21M.2.AHL.TZ1.8

pestleMathematicsAAHLPaper 221M· ahl-1-13-polar-and-euler-formsource ↗

Consider the complex numbers $z = 2 (\cos \frac{π}{5} + i \sin \frac{π}{5})$ and $w = 8 (\cos \frac{2 k π}{5} - i \sin \frac{2 k π}{5})$ , where $k \in ℤ^{+}$ .

Suppose that $z w \in ℤ$ .

Find the modulus of $z w$ .

[1]

Find the argument of $z w$ in terms of $k$ .

[2]

Find the minimum value of $k$ .

[3]

c.i.

For the value of $k$ found in part (i), find the value of $z w$ .

[1]

c.ii.

Markscheme / solution

$(|z w| =) 16$ A1

[1 mark]

attempt to find $arg (z) + arg (w)$ (M1)

$arg (z w) = arg (z) + arg (w)$

$= \frac{π}{5} - \frac{2 k π}{5} (= \frac{(1 - 2 k) π}{5})$ A1

[2 marks]

$z w \in ℤ \Rightarrow arg (z w)$ is a multiple of $π$ (M1)

$\Rightarrow 1 - 2 k$ is a multiple of $5$ (M1)

$k = 3$ A1

[3 marks]

c.i.

$z w = 16 (\cos (- π) + i \sin (- π))$

$- 16$ A1

[1 mark]

c.ii.

Examiners’ report

[N/A]

c.i.

[N/A]

c.ii.