5.17 Phase portraitsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The populations $x$ and $y$ (in hundreds) of two species of algae in a pond, $t$ weeks after they are introduced, satisfy $\frac{dx}{dt}=4x+y$ and $\frac{dy}{dt}=2x+3y$. A GDC may be used.

   (a)

   Find the eigenvalues of the matrix $\begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$ that represents the system.

   [1 mark]

   • A$4$ and $3$
   • B$2$ and $5$
   • C$7$ and $10$
   • D$-2$ and $-5$

   (b)

   Which statement describes the phase portrait of the system?

   [1 mark]

   • AThe origin is a saddle point.
   • BTrajectories form closed ellipses around the origin.
   • CTrajectories spiral towards the origin.
   • DTrajectories move away from the origin.

   (c)

   Find an eigenvector of the matrix corresponding to the eigenvalue $2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The deviations $x$ and $y$ (in thousands) of the populations of two interacting species from their equilibrium values satisfy $\frac{dx}{dt}=x+2y$ and $\frac{dy}{dt}=3x$. A GDC may be used.

   (a)

   What type of equilibrium point is the origin?

   [1 mark]

   • AA saddle point
   • BA stable node, where all trajectories move towards the origin
   • CAn unstable node, where all trajectories move away from the origin
   • DA centre, with closed elliptical trajectories

   (b)

   At $t=0$, $x=1$ and $y=1$. Which statement is correct?

   [1 mark]

   • AThe point moves along the line $y=x$ towards the origin.
   • BThe point moves along the line $y=-\frac32x$ away from the origin.
   • CThe point moves along the line $y=x$ away from the origin.
   • DThe point follows a closed elliptical path around the origin.

   (c)

   Find the equations of the two straight-line trajectories that pass through the origin.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The deviations $x$ and $y$ (in hundreds) of two interacting populations from their equilibrium values, $t$ years after a survey begins, satisfy $\frac{dx}{dt}=-x-4y$ and $\frac{dy}{dt}=x-y$.

   (a)

   Find the eigenvalues of the matrix $\begin{pmatrix} -1 & -4 \\ 1 & -1 \end{pmatrix}$ that represents the system.

   [3 marks]

   (b)

   Describe the long-term behaviour of $x$ and $y$, and state whether the paths around the origin are clockwise or anticlockwise. Justify your answers.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A drug is given to a patient. The amounts $x$ mg and $y$ mg in the blood and in the body tissue, $t$ hours later, satisfy $\frac{dx}{dt}=-2x+y$ and $\frac{dy}{dt}=x-2y$.

   (a)

   (i) Find the eigenvalues of the matrix $\begin{pmatrix} -2 & 1 \\ 1 & -2 \end{pmatrix}$ that represents the system.
   (ii) Find an eigenvector corresponding to the eigenvalue of smaller magnitude.
   (iii) State what happens to the amounts of drug in the long term, giving a reason.

   [6 marks]

   (b)

   Initially $x=5$ and $y=1$. Find $x$ and $y$ as functions of $t$.

   [6 marks]

   Total for question 4: 12 marks

End of questions