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Date	May 2022	Marks available	2	Reference code	22M.2.AHL.TZ2.7
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Show that	Question number	7	Adapted from	N/A

Question

An environmental scientist is asked by a river authority to model the effect of a leak from a power plant on the mercury levels in a local river. The variable $x$ measures the concentration of mercury in micrograms per litre.

The situation is modelled using the second order differential equation

$\frac{d^{2} x}{d t^{2}} + 3 \frac{d x}{d t} + 2 x = 0$

where $t \geq 0$ is the time measured in days since the leak started. It is known that when $t = 0, x = 0$ and $\frac{d x}{d t} = 1$ .

If the mercury levels are greater than $0.1$ micrograms per litre, fishing in the river is considered unsafe and is stopped.

The river authority decides to stop people from fishing in the river for $10 %$ longer than the time found from the model.

Show that the system of coupled first order equations:

$\frac{d x}{d t} = y$

$\frac{d y}{d t} = - 2 x - 3 y$

can be written as the given second order differential equation.

[2]

Find the eigenvalues of the system of coupled first order equations given in part (a).

[3]

Hence find the exact solution of the second order differential equation.

[5]

Sketch the graph of $x$ against $t$ , labelling the maximum point of the graph with its coordinates.

[2]

Use the model to calculate the total amount of time when fishing should be stopped.

[3]

Write down one reason, with reference to the context, to support this decision.

[1]

Markscheme

differentiating first equation. M1

$\frac{d^{2} x}{d t^{2}} = \frac{d y}{d t}$

substituting in for $\frac{d y}{d t}$ M1

$= - 2 x - 3 y = - 2 x - 3 \frac{d x}{d t}$

therefore $\frac{d^{2} x}{d t^{2}} + 3 \frac{d x}{d t} + 2 x = 0$ AG

Note: The AG line must be seen to award the final M1 mark.

[2 marks]

the relevant matrix is $(\begin{matrix} 0 & 1 \\ - 2 & - 3 \end{matrix})$ (M1)

Note: $(\begin{matrix} - 3 & - 2 \\ 1 & 0 \end{matrix})$ is also possible.

(this has characteristic equation) $- λ (- 3 - λ) + 2 = 0$ (A1)

$λ = - 1, - 2$ A1

[3 marks]

EITHER

the general solution is $x = A e^{- t} + B e^{- 2 t}$ M1

Note: Must have constants, but condone sign error for the M1.

so $\frac{d x}{d t} = - A e^{- t} - 2 B e^{- 2 t}$ M1A1

attempt to find eigenvectors (M1)

respective eigenvectors are $(\begin{matrix} 1 \\ - 1 \end{matrix})$ and $(\begin{matrix} 1 \\ - 2 \end{matrix})$ (or any multiple)

$(\begin{matrix} x \\ y \end{matrix}) = A e^{- t} (\begin{matrix} 1 \\ - 1 \end{matrix}) + B e^{- 2 t} (\begin{matrix} 1 \\ - 2 \end{matrix})$ (M1)A1

THEN

the initial conditions become:

$0 = A + B$

$1 = - A - 2 B$ M1

this is solved by $A = 1, B = - 1$

so the solution is $x = e^{- t} - e^{- 2 t}$ A1

[5 marks]

A1A1

Note: Award A1 for correct shape (needs to go through origin, have asymptote at $y = 0$ and a single maximum; condone $x < 0$ ). Award A1 for correct coordinates of maximum.

[2 marks]

intersecting graph with $y = 0.1$ (M1)

so the time fishing is stopped between $2.1830 \dots$ and $0.11957 \dots$ (A1)

$= 2.06 (343 \dots)$ days A1

[3 marks]

Any reasonable answer. For example:

There are greater downsides to allowing fishing when the levels may be dangerous than preventing fishing when the levels are safe.

The concentration of mercury may not be uniform across the river due to natural variation / randomness.

The situation at the power plant might get worse.

Mercury levels are low in water but still may be high in fish. R1

Note: Award R1 for a reasonable answer that refers to this specific context (and not a generic response that could apply to any model).

[1 mark]

Examiners report

Many candidates did not get this far, but the attempts at the question that were seen were generally good. The greater difficulties were seen in parts (e) and (f), but this could be a problem with time running out.

