Edexcel
Paper 1
Specimen
Q7
7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{96e004d9-c6b6-474b-9b67-06e1771c609e-14_604_1063_251_502}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Figure 4 shows a sketch of part of the curve with equation
$$y = 2 \mathrm { e } ^ { 2 x } - x \mathrm { e } ^ { 2 x } , \quad x \in \mathbb { R }$$
The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.
Use calculus to show that the exact area of \(R\) can be written in the form \(p \mathrm { e } ^ { 4 } + q\), where \(p\) and \(q\) are rational constants to be found.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel
Paper 1
Specimen
Q13
13. (a) Express \(2 \sin \theta - 1.5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
State the value of \(R\) and give the value of \(\alpha\) to 4 decimal places.
Tom models the depth of water, \(D\) metres, at Southview harbour on 18th October 2017 by the formula
$$D = 6 + 2 \sin \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t \leqslant 24$$
where \(t\) is the time, in hours, after 00:00 hours on 18th October 2017.
Use Tom's model to
(b) find the depth of water at 00:00 hours on 18th October 2017,
(c) find the maximum depth of water,
(d) find the time, in the afternoon, when the maximum depth of water occurs. Give your answer to the nearest minute.
Tom's model is supported by measurements of \(D\) taken at regular intervals on 18th October 2017. Jolene attempts to use a similar model in order to model the depth of water at Southview harbour on 19th October 2017.
Jolene models the depth of water, \(H\) metres, at Southview harbour on 19th October 2017 by the formula
$$H = 6 + 2 \sin \left( \frac { 4 \pi x } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi x } { 25 } \right) , \quad 0 \leqslant x \leqslant 24$$
where \(x\) is the time, in hours, after 00:00 hours on 19th October 2017.
By considering the depth of water at 00:00 hours on 19th October 2017 for both models,
(e) (i) explain why Jolene's model is not correct,
(ii) hence find a suitable model for \(H\) in terms of \(x\).