IB Mathematics Applications & Interpretation Topic AHL 5.17—phase Portrait Questionbank (HL)

IB Mathematics Applications & Interpretation Topic AHL 5.17—phase Portrait Questionbank (HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 5.17—phase Portrait with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 5.17—phase Portrait and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 2

A model for the spread of a rumor involves susceptible ( $x$ ) and informed ( $y$ ) individuals:

$\frac{d x}{d t} = - 0.4 x y, \frac{d y}{d t} = 0.4 x y - 0.1 y$

where $x$ and $y$ are in thousands and $t$ is in days.

Show that the system has a line of equilibrium points on the $x$ -axis (i.e. $y = 0$ ).

[3]

For the equilibrium point $(0, 0)$ , find the Jacobian matrix and its eigenvalues, and classify the stability.

[5]

Given initial conditions $x (0) = 8, y (0) = 1$ , use Euler's method with a step size of $0.1$ days to estimate the populations after $0.3$ days.

[5]

Sketch the phase portrait on $0 \leq x \leq 10$ , $0 \leq y \leq 3$ . Show the equilibrium line(s) and at least three trajectories with direction arrows.

[4]

Question 2

HLPaper 2

A model for the populations of two symbiotic algae species, $x$ and $y$ (in thousands per $m^{2}$ ), is given by:

$\frac{d x}{d t} = x + 2 y - 10, \frac{d y}{d t} = 2 x + y - 15$

Time $t$ is measured in arbitrary time units.

Find the equilibrium point.

[3]

Use the substitution $X = x - \frac{20}{3}, Y = y - \frac{5}{3}$ to transform the system into a homogeneous system, and find its eigenvalues.

[5]

Find the general solution of the original system, given the eigenvalues and eigenvectors from Part 1.

[6]

Given initial conditions $x (0) = 5, y (0) = 3$ , find the particular solution and determine the limit of $\frac{y}{x}$ as $t \to \infty$ .

[4]

Sketch the phase portrait for $0 \leq x \leq 6, 0 \leq y \leq 6$ , showing the equilibrium point and trajectories.

[4]

Question 3

HLPaper 2

The displacement of a damped spring-mounted platform, $q$ (in $cm$ ), is modeled by

$\frac{d ^{2} q}{d t ^{2}} + 2 \frac{d q}{d t} + 5 q = 15$

where $t$ is in seconds.

Convert the equation to a system of first-order differential equations using $y = \frac{d q}{d t}$ , and find the equilibrium point.

[4]

Find the eigenvalues of the homogeneous system's matrix.

[3]

Find the exact solution for $q (t)$ , given $q (0) = 0, y (0) = 0$ .

[6]

Sketch the graph of $q$ against $t$ for $0 \leq t \leq 5$ , labeling the equilibrium level.

[2]

Question 4

HLPaper 2

A predator-prey system models the populations of deer ( $D$ , in thousands) and wolves ( $W$ , in hundreds) in a national park, given by:

$\frac{d D}{d t} = 0.45 D - 0.03 D W, \frac{d W}{d t} = 0.02 D W - 0.4 W$

where $t$ is in years. Initially, $D = 22, W = 10$ .

Find the non-zero equilibrium point for the system.

[3]

Interpret the equilibrium point in the context of the model.

[1]

Use Euler's method with a step size of $0.2$ years to estimate $D$ and $W$ after $0.2$ years.

[2]

On a phase plane with $D$ on the horizontal axis and $W$ on the vertical axis, for $18 \leq D \leq 24$ and $8 \leq W \leq 12$ , sketch the initial direction of the trajectory from $(22, 10)$ .

[2]

Question 5

HLPaper 2

A biologist is studying the interaction between two species in an ecosystem, where the population dynamics can be modeled by the following system of differential equations. Let $x (t)$ and $y (t)$ be the population sizes (in individuals) at time $t$ days.

$\frac{d x}{d t} = 2 x + y, \frac{d y}{d t} = 4 x + 3 y .$

Determine the eigenvalues of the system.

[5]

Find the corresponding eigenvectors.

[6]

Classify the origin and describe the behaviour of solutions near it.

[4]

Suggest an interpretation of this system in terms of population dynamics.

[3]

IB Mathematics Applications & Interpretation AHL 5.17—phase Portrait Questionbank

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IB Mathematics Applications & Interpretation AHL 5.17—phase Portrait Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 5.17—phase Portrait Questionbank

Unlock all 24 questions with a free account