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Date May 2022 Marks available 3 Reference code 22M.2.AHL.TZ2.7
Level Additional Higher Level Paper Paper 2 Time zone Time zone 2
Command term Find Question number 7 Adapted from N/A

Question

An environmental scientist is asked by a river authority to model the effect of a leak from a power plant on the mercury levels in a local river. The variable x measures the concentration of mercury in micrograms per litre.

The situation is modelled using the second order differential equation

d2xdt2+3dxdt+2x=0

where t0 is the time measured in days since the leak started. It is known that when t=0, x=0 and dxdt=1.

If the mercury levels are greater than 0.1 micrograms per litre, fishing in the river is considered unsafe and is stopped.

The river authority decides to stop people from fishing in the river for 10% longer than the time found from the model.

Show that the system of coupled first order equations:

dxdt=y

dydt=-2x-3y

can be written as the given second order differential equation.

[2]
a.

Find the eigenvalues of the system of coupled first order equations given in part (a).

[3]
b.

Hence find the exact solution of the second order differential equation.

[5]
c.

Sketch the graph of x against t, labelling the maximum point of the graph with its coordinates.

[2]
d.

Use the model to calculate the total amount of time when fishing should be stopped.

[3]
e.

Write down one reason, with reference to the context, to support this decision.

[1]
f.

Markscheme

differentiating first equation.         M1

d2xdt2=dydt

substituting in for dydt         M1

=-2x-3y=-2x-3dxdt

therefore d2xdt2+3dxdt+2x=0         AG


Note: The AG line must be seen to award the final M1 mark.

 

[2 marks]

a.

the relevant matrix is 0  1-2  -3           (M1)


Note:  -3  -21  0 is also possible.


(this has characteristic equation) -λ-3-λ+2=0           (A1)

λ=-1, -2         A1

 

[3 marks]

b.

EITHER

the general solution is x=Ae-t+Be-2t             M1


Note: Must have constants, but condone sign error for the M1.


so dxdt=-Ae-t-2Be-2t             M1A1

 

OR

attempt to find eigenvectors           (M1)

respective eigenvectors are 1-1 and 1-2 (or any multiple)

xy=Ae-t1-1+Be-2t1-2           (M1)A1

 

THEN

the initial conditions become:

0=A+B

1=-A-2B             M1

this is solved by A=1, B=-1

so the solution is x=e-t-e-2t            A1

 

[5 marks]

c.

            A1A1

 

Note: Award A1 for correct shape (needs to go through origin, have asymptote at y=0 and a single maximum; condone x<0). Award A1 for correct coordinates of maximum.

 

[2 marks]

d.

intersecting graph with y=0.1         (M1)

so the time fishing is stopped between 2.1830 and 0.11957           (A1)

=2.06 343  days           A1

 

[3 marks]

e.

Any reasonable answer. For example:

There are greater downsides to allowing fishing when the levels may be dangerous than preventing fishing when the levels are safe.

The concentration of mercury may not be uniform across the river due to natural variation / randomness.

The situation at the power plant might get worse.

Mercury levels are low in water but still may be high in fish.           R1

 

Note: Award R1 for a reasonable answer that refers to this specific context (and not a generic response that could apply to any model).

 

[1 mark]

f.

Examiners report

Many candidates did not get this far, but the attempts at the question that were seen were generally good. The greater difficulties were seen in parts (e) and (f), but this could be a problem with time running out.

a.
[N/A]
b.
[N/A]
c.
[N/A]
d.
[N/A]
e.
[N/A]
f.

Syllabus sections

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

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