← Complex numbers: past papers

AS 91577 · 2024 paper · a surd fraction, a square on the Argand diagram, a conjugate equation, fourth roots, and a cubic identity

Question 3(b) Achieved

Practises Argand diagram & polar form

Five separate parts: a fraction of surds simplified, a complex number squared and plotted, an equation in a number and its conjugate, the fourth roots of a negative number, and a cube of x + 1/x.

On an Argand diagram, the point (−2, 3) represents the complex number z.

Show clearly, on an Argand diagram, the point representing w = z2.

z=−2 + 3i

The 2024 paper, page 9 ↗ Marking schedule, page 5 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

  1. 1 Write z as a complex number.

    z = −2 + 3i

  2. 2 Square it: a bracket of two terms.

    z2 = (−2)2 + 2(−2)(3i) + (3i)2 = 4 − 12i − 9 = −5 − 12i

  3. 3 Where does it go on the diagram?

    At the point (−5, −12): 5 to the left, 12 down.
Answer w = −5 − 12i, plotted at (−5, −12).

What the marker wanted

  • Achieved −5 − 12i plotted in the right place.

In our words, from NZQA’s 2024 marking schedule.

How did it go?

Got it moves it further out. Review brings it back tomorrow.