27 exam-style questions on Phase portraits; eigenvalue analysis of trajectories — topic 5.17 of Calculus in the IB Mathematics: Applications and Interpretation syllabus — with answers and explanations. Samples below; the full set is free with an account.
A coupled system has matrix M = [[3, 0], [0, -2]]. Determine the nature of the equilibrium point at the origin.
Answer: Saddle point
The eigenvalues are 3 and -2. Real eigenvalues of opposite sign give a saddle point.
The matrix of a linear system is M = [[-4, 0], [0, -1]]. Determine the long-term behaviour of trajectories.
Answer: Approach the origin: stable node
Both eigenvalues (-4 and -1) are real and negative, so every trajectory decays to the origin: a stable node.
For the system with matrix M = [[2, 1], [0, 2]] the eigenvalues are both 2. State the nature of the origin.
Answer: Unstable node
A repeated real eigenvalue that is positive gives an unstable node; trajectories move away from the origin.
A system has complex eigenvalues λ = 1 ± 3i. Determine the type of equilibrium at the origin.
Answer: Unstable spiral (source)
Complex eigenvalues with a positive real part (Re λ = 1 > 0) produce trajectories that spiral outward: an unstable spiral.
Find the eigenvalues of M = [[1, 2], [2, 1]].
Answer: λ = 3 and λ = -1
det(M − λI) = (1−λ)² − 4 = λ² − 2λ − 3 = (λ−3)(λ+1), giving λ = 3 and λ = −1.
The matrix M = [[0, -2], [2, 0]] has eigenvalues λ = ±2i. Determine the nature of the origin.
Answer: Centre (closed orbits)
Purely imaginary eigenvalues (real part 0) give a centre: trajectories are closed loops around the origin.
That was 6 of 27 Phase portraits; eigenvalue analysis of trajectories questions.
Practise all 27 freeFindAO2 · 2–6 marks
Obtain the answer and show the method; if the calculator does it, say what was entered.
CalculateAO2 · 2–6 marks
Show the formula, the substitution and the answer with units and sensible significant figures.
StateAO1 · 1–2 marks
One word or one short sentence. No explanation, no working: the mark is for the fact itself.
27 Phase portraits; eigenvalue analysis of trajectories questions — free with an account. Spaced repetition, streaks and full exam simulations included.
Practise them freeThis platform is independently developed and is not endorsed by, affiliated with, or sponsored by the International Baccalaureate Organization. “International Baccalaureate” and “IB” are registered trademarks of the IBO.