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Date	November 2021	Marks available	2	Reference code	21N.2.AHL.TZ0.6
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Show that	Question number	6	Adapted from	N/A

Question

A shock absorber on a car contains a spring surrounded by a fluid. When the car travels over uneven ground the spring is compressed and then returns to an equilibrium position.

The displacement, $x$ , of the spring is measured, in centimetres, from the equilibrium position of $x = 0$ . The value of $x$ can be modelled by the following second order differential equation, where $t$ is the time, measured in seconds, after the initial displacement.

$\ddot{x} + 3 \dot{x} + 1.25 x = 0$

The differential equation can be expressed in the form $(\begin{matrix} \dot{x} \\ \dot{y} \end{matrix}) = A (\begin{matrix} x \\ y \end{matrix})$ , where $A$ is a $2 \times 2$ matrix.

Given that $y = \dot{x}$ , show that $\dot{y} = - 1.25 x - 3 y$ .

[2]

Write down the matrix $A$ .

[1]

Find the eigenvalues of matrix $A$ .

[3]

c.i.

Find the eigenvectors of matrix $A$ .

[3]

c.ii.

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 $t$ .

[6]

Markscheme

$y = \dot{x} \Rightarrow \dot{y} = \ddot{x}$ A1

$\dot{y} + 3 (y) + 1.25 x = 0$ R1

Note: If no explicit reference is made to $\dot{y} = \ddot{x}$ , or equivalent, award A0R1 if second line is seen. If $\frac{d y}{d x}$ used instead of $\frac{d y}{d t}$ , award A0R0.

$\dot{y} = - 3 y - 1.25 x$ AG

[2 marks]

$A = (\begin{matrix} 0 & 1 \\ - 1.25 & - 3 \end{matrix})$ A1

[1 mark]

$|\begin{matrix} - λ & 1 \\ - 1.25 & - 3 - λ \end{matrix}| = 0$ (M1)

$λ (λ + 3) + 1.25 = 0$ (A1)

$λ = - 2.5; λ = - 0.5$ A1

[3 marks]

c.i.

$(\begin{matrix} 2.5 & 1 \\ - 1.25 & - 0.5 \end{matrix}) (\begin{matrix} a \\ b \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix})$ (M1)

$2.5 a + b = 0$

$v_{1} = (\begin{matrix} - 2 \\ 5 \end{matrix})$ A1

$(\begin{matrix} 0.5 & 1 \\ - 1.25 & - 2.5 \end{matrix}) (\begin{matrix} a \\ b \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix})$

$0.5 a + b = 0$

$v_{2} = (\begin{matrix} - 2 \\ 1 \end{matrix})$ A1

Note: Award M1 for a valid attempt to find either eigenvector. Accept equivalent forms of the eigenvectors.
Do not award FT for eigenvectors that do not satisfy both rows of the matrix.

[3 marks]

c.ii.

$(\begin{matrix} x \\ y \end{matrix}) = A e^{- 2.5 t} (\begin{matrix} - 2 \\ 5 \end{matrix}) + B e^{- 0.5 t} (\begin{matrix} - 2 \\ 1 \end{matrix})$ M1A1

$t = 0 \Rightarrow x = 8, \dot{x} = y = 0$ (M1)

$- 2 A - 2 B = 8$

$5 A + B = 0$ (M1)

$A = 1; B = - 5$ A1

$x = - 2 e^{- 2.5 t} + 10 e^{- 0.5 t}$ A1

Note: Do not award the final A1 if the answer is given in the form $(\begin{matrix} x \\ y \end{matrix}) = A e^{- 2.5 t} (\begin{matrix} - 2 \\ 5 \end{matrix}) + B e^{- 0.5 t} (\begin{matrix} - 2 \\ 1 \end{matrix})$ .

[6 marks]

Examiners report

There were many good attempts at this problem, although simple errors often complicated things. In part (a) an explicit statement of the relationship between the second derivative of $x$ and the first derivative of $y$ was often issing. Then in part (b) there seemed to be confusion about the matrix, with the correct values often placed in the wrong row or column of the matrix. Despite these errors, candidates made good attempts at finding eigenvalues and eigenvectors. It is to be noted that an error in solving the quadratic equation to find the eigenvectors means that follow-through marks are unlikely to be awarded since the eigenvectors are not reasonable answers and will not be consistent with the eigenvalues. Candidates need to take real care at this point of a question in part (c)(i). A significant number of candidates did not write down the final answer correctly, leaving their final answer in vector form, rather than “ $x =$ ….” as asked for in the question.

c.i.

c.ii.

Syllabus sections

Topic 1—Number and algebra » AHL 1.15—Eigenvalues and eigenvectors

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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 $(\begin{matrix} 0 & 1 \\ - 6 & - 5 \end{matrix})$ .
22M.2.AHL.TZ2.7b:
Find the eigenvalues of the system of coupled first order equations given in part (a).
EXN.2.AHL.TZ0.5a.i:
the eigenvalues.
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:
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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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Topic 5—Calculus » AHL 5.17—Phase portrait

Topic 1—Number and algebra

Topic 5—Calculus

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