Date | November Example question | Marks available | 3 | Reference code | EXN.2.AHL.TZ0.5 |
Level | Additional Higher Level | Paper | Paper 2 | Time zone | Time zone 0 |
Command term | Find | Question number | 5 | Adapted from | N/A |
Question
A change in grazing habits has resulted in two species of herbivore, and , competing for food on the same grasslands. At time environmentalists begin to record the sizes of both populations. Let the size of the population of be , and the size of the population be . The following model is proposed for predicting the change in the sizes of the two populations:
for
For this system of coupled differential equations find
When has a population of .
It is known that has an initial population of .
the eigenvalues.
the eigenvectors.
Hence write down the general solution of the system of equations.
Sketch the phase portrait for this system, for .
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
Write down a condition on the size of the initial population of if it is to avoid its population reducing to zero.
Find the value of at which .
Find the population of at this value of . Give your answer to the nearest herbivores.
Markscheme
* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.
(M1)(A1)
and A1
[3 marks]
Attempt to solve either
or
or equivalent (M1)
or A1A1
Note: accept equivalent forms
[3 marks]
A1
[1 mark]
A1A1A1
Note: A1 for correctly labelled, A1 for at least two trajectories above and A1 for at least two trajectories below , including arrows.
[3 marks]
A1
[1 mark]
At M1A1
Note: Award M1 for the substitution of and
Hence A1A1
M1
(years) A1
[6 marks]
(M1)
(to the nearest animals) A1
[2 marks]