EXN.1.SL.TZ0.6

pestleMathematicsAISLPaper 1EXN· sl-2-5-modelling-functionssource ↗

The size of the population $(P)$ of migrating birds in a particular town can be approximately modelled by the equation $P = a \sin (b t) + c, a, b, c \in ℝ^{+}$ , where $t$ is measured in months from the time of the initial measurements.

In a $12$ month period the maximum population is $2600$ and occurs when $t = 3$ and the minimum population is $800$ and occurs when $t = 9$ .

This information is shown on the graph below.

Find the value of $a$ .

[2]

a.i.

Find the value of $b$ .

[2]

a.ii.

Find the value of $c$ .

[1]

a.iii.

Find the value of $t$ at which the population first reaches $2200$ .

[2]

Markscheme / solution

* 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.

$\frac{2600 - 800}{2} = 900$ (M1)A1

[2 marks]

a.i.

$\frac{360}{12} = 30$ (M1)A1

Note: Accept $\frac{2 π}{12} = 0.524 (0.523598 \dots)$ .

[2 marks]

a.ii.

$\frac{2600 + 800}{2} = 1700$ A1

[1 mark]

a.iii.

Solve $900 \sin (30 t) + 1700 = 2200$ (M1)

$t = 1.12 (1.12496 \dots)$ A1

[2 marks]

Examiners’ report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

a.iii.

[N/A]