← Complex numbers: past papers

AS 91577 · 2024 paper · a factor, a power in polar form, a surd equation, a locus gradient, and a line a quotient never meets

Question 1(b) Achieved

Practises De Moivre’s theorem & powers

Five separate parts: a factor of a cubic, a power of a number in polar form, an equation with two square roots, the gradient of a straight-line locus, and a proof that a quotient never lies on y = x.

If z = m cis(nπ/5), where m and n are positive real constants, find z15, giving your answer in polar form, in terms of m and n.

The 2024 paper, page 2 ↗ Marking schedule, page 1 ↗

See it move: The Argand plane, and what a power does to a point

Work it out

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

  1. 1 De Moivre: raise the modulus to the power, and multiply the argument by it.

    z15 = m15 cis(15 × nπ/5)

  2. 2 Simplify the angle.

    15 × nπ/5 = 3nπ, so z15 = m15 cis(3nπ). n is any positive real number, not a whole number, so the angle stays as it is.
Answer z15 = m15 cis(3nπ)

What the marker wanted

  • Achieved The correct power, in polar form.

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

How did it go?

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