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Date	May 2022	Marks available	3	Reference code	22M.2.AHL.TZ2.7
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Find	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 1—Number and algebra » AHL 1.15—Eigenvalues and eigenvectors

Show 60 related questions

22M.2.AHL.TZ1.4a.iii:
Hence state the long-term velocity of the particle.
22M.2.AHL.TZ2.5b:
Find the eigenvalues of $M$ .
22M.2.AHL.TZ2.5c:
Find the eigenvectors of $M$ .
22M.1.AHL.TZ1.11a:
Find an eigenvector corresponding to the eigenvalue of $1$ . Give your answer in the form $(\begin{matrix} a \\ b \end{matrix})$ , where $a, b \in ℤ$ .
EXM.3.AHL.TZ0.5g:
State geometrically what transformation the matrix ${\mathbf{R}}$ represents.
EXM.2.AHL.TZ0.23a:
Find the values of $\lambda$ for which the matrix (A − $\lambda$ I) is singular.
22M.2.AHL.TZ1.4a.ii:
Find the eigenvalues for the matrix $(\begin{matrix} 0 & 1 \\ - 6 & - 5 \end{matrix})$ .
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.3.AHL.TZ0.5i:
Write down the equations of two lines of symmetry for the curve in $x$ and $y$ in part (b).
EXM.2.AHL.TZ0.23b.iii:
Use the result from part (b) (ii) to explain why A is non-singular.
EXM.3.AHL.TZ0.5h.i:
the curve in $X$ and $Y$ in part (e) (ii), giving a reason.
EXM.3.AHL.TZ0.5d.i:
Find matrix R.
EXM.1.AHL.TZ0.42:
Given the matrix A = $\left( {\begin{array}{*{20}{c}} 3&2 \\ { - 1}&0 \end{array}} \right)$ find the values of the real number $k$ for which ${\text{det}}\left( {A - kI} \right) = 0$ where $I = \left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right)$
21M.2.AHL.TZ1.5a:
It is sunny today. Find the probability that it will be sunny in three days’ time.
21M.2.AHL.TZ1.5b:
Find the eigenvalues and eigenvectors of $T$ .
21M.2.AHL.TZ1.5c.ii:
Write down the matrix $D$ .
EXM.3.AHL.TZ0.5h.ii:
the curve in $x$ and $y$ in part (b).
EXM.3.AHL.TZ0.5f:
Find the Cartesian equation satisfied by $u$ and $v$ and state the geometric shape that this curve represents.
19M.1.AHL.TZ0.F_3a.i:
Show that the eigenvalues of A are real if ${\left( {a - d} \right)^2} + 4bc \geqslant 0$ .
19M.1.AHL.TZ0.F_3b.i:
Determine the eigenvalues of B.
EXM.3.AHL.TZ0.5a:
Find the eigenvalues for $M$ . For each eigenvalue find the set of associated eigenvectors.
EXM.3.AHL.TZ0.5b:
Show that the matrix equation $\left( {x\,\,\,y} \right){\mathbf{M}}\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( 6 \right)$ is equivalent to the Cartesian equation $\frac{5}{2}{x^2} + xy + \frac{5}{2}{y^2} = 6$ .
EXM.3.AHL.TZ0.5c.i:
Show that $\left( {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}} \\ {\frac{{ - 1}}{{\sqrt 2 }}} \end{array}} \right)$ and $\left( {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}} \\ {\frac{1}{{\sqrt 2 }}} \end{array}} \right)$ are unit eigenvectors and that they correspond to different eigenvalues.
EXM.2.AHL.TZ0.23c:
Use the values from part (b) (i) to express A⁴ in the form $p$ A+ $q$ I where $p$ , $q \in \mathbb{Z}$ .
19M.1.AHL.TZ0.F_3b.ii:
Determine the corresponding eigenvectors.
20N.1.AHL.TZ0.F_4b.ii:
Assuming that $A$ is non-singular, use the result in part (b)(i) to show that

$A^{- 1} = \frac{1}{β} (α I - A)$ .
EXM.2.AHL.TZ0.23b.i:
Find the value of $m$ and of $n$ .
EXM.1.AHL.TZ0.47:
Given that A = $\left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ { - 3}&4 \end{array}} \right)$ and I = $\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right)$ , find the values of $\lambda$ for which (A – $\lambda$ I) is a singular matrix.
EXM.3.AHL.TZ0.5d.ii:
Show that ${\mathbf{M}} = {{\mathbf{R}}^{ - 1}}\left( {\begin{array}{*{20}{c}} 2&0 \\ 0&3 \end{array}} \right){\mathbf{R}}$ .
19M.1.AHL.TZ0.F_3a.ii:
Deduce that the eigenvalues are real if A is symmetric.
20N.1.AHL.TZ0.F_4b.i:
Verify that

$A^{2} - α A + β I =$ 0.
21M.2.AHL.TZ2.7a:
Find the eigenvalues and corresponding eigenvectors of the matrix $(\begin{matrix} - 4 & 0 \\ 3 & - 2 \end{matrix})$ .
EXM.3.AHL.TZ0.5e.ii:
Hence, find the Cartesian equation satisfied by $X$ and $Y$ .
EXM.3.AHL.TZ0.5e.i:
Verify that $\left( {\begin{array}{*{20}{c}} X&Y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} x&y \end{array}} \right){{\mathbf{R}}^{ - 1}}$ .
21M.2.AHL.TZ1.5d:
Hence find the long-term percentage of sunny days in Vokram.
EXM.2.AHL.TZ0.23b.ii:
Hence show that I = $\frac{1}{5}$ A (6I – A).
EXM.3.AHL.TZ0.5c.ii:
Hence, show that $M\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}} \end{array}} \\ {\begin{array}{*{20}{c}} {\frac{{ - 1}}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}} \end{array}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}} \end{array}} \\ {\begin{array}{*{20}{c}} {\frac{{ - 1}}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}} \end{array}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} 2&0 \\ 0&3 \end{array}} \right)$ .
20N.1.AHL.TZ0.F_4a:
By considering the determinant of a relevant matrix, show that the eigenvalues, $λ$ , of $A$ satisfy the equation

$λ^{2} - α λ + β = 0$ ,

where $α$ and $β$ are functions of $a, b, c, d$ to be determined.
21M.2.AHL.TZ2.4a:
Write down a transition matrix $T$ indicating the annual population movement between clinics.
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.2.AHL.TZ2.7c:
Determine, with justification, whether the equilibrium point $(0, 0)$ is stable or unstable.
21M.2.AHL.TZ2.4b:
Find a prediction for the ratio of the number of patients Doctor Black will have, compared to Doctor Green, after two years.
21M.2.AHL.TZ1.5c.i:
Write down the matrix $P$ .
21M.2.AHL.TZ2.4c:
Find a matrix $P$ , with integer elements, such that $T = P D P^{- 1}$ , where $D$ is a diagonal matrix.
21M.2.AHL.TZ2.4d:
Hence, show that the long-term transition matrix $T^{\infty}$ is given by $T^{\infty} = (\begin{matrix} \frac{10}{17} & \frac{10}{17} \\ \frac{7}{17} & \frac{7}{17} \end{matrix})$ .
21M.2.AHL.TZ2.4e:
Hence, or otherwise, determine the expected ratio of the number of patients Doctor Black would have compared to Doctor Green in the long term.
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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