IB Maths AI HL · Unit 5: Calculus
IB Maths AI HL Phase Portraits Questions
Exam-style IB Maths AI HL phase portraits 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.
- 12 questions
- Paper 1: 12
- 12 starter
- 3 worked examples
Practise Phase Portraits questions →
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What's examined in AI HL phase portraits
The question bank covers these phase portraits question types (number of questions in brackets):
- Matrix Equilibrium Analysis (8)
- Phase Trajectories & Fields (4)
Phase Portraits worked examples
Worked example 1: Finding the Equilibrium Point · easy
A system of differential equations is given by $\frac{dx}{dt} = 3x - y - 5$ and $\frac{dy}{dt} = 2x + y - 5$. Find the exact coordinates of the equilibrium point.
1. State the condition for an equilibrium point: both $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} = 0$.
2. Set up the system of linear equations: $3x - y = 5$ and $2x + y = 5$.
3. Add the two equations to eliminate $y$: $5x = 10 \implies x = 2$.
4. Substitute $x = 2$ back into the second equation: $2(2) + y = 5 \implies y = 1$.
5. The exact coordinates of the equilibrium point are $(2, 1)$.
Examiner tip: Equilibrium points (or critical points) on a phase portrait occur where both the horizontal and vertical rates of change are simultaneously zero, meaning a particle at this point remains completely stationary.
Worked example 2: Classifying Equilibrium via Eigenvalues · medium
The behavior of a linear system is governed by the matrix $M = \begin{pmatrix} 1 & 4 \\ 2 & -1 \end{pmatrix}$. Find the eigenvalues of $M$ and hence classify the nature of the equilibrium point at the origin.
1. Set up the characteristic equation $\det(M - \lambda I) = 0$.
2. Evaluate the determinant: $(1-\lambda)(-1-\lambda) - (4)(2) = 0$.
3. Expand and simplify the quadratic: $-1 - \lambda + \lambda + \lambda^2 - 8 = 0 \implies \lambda^2 - 9 = 0$.
4. Solve for the eigenvalues: $\lambda = 3$ and $\lambda = -3$.
5. Classify the point: Since the eigenvalues are real and have opposite signs, the origin is a saddle point.
Examiner tip: A saddle point is always unstable. Trajectories will approach the origin along one eigenvector's direction but diverge away to infinity along the other eigenvector's direction.
Worked example 3: Interpreting Complex Eigenvalues · hard
A coupled system of differential equations is given by $x' = -x - y$ and $y' = x - y$. By finding the eigenvalues of the system's matrix, determine whether the phase portrait represents a stable or unstable spiral, and deduce the direction of rotation.
1. Extract the coefficient matrix $M = \begin{pmatrix} -1 & -1 \\ 1 & -1 \end{pmatrix}$.
2. Form the characteristic equation: $(-1-\lambda)^2 - (-1)(1) = 0 \implies \lambda^2 + 2\lambda + 2 = 0$.
3. Solve using the quadratic formula: $\lambda = \frac{-2 \pm \sqrt{4 - 8}}{2} = -1 \pm i$.
4. Analyse the real part of the complex eigenvalues: The real part is $-1$. Since it is negative, trajectories spiral inwards, so it is a stable spiral.
5. Determine the direction of rotation by testing a coordinate on the positive $x$-axis, e.g., $(1, 0)$. Here, $y' = 1 - 0 = 1$, which means the vector points upwards. Thus, the rotation is anti-clockwise.
Examiner tip: For complex eigenvalues $\lambda = a \pm bi$, the sign of the real part '$a$' completely determines stability: $a<0$ means stable (attracting inward) and $a>0$ means unstable (repelling outward).
Try these IB Maths AI HL phase portraits questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · starter · 4 marks · Paper 1
For the system of coupled linear differential equations \(\mathbf{x}' = \begin{pmatrix} 3 & -2 \\ 1 & 0 \end{pmatrix} \mathbf{x}\), classify the phase portrait equilibrium point at the origin. Justify your answer mathematically.
Question 2 · starter · 3 marks · Paper 1
The system of coupled differential equations \(\mathbf{x}' = A\mathbf{x}\) has the coefficient matrix \(A = \begin{pmatrix} -2 & 0 \\ 0 & -3 \end{pmatrix}\). Classify the equilibrium point at the origin and state its stability.
Question 3 · starter · 5 marks · Paper 1
A system of coupled differential equations is given by \(\frac{dx}{dt} = -y\) and \(\frac{dy}{dt} = x\). By finding an expression for \(\frac{dy}{dx}\) and integrating, show exactly that the trajectories of the phase portrait are circles centered at the origin.
All 12 phase portraits questions with mark schemes →
FAQ
How many IB Maths AI HL phase portraits questions are there?
There are 12 exam-style phase portraits questions in the AI HL question bank (Paper 1: 12), graded 12 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is phase portraits 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 5 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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