- Show that \(\int x \mathrm { e } ^ { - 2 x } \mathrm {~d} x = - \frac { 1 } { 4 } \mathrm { e } ^ { - 2 x } ( 1 + 2 x ) + c\).
A vase is made in the shape of the volume of revolution of the curve \(y = x ^ { 1 / 2 } \mathrm { e } ^ { - x }\) about the \(x\)-axis between \(x = 0\) and \(x = 2\) (see Fig. 5).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-3_716_741_1233_662}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{figure} - Show that this volume of revolution is \(\frac { 1 } { 4 } \pi \left( 1 - \frac { 5 } { \mathrm { e } ^ { 4 } } \right)\).
Fig. 6 shows the arch ABCD of a bridge.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-4_378_1630_461_214}
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{figure}
The section from B to C is part of the curve OBCE with parametric equations
$$x = a ( \theta - \sin \theta ) , y = a ( 1 - \cos \theta ) \text { for } 0 \leqslant \theta \leqslant 2 \pi$$
where \(a\) is a constant. - Find, in terms of \(a\),
(A) the length of the straight line OE,
(B) the maximum height of the arch. - Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\).
The straight line sections AB and CD are inclined at \(30 ^ { \circ }\) to the horizontal, and are tangents to the curve at B and C respectively. BC is parallel to the \(x\)-axis. BF is parallel to the \(y\)-axis.
- Show that at the point B the parameter \(\theta\) satisfies the equation
$$\sin \theta = \frac { 1 } { \sqrt { 3 } } ( 1 - \cos \theta )$$
Verify that \(\theta = \frac { 2 } { 3 } \pi\) is a solution of this equation.
Hence show that \(\mathrm { BF } = \frac { 3 } { 2 } a\), and find OF in terms of \(a\), giving your answer exactly. - Find BC and AF in terms of \(a\).
Given that the straight line distance AD is 20 metres, calculate the value of \(a\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-5_748_1306_319_367}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{figure}
Fig. 7 illustrates a house. All units are in metres. The coordinates of A, B, C and E are as shown. BD is horizontal and parallel to AE . - Find the length AE .
- Find a vector equation of the line BD . Given that the length of BD is 15 metres, find the coordinates of D.
- Verify that the equation of the plane ABC is
$$- 3 x + 4 y + 5 z = 30$$
Write down a vector normal to this plane.
- Show that the vector \(\left( \begin{array} { l } 4
3
5 \end{array} \right)\) is normal to the plane ABDE . Hence find the equation of the plane ABDE . - Find the angle between the planes ABC and ABDE .
RECOGNISING ACHIEVEMENT
\section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS}
Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education
\section*{MEI STRUCTURED MATHEMATICS}
Applications of Advanced Mathematics (C4)
\section*{Paper B: Comprehension}
Monday 12 JUNE 2006 Afternoon Up to 1 hour
Additional materials:
Rough paper
MEI Examination Formulae and Tables (MF2)
TIME Up to 1 hour
- Write your name, centre number and candidate number in the spaces at the top of this page.
- Answer all the questions.
- Write your answers in the spaces provided on the question paper.
- You are permitted to use a graphical calculator in this paper.
- The number of marks is given in brackets [ ] at the end of each question or part question.
- The insert contains the text for use with the questions.
- You may find it helpful to make notes and do some calculations as you read the passage.
- You are not required to hand in these notes with your question paper.
- You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
- The total number of marks for this paper is 18.
| For Examiner's Use |
| Qu. | Mark |
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 | |
| 6 | |
| Total | |
2 According to the linear model, in which calendar year would the record for the men's mile first become negative?
3 Explain the statement in line 93 "According to this model the 2-hour marathon will never be run."
4 Explain how the equation in line 49,
$$R = L + ( U - L ) \mathrm { e } ^ { - k t } ,$$
is consistent with Fig. 2 - initially,
- for large values of \(t\).
- \(\_\_\_\_\)
5 A model for an athletics record has the form
$$R = A - ( A - B ) \mathrm { e } ^ { - k t } \text { where } A > B > 0 \text { and } k > 0 .$$ - Sketch the graph of \(R\) against \(t\), showing \(A\) and \(B\) on your graph.
- Name one event for which this might be an appropriate model.
\includegraphics[max width=\textwidth, alt={}, center]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-9_803_808_721_575}- \(\_\_\_\_\)