[N/A]

Syllabus sections

Topic 5—Calculus » AHL 5.17—Phase portrait

Show 63 related questions

22M.2.AHL.TZ1.4a.iii:
Hence state the long-term velocity of the particle.
EXM.3.AHL.TZ0.2f.ii:
State the direction of the trajectory, including the quadrant it is in as it approaches this asymptote.
EXM.2.AHL.TZ0.1c:
As $t \to \infty$ , find an asymptote to the trajectory of the particular solution found in (b)(i) and state if this trajectory will be moving towards or away from the origin.
22M.2.AHL.TZ1.4a.ii:
Find the eigenvalues for the matrix $(\begin{matrix} 0 & 1 \\ - 6 & - 5 \end{matrix})$ .
22M.2.AHL.TZ2.7c:
Hence find the exact solution of the second order differential equation.
21M.3.AHL.TZ2.2f.i:
Use Euler’s method to sketch the trajectory of the phase portrait, for $4 \leq M \leq 7$ and $1.5 \leq S \leq 3$ , over the first nine years.
EXN.2.AHL.TZ0.5a.i:
the eigenvalues.
21N.2.AHL.TZ0.6a:
Given that $y = \dot{x}$ , show that $\dot{y} = - 1.25 x - 3 y$ .
EXN.2.AHL.TZ0.5d:
Write down a condition on the size of the initial population of $Y$ if it is to avoid its population reducing to zero.
EXN.2.AHL.TZ0.5e.i:
Find the value of $t$ at which $x = 0$ .
21N.2.AHL.TZ0.6c.i:
Find the eigenvalues of matrix $A$ .
22M.2.AHL.TZ1.4a.i:
Use the substitution $y = \frac{d x}{d t}$ to show that this equation can be written as

$(\begin{matrix} \frac{d x}{d t} \\ \frac{d y}{d t} \end{matrix}) = (\begin{matrix} 0 & 1 \\ - 6 & - 5 \end{matrix}) (\begin{matrix} x \\ y \end{matrix})$ .
21N.2.AHL.TZ0.6c.ii:
Find the eigenvectors of matrix $A$ .
EXM.2.AHL.TZ0.1b.i:
Given the initial conditions that when $t = 0$ , $x = 150$ , $y = 50$ , find the particular solution.
EXM.3.AHL.TZ0.3i:
With the initial conditions as given in part (b) state if the equilibrium point is stable or unstable.
EXM.3.AHL.TZ0.3k:
With the initial conditions as given in part (j), determine if the equilibrium point is stable or unstable.
EXM.3.AHL.TZ0.2b:
Hence, verify that $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right){e^{2t}}$ is a solution to the above system.
SPM.1.AHL.TZ0.13a:
Find the value of $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ when $t = 0$ .
SPM.1.AHL.TZ0.13b:
On the following axes, sketch a possible trajectory for the growth of the two fungi, making clear any asymptotic behaviour.
EXM.3.AHL.TZ0.3j:
If instead the initial conditions were given as $x = 20$ and $y = 10$ , find the particular solution for $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$ of the original system, in this case.
EXM.3.AHL.TZ0.1b.ii:
Explain why $b$ and $c$ must have opposite signs.
EXM.3.AHL.TZ0.1a:
Show that if the system has two distinct real eigenvalues then ${\left( {a - d} \right)^2} + 4bc > 0$ .
EXM.3.AHL.TZ0.3e:
Solve this system of equations by the eigenvalue method and hence find the general solution for $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$ of the original system.
EXM.3.AHL.TZ0.1b.i:
Find two conditions that must be satisfied by $a$ , $b$ , $c$ , $d$ .
EXM.3.AHL.TZ0.3b:
If initially $x = 100$ and $y = 50$ use Euler’s method with an time increment of 0.1 to find an approximation for the values of $x$ and $y$ when $t = 1$ .
EXM.2.AHL.TZ0.1a:
Use the eigenvalue method to find the general solution to this system of equations.
EXM.3.AHL.TZ0.2a:
Show that the matrix $\left( {\begin{array}{*{20}{c}} 3&1 \\ { - 1}&1 \end{array}} \right)$ has (sadly) only one eigenvalue. Find this eigenvalue and an associated eigenvector.
EXM.3.AHL.TZ0.3g:
Hence find the exact values of $x$ and $y$ when $t = 1$ , giving the answers to 4 significant figures.
EXM.3.AHL.TZ0.3d:
Show how using the substitution $X = x - 10{\text{,}}\,\,Y = y - 20$ transforms the system of differential equations into $\begin{array}{*{20}{c}} {\dot X = X + 2Y} \\ {\dot Y = 2X + Y} \end{array}$ .
EXM.3.AHL.TZ0.1c:
In the case when there is a pair of purely imaginary eigenvalues you are told that the solution will form an ellipse. You are also told that the initial conditions are such that the ellipse is large enough that it will cross both the positive and negative $x$ axes and the positive and negative $y$ axes.

