Edexcel
Paper 1
Specimen
Q13
11 marks
Standard +0.3
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 \(x\).
Edexcel
Paper 2
2018
June
Q13
10 marks
Challenging +1.2
13.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{580fc9b9-d78c-4a86-91fc-22638cb5186d-38_714_826_251_621}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x \ln x , x > 0\)
The line \(l\) is the normal to \(C\) at the point \(P ( \mathrm { e } , \mathrm { e } )\)
The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(x\)-axis.
Show that the exact area of \(R\) is \(A e ^ { 2 } + B\) where \(A\) and \(B\) are rational numbers to be found.
(10)
Edexcel
Paper 2
2019
June
Q4
6 marks
Standard +0.3
4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa4afaf4-fe5d-4f3a-b3de-9600d5502a49-08_620_679_251_740}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
The curve \(C _ { 1 }\) with parametric equations
$$x = 10 \cos t , \quad y = 4 \sqrt { 2 } \sin t , \quad 0 \leqslant t < 2 \pi$$
meets the circle \(C _ { 2 }\) with equation
$$x ^ { 2 } + y ^ { 2 } = 66$$
at four distinct points as shown in Figure 2.
Given that one of these points, \(S\), lies in the 4th quadrant, find the Cartesian coordinates of \(S\).