21M.1.AHL.TZ1.9

pestleMathematicsAIHLPaper 121M· ahl-1-12-complex-numbers-introductionsource ↗

Consider $w = i z + 1$ , where $w, z \in ℂ$ .

Find $w$ when

Point $z$ on the Argand diagram can be transformed to point $w$ by two transformations.

$z = 2 i$ .

[2]

a.i.

$z = 1 + i$ .

[1]

a.ii.

Describe these two transformations and give the order in which they are applied.

[3]

Hence, or otherwise, find the value of $z$ when $w = 2 - i$ .

[2]

Markscheme / solution

$i^{2} = - 1$ (M1)

$w = - 2 + 1 = - 1$ A1

[2 marks]

a.i.

$w = - 1 + i + 1 = i$ A1

[1 mark]

a.ii.

EITHER

rotation of $90 °$ (anticlockwise, centre at the origin) A1A1

Note: Award A1 for “rotation” and A1 for “ $90 °$ ”.

followed by a translation of $(\begin{matrix} 1 \\ 0 \end{matrix})$    A1

OR
translation of $(\begin{matrix} 0 \\ - 1 \end{matrix})$    A1

followed by rotation of $90 °$ (anticlockwise, centre at the origin)    A1A1

Note: Award A1 for “rotation” and A1 for “ $90 °$ ”.

[3 marks]

EITHER

move $1$ to left to $1 - i$ (M1)

then rotate by $- 90 °$ to

$- 1 - i$ A1

$i z + 1 = 2 - i$

$i z = 1 - i$

$z = \frac{1 - i}{i}$ (M1)

$- 1 - i$ A1

[2 marks]

Examiners’ report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]