Standard +0.8 This question requires converting a complex number to modulus-argument form, applying De Moivre's theorem to find fourth roots, and recognizing that all roots share the same modulus (which defines the circle's radius). While the steps are standard for Further Maths, the calculation involves non-trivial values (modulus 2√3, argument -π/6) and requires careful execution across multiple techniques, placing it moderately above average difficulty.
When plotted on an Argand diagram, the four fourth roots of the complex number \(9 - 3\sqrt{3}i\) lie on a circle. Find the equation of this circle. [4]
e.g. Radius of circle = \(\sqrt[4]{108}\) or \(108^{\frac{1}{8}} = 1.795...\)
A1
Circle: \(x^2 + y^2 = 3.22\) or \(1.795^2\)
A1
FT their radius. Allow \(1.8^2\). Allow \(
Let $z^4 = 9 - 3\sqrt{3}i$
$|z^4| = \sqrt{9^2 + (3\sqrt{3})^2} = \sqrt{108} = 6\sqrt{3}$ | B1 | si
Finding the radius of the circle | M1 | FT their $|z^4|$
e.g. Radius of circle = $\sqrt[4]{108}$ or $108^{\frac{1}{8}} = 1.795...$ | A1 |
Circle: $x^2 + y^2 = 3.22$ or $1.795^2$ | A1 | FT their radius. Allow $1.8^2$. Allow $|z| = 108^{1/8}$
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When plotted on an Argand diagram, the four fourth roots of the complex number $9 - 3\sqrt{3}i$ lie on a circle. Find the equation of this circle. [4]
\hfill \mbox{\textit{WJEC Further Unit 4 2022 Q2 [4]}}