Question 10
The Argand diagram that represents the solutions to $z^4=16\operatorname{cis}\left(\dfrac{2\pi}{3}\right)$, $z\in\mathbb{C}$, is
In each diagram the dashed rays are $30^\circ$ apart and the circles are drawn at the labelled value and half of it.
Not quite. The angles are right, but the radius is $4=\sqrt{16}$. For a fourth root you need $\sqrt[4]{16}=2$. In 2023, 11.2% of Queensland students chose (A).
Not quite. Both the radius and the angles are off. The radius is $\sqrt[4]{16}=2$, and the first root is at $\frac{2\pi}{3}\div4=\frac{\pi}{6}$. In 2023, 12.9% of Queensland students chose (B).
That’s it. Step through the hints below to see why. In 2023, 57.8% of Queensland students chose (C).
Not quite. The radius is right, but the first root is at $\frac{\pi}{3}$. Divide the argument $\frac{2\pi}{3}$ by $4$: the first root is at $\frac{\pi}{6}$. In 2023, 17.2% of Queensland students chose (D).
The first solution is at $\frac{\pi}{6}$, which is $30^\circ$ above the positive real axis, on a circle of radius $2$. That matches option (C).
Each solution is a quarter turn from the last, because adding $2\pi$ to $4\theta$ adds $\frac{\pi}{2}$ to $\theta$.
Write $z=r\operatorname{cis}\theta$ and use De Moivre’s theorem.
$$r^4\operatorname{cis}(4\theta)=16\operatorname{cis}\left(\frac{2\pi}{3}\right)$$
So $r^4=16$, which gives $r=2$. That rules out (A) and (B).
The arguments match up to a multiple of $2\pi$.
$$4\theta=\frac{2\pi}{3}+2k\pi\quad\Rightarrow\quad\theta=\frac{\pi}{6}+\frac{k\pi}{2}$$
The first solution is at $\frac{\pi}{6}=30^\circ$, and the others are $90^\circ$ apart. Option (D) starts at $60^\circ$, so the answer is (C).
With $z=r\operatorname{cis}\theta$, $r^4=16$ and $4\theta=\frac{2\pi}{3}+2k\pi$.
$$z=2\operatorname{cis}\left(\frac{\pi}{6}\right),\ 2\operatorname{cis}\left(\frac{2\pi}{3}\right),\ 2\operatorname{cis}\left(-\frac{5\pi}{6}\right),\ 2\operatorname{cis}\left(-\frac{\pi}{3}\right)$$
These sit on a circle of radius $2$ at $30^\circ$, $120^\circ$, $210^\circ$ and $300^\circ$.
How Queensland students answered in 2023
From the QCAA Specialist Mathematics subject report for 2023. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2023 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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