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Exam-Style Question on Complex Numbers

A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 659. This question is similar to one that appeared on an IB AA Higher paper in 2023. The use of a calculator is allowed.

A complex number, \(c\), has a real part \(1\) and an imaginary part \( -\sqrt{3} \).

(a) Show that \(c = 2e^{i\frac{5 \pi}{3}}\).

(b) Find the smallest positive integer \(n\) such that \( c^n\) is a real number.

(c) Find the value of \(c^n\) when \(n\) takes the value found in part (b).

Consider the equation \(z^3-9z^2+18z-28=0\), where \(z \in \mathbb{C} \).

(d) Given that \(c\) is a root of \(z^3-9z^2+18z-28=0\), find the other roots.

(e) By using a suitable transformation from \(z\) to \(w\), or otherwise, find the roots of the equation \(1-9w+18w^2-28w^3=0\), where \(w \in \mathbb{C} \).

Consider the equation \(z^2 = 2z^*\) , where \(z \in \mathbb{C}, z \neq 0 \).

(f) By expressing \(z\) in the form \(a + bi\), find the roots of the equation.

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