IB Maths AI HL · Unit 1: Number and Algebra
IB Maths AI HL Complex Numbers and Argand Diagrams Questions
Exam-style IB Maths AI HL complex numbers and argand diagrams questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 29 questions
- Paper 1: 29
- 10 easy
- 10 medium
- 5 hard
- 4 starter
- 3 worked examples
Practise Complex Numbers and Argand Diagrams questions →
AI HL formula booklet
What's examined in AI HL complex numbers and argand diagrams
The question bank covers these complex numbers and argand diagrams question types (number of questions in brackets):
- Complex Number Algebra (20)
- Argand Diagrams and Loci (5)
- De Moivre's Theorem and Roots (4)
Key formulas
- Modulus of a complex number
- \(|z_1 z_2| = |z_1| \cdot |z_2|,\ \left|\tfrac{z_1}{z_2}\right| = \tfrac{|z_1|}{|z_2|}\)
- Complex numbers — Cartesian
- \(z = a + bi,\ |z| = \sqrt{a^2 + b^2},\ \arg z = \arctan\!\left(\tfrac{b}{a}\right)\)
In the same notation as the IB formula booklet. All AI HL formulas →
Complex Numbers and Argand Diagrams worked examples
Worked example 1: Converting Cartesian to Euler Form · easy
Convert the complex number $z = -1 + i\sqrt{3}$ to Euler form $re^{i\theta}$, where $r > 0$ and $-\pi < \theta \le \pi$.
1. Calculate the modulus $r$: $r = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2$.
2. Calculate the reference angle: $\alpha = \arctan\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}$.
3. Identify the correct quadrant: Since the real part is negative and the imaginary part is positive, $z$ lies in the second quadrant.
4. Find the principal argument $\theta$: $\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$.
5. The Euler form is $z = 2e^{i\frac{2\pi}{3}}$.
Examiner tip: Always sketch a quick Argand diagram to visually confirm which quadrant your complex number lies in, preventing argument errors.
Worked example 2: Solving Quadratics with Complex Roots · medium
Find the complex roots of the quadratic equation $x^2 + 4x + 13 = 0$. Give your answers in Cartesian form $a + bi$.
1. Apply the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
2. Substitute $a=1, b=4, c=13$: $x = \frac{-4 \pm \sqrt{16 - 4(1)(13)}}{2}$.
3. Simplify the discriminant: $x = \frac{-4 \pm \sqrt{16 - 52}}{2} = \frac{-4 \pm \sqrt{-36}}{2}$.
4. Convert the negative square root to an imaginary number: $\sqrt{-36} = 6i$.
5. Divide by $2$ to find the final conjugate roots: $x = -2 \pm 3i$.
Examiner tip: Notice that for polynomials with real coefficients, complex roots will always appear in conjugate pairs.
Worked example 3: Applying De Moivre's Theorem · hard
Given the complex number $z = 2e^{i\frac{\pi}{6}}$, use De Moivre's Theorem to find the exact value of $z^5$ in Cartesian form.
1. Apply De Moivre's Theorem to the modulus and argument: $z^5 = (2)^5 e^{i(5 \times \frac{\pi}{6})}$.
2. Evaluate the new modulus and argument: $z^5 = 32e^{i\frac{5\pi}{6}}$.
3. Convert the Euler form into trigonometric form: $32\left(\cos\left(\frac{5\pi}{6}\right) + i\sin\left(\frac{5\pi}{6}\right)\right)$.
4. Substitute the exact values from the unit circle for the second quadrant: $\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$ and $\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}$.
5. Expand the brackets: $32\left(-\frac{\sqrt{3}}{2} + \frac{1}{2}i\right) =$ $-16\sqrt{3} + 16i$.
Examiner tip: Operations involving high integer powers or roots are significantly faster and less prone to algebraic error when performed in Euler or polar form rather than Cartesian form.
Try these IB Maths AI HL complex numbers and argand diagrams questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 4 marks · Paper 1
Given the complex numbers $z_1 = 2+3i$ and $z_2 = 1-i$, find $z_1 z_2$.
Attempt it and see the mark scheme →
Question 2 · medium · 4 marks · Paper 1
Let $z = 3+i$.
(a) Write down the complex conjugate of $z$. [1 mark]
(b) Given that $w = \frac{z^*}{1+i}$, express $w$ in the form $p+qi$, where $p, q \in \mathbb{R}$. [3 marks]
Question 3 · hard · 7 marks · Paper 1
Solve the following system of simultaneous linear equations for the complex variables \(z\) and \(w\):
\[\begin{aligned}
(2 + i)z + w &= 5i \\
z - iw &= 3\end{aligned}\]
Give your final answers for \(z\) and \(w\) in Cartesian form.
All 29 complex numbers and argand diagrams questions with mark schemes →
FAQ
How many IB Maths AI HL complex numbers and argand diagrams questions are there?
There are 29 exam-style complex numbers and argand diagrams questions in the AI HL question bank (Paper 1: 29), graded 10 easy, 10 medium, 5 hard, 4 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is complex numbers and argand diagrams on Paper 1 or Paper 2?
In the question bank these questions are set as Paper 1 questions.
Where can I get the mark schemes?
Open the AI HL Unit 1 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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