Free Coupled systems & phase portraits questions for IB Mathematics: Applications & Interpretation (SL & HL), HL only, part of Calculus. Each question has the answer and the marking points an examiner would look for.
Generate a full Coupled Systems & Phase Portraits worksheet →All Calculus questionsConsider the system dxdt = −2x − 3y, dydt = 3x − 2y.
Find the eigenvalues of the coefficient matrix.
Describe the trajectories near the origin, and their direction.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | −2 ± 3i det(M − λI) = 0 M1 (−2 − λ)2 + 9 = 0 A1 λ = −2 ± 3i A1 | 3 |
| (b) | stable spiral real part negative: spiral inwards R1 stable spiral A1 at (1, 0), dydt = 3 > 0 so anticlockwise R1 | 3 |
Consider the system of differential equations dxdt = y, dydt = −3x + 4y.
Find the eigenvalues of the matrix of coefficients.
Hence classify the equilibrium point at the origin, giving a reason.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | λ = 1, 3 det(M − λI) = 0 M1 λ2 − 4λ + 3 = 0 A1 λ = 1, 3 A1 | 3 |
| (b) | unstable node both eigenvalues real and positive R1 unstable node A1 | 2 |
Consider the system dxdt = −x − 2y, dydt = 2x − y.
Find the eigenvalues of the coefficient matrix.
Describe the trajectories near the origin, and their direction.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | −1 ± 2i det(M − λI) = 0 M1 (−1 − λ)2 + 4 = 0 A1 λ = −1 ± 2i A1 | 3 |
| (b) | stable spiral real part negative: spiral inwards R1 stable spiral A1 at (1, 0), dydt = 2 > 0 so anticlockwise R1 | 3 |
Consider the system of differential equations dxdt = 13x − 5y, dydt = 30x − 12y.
Find the eigenvalues of the matrix of coefficients.
Hence classify the equilibrium point at the origin, giving a reason.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | λ = 3, −2 det(M − λI) = 0 M1 λ2 − λ − 6 = 0 A1 λ = 3, −2 A1 | 3 |
| (b) | saddle point real eigenvalues of opposite sign R1 saddle point A1 | 2 |
Consider the system dxdt = −x − 3y, dydt = 3x − y.
Find the eigenvalues of the coefficient matrix.
Describe the trajectories near the origin, and their direction.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | −1 ± 3i det(M − λI) = 0 M1 (−1 − λ)2 + 9 = 0 A1 λ = −1 ± 3i A1 | 3 |
| (b) | stable spiral real part negative: spiral inwards R1 stable spiral A1 at (1, 0), dydt = 3 > 0 so anticlockwise R1 | 3 |
Consider the system of differential equations dxdt = 8x − 3y, dydt = 18x − 7y.
Find the eigenvalues of the matrix of coefficients.
Hence classify the equilibrium point at the origin, giving a reason.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | λ = 2, −1 det(M − λI) = 0 M1 λ2 − λ − 2 = 0 A1 λ = 2, −1 A1 | 3 |
| (b) | saddle point real eigenvalues of opposite sign R1 saddle point A1 | 2 |
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Meet Riyaz →Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in