7 Data suggest that the number of cases of infection from a particular disease tends to oscillate between two values over a period of approximately 6 months.
- Suppose that the number of cases, \(P\) thousand, after time \(t\) months is modelled by the equation \(P = \frac { 2 } { 2 - \sin t }\). Thus, when \(t = 0 , P = 1\).
- By considering the greatest and least values of \(\sin t\), write down the greatest and least values of \(P\) predicted by this model.
- Verify that \(P\) satisfies the differential equation \(\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 2 } P ^ { 2 } \cos t\).
- An alternative model is proposed, with differential equation
$$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 2 } \left( 2 P ^ { 2 } - P \right) \cos t$$
As before, \(P = 1\) when \(t = 0\).
- Express \(\frac { 1 } { P ( 2 P - 1 ) }\) in partial fractions.
- Solve the differential equation (*) to show that
$$\ln \left( \frac { 2 P - 1 } { P } \right) = \frac { 1 } { 2 } \sin t$$
This equation can be rearranged to give \(P = \frac { 1 } { 2 - \mathrm { e } ^ { \frac { 1 } { 2 } \sin t } }\).
- Find the greatest and least values of \(P\) predicted by this model.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9296c786-a42a-4aa5-b326-39adbb544cbc-05_609_622_301_719}
\captionsetup{labelformat=empty}
\caption{Fig. 8}
\end{figure}
In a theme park ride, a capsule C moves in a vertical plane (see Fig. 8). With respect to the axes shown, the path of C is modelled by the parametric equations
$$x = 10 \cos \theta + 5 \cos 2 \theta , \quad y = 10 \sin \theta + 5 \sin 2 \theta , \quad ( 0 \leqslant \theta < 2 \pi )$$
where \(x\) and \(y\) are in metres. - Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { \cos \theta + \cos 2 \theta } { \sin \theta + \sin 2 \theta }\).
Verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(\theta = \frac { 1 } { 3 } \pi\). Hence find the exact coordinates of the highest point A on the path of C .
- Express \(x ^ { 2 } + y ^ { 2 }\) in terms of \(\theta\). Hence show that
$$x ^ { 2 } + y ^ { 2 } = 125 + 100 \cos \theta$$
- Using this result, or otherwise, find the greatest and least distances of C from O .
You are given that, at the point B on the path vertically above O ,
$$2 \cos ^ { 2 } \theta + 2 \cos \theta - 1 = 0$$
- Using this result, and the result in part (ii), find the distance OB. Give your answer to 3 significant figures.
\section*{ADVANCED GCE UNIT MATHEMATICS (MEI)}
Applications of Advanced Mathematics (C4)
\section*{Paper B: Comprehension}
\section*{THURSDAY 14 JUNE 2007}
Afternoon
Time: Up to 1 hour
Additional materials:
Rough paper
MEI Examination Formulae and Tables (MF2)
Candidate
Name □
Centre
Number
sufficient detail of the working to indicate that a correct method is being used.
1 This basic cycloid has parametric equations
$$x = a \theta - a \sin \theta , \quad y = a - a \cos \theta$$
\includegraphics[max width=\textwidth, alt={}]{9296c786-a42a-4aa5-b326-39adbb544cbc-10_307_1138_445_411}
Find the coordinates of the points M and N , stating the value of \(\theta\) at each of them.
Point M
Point N
2 A sea wave has parametric equations (in suitable units)
$$x = 7 \theta - 0.25 \sin \theta , \quad y = 0.25 \cos \theta$$
Find the wavelength and height of the wave.
3 The graph below shows the profile of a wave. - Assuming that it has parametric equations of the form given on line 68 , find the values of \(a\) and \(b\).
- Investigate whether the ratio of the trough length to the crest length is consistent with this shape.
\includegraphics[max width=\textwidth, alt={}, center]{9296c786-a42a-4aa5-b326-39adbb544cbc-11_312_1141_623_415} - \(\_\_\_\_\)
- \(\_\_\_\_\)
4 This diagram illustrates two wave shapes \(U\) and \(V\). They have the same wavelength and the same height.
\includegraphics[max width=\textwidth, alt={}, center]{9296c786-a42a-4aa5-b326-39adbb544cbc-12_423_1552_356_205}
One of the curves is a sine wave, the other is a curtate cycloid. - State which is which, justifying your answer.
- \(\_\_\_\_\)
The parametric equations for the curves are:
$$x = a \theta , \quad y = b \cos \theta ,$$
and
$$x = a \theta - b \sin \theta , \quad y = b \cos \theta .$$ - Show that the distance marked \(d\) on the diagram is equal to \(b\).
- Hence justify the statement in lines 109 to 111: "In such cases, the curtate cycloid and the sine curve with the same wavelength and height are very similar and so the sine curve is also a good model."
- \(\_\_\_\_\)
- \(\_\_\_\_\)
5 The diagram shows a curtate cycloid with scales given. Show that this curve could not be a scale drawing of the shape of a stable sea wave.
\includegraphics[max width=\textwidth, alt={}, center]{9296c786-a42a-4aa5-b326-39adbb544cbc-13_289_1310_397_331}