5
\begin{enumerate}[label=(\roman*)]
\item 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]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-3_716_741_1233_662}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{center}
\end{figure}
\item 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]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-4_378_1630_461_214}
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{center}
\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.
\item Find, in terms of $a$,\\
(A) the length of the straight line OE,\\
(B) the maximum height of the arch.
\item 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.
\item 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.
\item 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]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-5_748_1306_319_367}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{center}
\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 .
\item Find the length AE .
\item Find a vector equation of the line BD . Given that the length of BD is 15 metres, find the coordinates of D.
\item Verify that the equation of the plane ABC is

$$- 3 x + 4 y + 5 z = 30$$

Write down a vector normal to this plane.
\item 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 .
\item 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

\begin{itemize}
  \item Write your name, centre number and candidate number in the spaces at the top of this page.
  \item Answer all the questions.
  \item Write your answers in the spaces provided on the question paper.
  \item You are permitted to use a graphical calculator in this paper.
\end{itemize}

\begin{itemize}
  \item The number of marks is given in brackets [ ] at the end of each question or part question.
  \item The insert contains the text for use with the questions.
  \item You may find it helpful to make notes and do some calculations as you read the passage.
  \item You are not required to hand in these notes with your question paper.
  \item 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.
  \item The total number of marks for this paper is 18.
\end{itemize}

\begin{center}
\begin{tabular}{ | c | c | }
\hline
\multicolumn{2}{|c|}{For Examiner's Use} \\
\hline
Qu. & Mark \\
\hline
1 &  \\
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2 &  \\
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3 &  \\
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4 &  \\
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5 &  \\
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6 &  \\
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Total &  \\
\hline
\end{tabular}
\end{center}

1 The marathon is 26 miles and 385 yards long ( 1 mile is 1760 yards). There are now several men who can run 2 miles in 8 minutes. Imagine that an athlete maintains this average speed for a whole marathon. How long does the athlete take?\\

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
\item initially,
\item for large values of $t$.
\item 
\item $\_\_\_\_$\\

5 A model for an athletics record has the form

$$R = A - ( A - B ) \mathrm { e } ^ { - k t } \text { where } A > B > 0 \text { and } k > 0 .$$
\item Sketch the graph of $R$ against $t$, showing $A$ and $B$ on your graph.
\item Name one event for which this might be an appropriate model.
\item \\
\includegraphics[max width=\textwidth, alt={}, center]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-9_803_808_721_575}
\item $\_\_\_\_$
\end{enumerate}

\hfill \mbox{\textit{OCR MEI C4 2006 Q5 [11]}}