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Date	May 2022	Marks available	3	Reference code	22M.2.AHL.TZ2.5
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Find	Question number	5	Adapted from	N/A

Question

A geneticist uses a Markov chain model to investigate changes in a specific gene in a cell as it divides. Every time the cell divides, the gene may mutate between its normal state and other states.

The model is of the form

$(\begin{matrix} X_{n + 1} \\ Z_{n + 1} \end{matrix}) = M (\begin{matrix} X_{n} \\ Z_{n} \end{matrix})$

where $X_{n}$ is the probability of the gene being in its normal state after dividing for the $n th$ time, and $Z_{n}$ is the probability of it being in another state after dividing for the $n th$ time, where $n \in ℕ$ .

Matrix $M$ is found to be $(\begin{matrix} 0.94 & b \\ 0.06 & 0.98 \end{matrix})$ .

The gene is in its normal state when $n = 0$ . Calculate the probability of it being in its normal state

Write down the value of $b$ .

[1]

a.i.

What does $b$ represent in this context?

[1]

a.ii.

Find the eigenvalues of $M$ .

[3]

Find the eigenvectors of $M$ .

[3]

when $n = 5$ .

[2]

d.i.

in the long term.

[2]

d.ii.

Markscheme

$0.02$ A1

[1 mark]

a.i.

the probability of mutating from ‘not normal state’ to ‘normal state’ A1

Note: The A1 can only be awarded if it is clear that transformation is from the mutated state.

[1 mark]

a.ii.

$det (\begin{matrix} 0.94 - λ & 0.02 \\ 0.06 & 0.98 - λ \end{matrix}) = 0$ (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).

$(0.94 - λ) (0.98 - λ) - 0.0012 = 0$ OR $λ^{2} - 1.92 λ + 0.92 = 0$ (A1)

$λ = 1, 0.92 (\frac{23}{25})$ A1

[3 marks]

$(\begin{matrix} 0.94 & 0.02 \\ 0.06 & 0.98 \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} x \\ y \end{matrix})$ OR $(\begin{matrix} 0.94 & 0.02 \\ 0.06 & 0.98 \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = 0.92 (\begin{matrix} x \\ y \end{matrix})$ (M1)

Note: Award M1 can be awarded for attempting to find either eigenvector.

$0.02 y - 0.06 x = 0$ OR $0.02 y + 0.02 x = 0$

$(\begin{matrix} 1 \\ 3 \end{matrix})$ and $(\begin{matrix} 1 \\ - 1 \end{matrix})$ A1A1

Note: Accept any multiple of the given eigenvectors.

[3 marks]

${(\begin{matrix} 0.94 & 0.02 \\ 0.06 & 0.98 \end{matrix})}^{5} (\begin{matrix} 1 \\ 0 \end{matrix})$ OR $(\begin{matrix} 0.744 & 0.0852 \\ 0.256 & 0.915 \end{matrix}) (\begin{matrix} 1 \\ 0 \end{matrix})$ (M1)

Note: Condone omission of the initial state vector for the M1.

$0.744 (0.744311 \dots)$ A1

[2 marks]

d.i.

$(\begin{matrix} 0.25 \\ 0.75 \end{matrix})$ (A1)

Note: Award A1 for $(\begin{matrix} 0.25 \\ 0.75 \end{matrix})$ OR $(\begin{matrix} 0.25 & 0.25 \\ 0.75 & 0.75 \end{matrix})$ seen.

$0.25$ A1

[2 marks]

d.ii.

Examiners report

There was some difficulty in interpreting the meaning of the values in the transition matrix, but most candidates did well with the rest of the question. In part (d) there was frequently evidence of a correct method, but a failure to identify the correct probabilities. It was surprising to see a significant number of candidates diagonalizing the matrix in part (d) and this often led to errors. Clearly this was not necessary.

a.i.

[N/A]

a.ii.

[N/A]

d.i.

[N/A]

d.ii.

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.TZ2.5c:
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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