Question 2(b) Achieved

Five separate parts: a combination of two points read off an Argand diagram, a power of a number in polar form, an equation with a modulus in it, two unknowns found by comparing real and imaginary parts, and an unknown found from an argument.

If u = m cis(3π/10), write u⁵ in the form a + bi, where a and b are both real numbers, giving your answer in terms of m.

The 2025 paper, page 5 ↗ Marking schedule, page 3 ↗

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: what happens to the modulus and the argument when you raise to the fifth power?

   u⁵ = m⁵ cis(5 × 3π/10) = m⁵ cis(3π/2)

2. 2 The question wants a + bi. What are cos3π/2 and sin3π/2?

   0 and −1: the angle points straight down the imaginary axis.
   u⁵ = m⁵(0 + i × (−1)) = −m⁵ i

What the marker wanted

• Achieved The correct value, u⁵ = −m⁵ i.

In our words, from NZQA’s 2025 marking schedule.

Common mistake

How did it go?

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