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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | IB Analysis and Approaches |
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.
2. | IB Analysis and Approaches |
Consider \( P(z) = z^3 +2nz - \frac{40z^2}{n} - 10n\), where \( z \in \mathbb{C} \) and \( n \in \mathbb{R}^+ \).
Given that \( z - 4i \) is a factor of \( P(z) \), find the roots of \( P(z) = 0 \).
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