Exercises for Question Type 6: Determining the type of eigenvalue given a phase portrait diagram

Question 1

Skill question

The following question relates to the analysis of linear systems of differential equations in the form $\frac{d x}{d t} = Ax$ , where $x \in R^{2}$ and $A$ is a $2 \times 2$ real constant matrix.

A linear system of differential equations is represented by the phase portrait shown below:

Identify the type of equilibrium point at the origin and characterize the nature of the eigenvalues of the system matrix $A$ .

[3]

Question 2

Skill question

This question involves the analysis of a phase portrait for a system of linear differential equations.

The following phase portrait shows the trajectories of a system of linear differential equations $\frac{d x}{d t} = Ax$ near the origin.

Identify the nature of the eigenvalues of matrix $A$ and the type of equilibrium point at the origin. Justify your answers.

[4]

Question 3

Skill question

This question tests the ability to qualitatively analyze a phase portrait of a system of linear differential equations $\frac{d x}{d t} = Ax$ to identify the classification of an equilibrium point and the nature of the associated eigenvalues and eigenvectors.

The following phase portrait shows the trajectories of a system of linear differential equations $\frac{d x}{d t} = Ax$ near an equilibrium point at the origin.

Classify the equilibrium point and describe the nature of the eigenvalues and eigenvectors of matrix $A$ .

[4]

Question 4

Skill question

The following phase portrait represents the behavior of a linear system $\dot{x} = A x$ in the plane.

Examine the following phase portrait:

Determine the equilibrium type at the origin and characterize the eigenvalues of the corresponding system.

[5]

Question 5

Skill question

Identify the type of equilibrium point and describe its eigenvalues based on the given phase portrait.

Consider the following phase portrait for a linear system of differential equations:

Identify the equilibrium type at the origin and describe the nature of the eigenvalues, $λ$ , associated with this system.

[3]

Question 6

Skill question

Consider the following phase portrait for a system of linear differential equations:

Classify the equilibrium point at the origin and describe the nature of its corresponding eigenvalues.

[3]

Question 7

Skill question

Examine the following phase portrait:

Determine the type of equilibrium point at the origin and describe the characteristics of its eigenvalues.

[3]

Question 8

Skill question

The following phase portrait is given for a system of linear differential equations:

Classify the equilibrium at the origin and state the nature of its eigenvalues.

[4]

Question 9

Skill question

Determine the type of equilibrium and the nature of the eigenvalues from a given phase portrait.

Consider the phase portrait shown below:

Determine the type of equilibrium at the origin and describe the eigenvalues (real or complex, distinct or repeated, and sign).

[4]

Question 10

Skill question

Level: HL Paper: 2

Analyze the phase portrait shown below for a system of linear differential equations:

Identify the type of equilibrium point at the origin and describe the nature of its eigenvalues.

[5]

Question 11

Skill question

The diagram shows the phase portrait of a linear system of differential equations near the origin.

Identify the type of equilibrium point at the origin and describe the nature of its eigenvalues.

[3]

IB Mathematics Applications & Interpretation Question Type 6: Determining the Type of Eigenvalue Given a Phase Portrait Diagram Exercises