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Exam-Style Questions on Complex NumbersProblems on Complex Numbers adapted from questions set in previous Mathematics exams. |
1. | IB Analysis and Approaches |
Consider \( w = a - bi \) where \( |w| = 1 \)
Show that \( \text{Re} \left( \frac{w + i}{w - i} \right) \) is always equal to 0.
2. | 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.
3. | 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 \( P(z) = 0 \).
4. | IB Analysis and Approaches |
Consider the complex numbers \(z_1 = c + 3i\) and \(z_2 = c^2-2-ci\)
(a) Find an expression for \(z_1z_2 \) in terms of \(c\).
(b) Hence, given that arg(\(z_1z_2 \)) = \( \tan^{-1} 2 \) show that:
$$c^3 - c^2 + c + 3 = 0 $$If you would like space on the right of the question to write out the solution try this Thinning Feature. It will collapse the text into the left half of your screen but large diagrams will remain unchanged.
The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.
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