SPM.1.AHL.TZ0.13

pestleMathematicsAIHLPaper 1SPM· ahl-5-17-phase-portraitsource ↗

The rates of change of the area covered by two types of fungi, X and Y, on a particular tree are given by the following equations, where $x$ is the area covered by X and $y$ is the area covered by Y.

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

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

The matrix $\left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ 2&{ - 2} \end{array}} \right)$ has eigenvalues of 2 and −1 with corresponding eigenvectors $\left( {\begin{array}{*{20}{c}} 2 \\ 1 \end{array}} \right)$ and $\left( {\begin{array}{*{20}{c}} 1 \\ 2 \end{array}} \right)$ .

Initially $x$ = 8 cm² and $y$ = 10 cm².

Find the value of $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ when $t = 0$ .

[2]

On the following axes, sketch a possible trajectory for the growth of the two fungi, making clear any asymptotic behaviour.

[4]

Markscheme / solution

$\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{16 - 20}}{{24 - 20}}$ M1

= −1 A1

[2 marks]

asymptote of trajectory along r $= k\left( {\begin{array}{*{20}{c}} 2 \\ 1 \end{array}} \right)$ M1A1

Note: Award M1A0 if asymptote along $\left( {\begin{array}{*{20}{c}} 1 \\ 2 \end{array}} \right)$ .

trajectory begins at (8, 10) with negative gradient A1A1

[4 marks]

Examiners’ report

[N/A]