2}$ , ${w^3}$ , and ${w^4}$ on the following Argand diagram.

[3]

a.ii.

Let $z = \frac{w}{{2 - {\text{i}}}}$ .

Find the value of $a$ for which successive powers of $z$ lie on a circle.

[2]

Markscheme / solution

$4{{\text{e}}^{\frac{\pi }{2}{\text{i}}}}$ , $8{{\text{e}}^{\frac{{3\pi }}{4}{\text{i}}}}$ , $16{{\text{e}}^{\pi {\text{i}}}}$ ( $= 4{\text{i}}$ , $- {\text{4}}\sqrt 2 + 4\sqrt 2 {\text{i}}$ , $- 16$ ) (M1)A1

[2 marks]

a.i.

Note: Award A1 for correct arguments, award A1 for $4{\text{i}}$ and −16 clearly indicated, award A1 for | $w$ | < 4 and 4 < | ${w^3}$ | < 16.

[3 marks]

a.ii.

${2^2} + {1^2} = {a^2}$ M1

$a = \sqrt 5 \,\,\left( { = 2.24} \right)$ A1

[2 marks]

Examiners’ report

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a.i.

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a.ii.

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