21M.1.AHL.TZ2.12

pestleMathematicsAIHLPaper 121M· ahl-1-13-polar-and-euler-formsource ↗

It is given that $z_{1} = 3 cis (\frac{3 π}{4})$ and $z_{2} = 2 cis (\frac{n π}{16}), n \in ℤ^{+}$ .

In parts (a)(i) and (a)(ii), give your answers in the form $r e^{i θ}, r \geq 0, - π < θ \leq π$ .

Find the value of ${z_{1}}^{3}$ .

[2]

a.i.

Find the value of ${(\frac{z_{1}}{z_{2}})}^{4}$ for $n = 2$ .

[3]

a.ii.

Find the least value of $n$ such that $z_{1} z_{2} \in ℝ^{+}$ .

[3]

Markscheme / solution

${z_{1}}^{3} = 27 e^{\frac{iπ}{4}} (= 27 e^{0.785398 \dots i})$ A1A1

Note: Award A1 for $27$ and A1 for the angle in the correct form.

[2 marks]

a.i.

${(\frac{z_{1}}{z_{2}})}^{4} = (\frac{81}{16}) e^{\frac{iπ}{2}} (= 5.0625 e^{1.57079 \dots i})$ A1A2

Note: Award A1 for $\frac{81}{16}$ , A2 for the angle in the correct form and A1 for the angle in incorrect form e.g. $cis \frac{π}{2}$ and/or $\frac{5 π}{2}$ . Award A1 if $i$ is given in place of $cis \frac{π}{2}$ .

[3 marks]

a.ii.

$z_{1} z_{2} = 6 cis (\frac{3 π}{4} + \frac{n π}{16})$ (M1)

$= 6 cis (\frac{12 π + n π}{16})$

$12 π + n π = 32 π$ (M1)

$n = 20$ A1

[3 marks]

Examiners’ report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]