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Date	May 2022	Marks available	1	Reference code	22M.2.AHL.TZ1.4
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 1
Command term	State and Hence	Question number	4	Adapted from	N/A

Question

A particle moves such that its displacement, $x$ metres, from a point $O$ at time $t$ seconds is given by the differential equation

$\frac{d^{2} x}{d t^{2}} + 5 \frac{d x}{d t} + 6 x = 0$

The equation for the motion of the particle is amended to

$\frac{d^{2} x}{d t^{2}} + 5 \frac{d x}{d t} + 6 x = 3 t + 4$ .

When $t = 0$ the particle is stationary at $O$ .

Use the substitution $y = \frac{d x}{d t}$ to show that this equation can be written as

$(\begin{matrix} \frac{d x}{d t} \\ \frac{d y}{d t} \end{matrix}) = (\begin{matrix} 0 & 1 \\ - 6 & - 5 \end{matrix}) (\begin{matrix} x \\ y \end{matrix})$ .

[1]

a.i.

Find the eigenvalues for the matrix $(\begin{matrix} 0 & 1 \\ - 6 & - 5 \end{matrix})$ .

[3]

a.ii.

Hence state the long-term velocity of the particle.

[1]

a.iii.

Use the substitution $y = \frac{d x}{d t}$ to write the differential equation as a system of coupled, first order differential equations.

[2]

b.i.

Use Euler’s method with a step length of $0.1$ to find the displacement of the particle when $t = 1$ .

[5]

b.ii.

Find the long-term velocity of the particle.

[1]

b.iii.

Markscheme

$y = \frac{d x}{d t} \Rightarrow \frac{d y}{d t} + 5 \frac{d x}{d t} + 6 x = 0$ OR $\frac{d y}{d t} + 5 y + 6 x = 0$ M1

Note: Award M1 for substituting $\frac{d y}{d t}$ for $\frac{d^{2} x}{d t^{2}}$ .

$(\begin{matrix} \frac{d x}{d t} \\ \frac{d y}{d t} \end{matrix}) = (\begin{matrix} 0 & 1 \\ - 6 & - 5 \end{matrix}) (\begin{matrix} x \\ y \end{matrix})$ AG

[1 mark]

a.i.

$det (\begin{matrix} - λ & 1 \\ - 6 & - 5 - λ \end{matrix}) = 0$ (M1)

Note: Award M1 for an attempt to find eigenvalues. Any indication that $det (M - λ I) = 0$ has been used is sufficient for the (M1).

$- λ (- 5 - λ) + 6 = 0$ OR $λ^{2} + 5 λ + 6 = 0$ (A1)

$λ = - 2, - 3$ A1

[3 marks]

a.ii.

(on a phase portrait the particle approaches $(0, 0)$ as $t$ increases so long term velocity ( $y$ ) is)

$0$ A1

Note: Only award A1 for $0$ if both eigenvalues in part (a)(ii) are negative. If at least one is positive accept an answer of ‘no limit’ or ‘infinity’, or in the case of one positive and one negative also accept ‘no limit or $0$ (depending on initial conditions)’.

[1 mark]

a.iii.

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

$\frac{d^{2} x}{d t^{2}} = \frac{d y}{d t}$ (A1)

$\frac{d y}{d t} + 5 y + 6 x = 3 t + 4$ A1

[2 marks]

b.i.

recognition that $h = 0.1$ in any recurrence formula (M1)

$(t_{n + 1} = t_{n} + 0.1)$

$x_{n + 1} = x_{n} + 0.1 y_{n}$ (A1)

$y_{n + 1} = y_{n} + 0.1 (3 t_{n} + 4 - 5 y_{n} - 6 x_{n})$ (A1)

(when $t = 1$ ,) $x = 0.64402 \dots \approx 0.644 m$ A2

[5 marks]

b.ii.

recognizing that $y$ is the velocity

$0.5 {m s}^{- 1}$ A1

[2 marks]

b.iii.

Examiners report

It was clear that second order differential equations had not been covered by many schools. Fortunately, many were able to successfully answer part (ii) as this was independent of the other two parts. For part (iii) it was expected that candidates would know that two negative eigenvalues mean the system tends to the origin and so the long-term velocity is 0. Some candidates tried to solve the system. It should be noted that when the command term is ‘state’ then no further working out is expected to be seen.

a.i.

[N/A]

a.ii.

[N/A]

a.iii.

Forming a coupled system from a second order differential equation and solving it using Euler’s method is a technique included in the course guide. Candidates who had learned this technique were successful in this question.

b.i.

[N/A]

b.ii.

[N/A]

b.iii.

Syllabus sections

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

Show 60 related questions

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.
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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Topic 5—Calculus » AHL 5.17—Phase portrait

Topic 1—Number and algebra

Topic 5—Calculus

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