By considering the differential equations at these four crossing point investigate if the trajectory is in a clockwise or anticlockwise direction round the ellipse. Give your decision in terms of $b$ and $c$ . Using part (b) (ii) show that your conclusions are consistent.
EXM.3.AHL.TZ0.3f:
Find the particular solution to the original system, given the initial conditions of part (b).
EXM.3.AHL.TZ0.3c:
Extend this method to conjecture the limit of the ratio $\frac{y}{x}$ as $t \to \infty$ .
21M.2.AHL.TZ2.7a:
Find the eigenvalues and corresponding eigenvectors of the matrix $(\begin{matrix} - 4 & 0 \\ 3 & - 2 \end{matrix})$ .
21M.1.SL.TZ1.12a.i:
Find $A$ in terms of $p$ and $q$ .
21M.1.SL.TZ1.12a.ii:
Show that $A = \frac{192}{q} + 3 q + 48$ .
21M.1.SL.TZ1.12b:
Find $\frac{d A}{d q}$ .
EXM.2.AHL.TZ0.1b.ii:
Hence find the solution when $t = 1$ .
21M.1.SL.TZ1.12c.ii:
Hence, or otherwise, find this value of $q$ .
21M.1.SL.TZ1.12c.i:
Write down an equation Ellis could solve to find this value of $q$ .
EXM.3.AHL.TZ0.2f.i:
Find the equation of this asymptote.
EXM.3.AHL.TZ0.3h:
Use part (f) to find limit of the ratio $\frac{y}{x}$ as $t \to \infty$ .
EXM.3.AHL.TZ0.3a:
Find the equilibrium point for this system.
21M.3.AHL.TZ2.2b.i:
show that, at the equilibrium point, the value of the mackerel population density is $\frac{δ}{γ}$ ;
EXM.3.AHL.TZ0.2c:
Verify that $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} t \\ { - t + 1} \end{array}} \right){e^{2t}}$ is also a solution.
21M.2.AHL.TZ2.7e:
Sketch a phase portrait for the general solution to the system of coupled differential equations for $- 6 \leq x \leq 6$ , $- 6 \leq y \leq 6$ .
21M.2.AHL.TZ2.7b:
Hence, write down the general solution of the system.
21M.3.AHL.TZ2.2a.i:
$γ M S$
21M.3.AHL.TZ2.2a.ii:
$δ S$
21M.3.AHL.TZ2.2b.ii:
find the value of the shark population density at the equilibrium point.
21M.3.AHL.TZ2.2c.ii:
Global warming increases the temperature of the Mediterranean Sea. Alessia claims that this promotes the mackerel population growth rate and hence the value of $α$ is doubled. No other parameter changes.
21M.2.AHL.TZ2.7c:
Determine, with justification, whether the equilibrium point $(0, 0)$ is stable or unstable.
21M.3.AHL.TZ2.2c.i:
Toxic sewage is added to the Mediterranean Sea. Alessia claims this reduces the shark population growth rate and hence the value of $γ$ is halved. No other parameter changes.
21M.3.AHL.TZ2.2f.iii:
State two reasons why Alessia’s conclusion, found in part (f)(ii), might not be valid.
EXM.3.AHL.TZ0.2d:
If initially at $t = 0{\text{,}}\,\,x = 20{\text{,}}\,\,y = 10$ find the particular solution.
EXM.3.AHL.TZ0.2e:
Find the values of $x$ and $y$ when $t = 2$ .
21M.3.AHL.TZ2.2f.ii:
Using your phase portrait, or otherwise, determine whether the mackerel population density would be sufficient to support sustainable fishing during the first nine years.
21M.2.AHL.TZ2.7d:
(i) at $(4, 0)$ .

(ii) at $(- 4, 0)$ .
EXN.2.AHL.TZ0.5e.ii:
Find the population of $Y$ at this value of $t$ . Give your answer to the nearest $10$ herbivores.
EXN.2.AHL.TZ0.5a.ii:
the eigenvectors.
EXN.2.AHL.TZ0.5b:
Hence write down the general solution of the system of equations.
EXN.2.AHL.TZ0.5c:
Sketch the phase portrait for this system, for $x, y > 0$ .

On your sketch show
- the equation of the line defined by the eigenvector in the first quadrant
- at least two trajectories either side of this line using arrows on those trajectories to represent the change in populations as t increases
21N.2.AHL.TZ0.6b:
Write down the matrix $A$ .
21N.2.AHL.TZ0.6d:
Given that when $t = 0$ the shock absorber is displaced $8 cm$ and its velocity is zero, find an expression for $x$ in terms of $t$ .

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