2022HSCExtended response

Q11 (c) (ii)

2 marks

Question

Hence, find the exact value of $(-\sqrt3+i)^{10}$ , giving your answer in the form $x+iy$ .

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Solution

Criteria	Marks
Provides correct solution	2
Attempts to apply de Moivre’s theorem, or equivalent merit	1

Sample answer:

(-\sqrt3+i)^{10}=\left(2e^{i5\pi/6}\right)^{10}\quad\text{from part (i)}

=2^{10}e^{i50\pi/6}=2^{10}e^{i\pi/3} =2^{10}\left(\frac12+\frac{\sqrt3}{2}i\right) =2^9+2^9\sqrt3i

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Syllabus alignment

Complex Numbers

Solving equations with complex numbersUse De Moivre's theorem and solve equations and polynomial problems involving complex and conjugate roots.

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