14
- 6
- 13
\end{array} \right) + \lambda \left( \begin{array} { r }
- 2
1
4
\end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r }
p
- 7
4
\end{array} \right) + \mu \left( \begin{array} { l }
q
2
1
\end{array} \right)$$
where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) and \(q\) are constants.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular,
- show that \(q = 3\)
Given further that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at point \(X\), find
- the value of \(p\),
- the coordinates of \(X\).
The point \(A\) lies on \(l _ { 1 }\) and has position vector \(\left( \begin{array} { r } 6
- 2
3 \end{array} \right)\)
Given that point \(B\) also lies on \(l _ { 1 }\) and that \(A B = 2 A X\) - find the two possible position vectors of \(B\).
12.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{03548211-79cb-4629-b6ca-aa9dfcc77a33-21_615_732_233_605}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Figure 4 shows a sketch of part of the curve \(C\) with equation
$$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 4 , \quad x > 0$$
The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 3\) - Complete the table below with the value of \(y\) corresponding to \(x = 2\). Give your answer to 4 decimal places.
| \(x\) | 1 | 1.5 | 2 | 2.5 | 3 |
| \(y\) | 2 | 1.3041 | | 0.9089 | 1.2958 |
- Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(S\), giving your answer to 3 decimal places.
- Use calculus to find the exact area of \(S\).
Give your answer in the form \(\frac { a } { b } + \ln c\), where \(a , b\) and \(c\) are integers.
- Hence calculate the percentage error in using your answer to part (b) to estimate the area of \(S\). Give your answer to one decimal place.
- Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of \(S\).
13. (a) Express \(10 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\)
Give the exact value of \(R\) and give the value of \(\alpha\) to 2 decimal places.
Alana models the height above the ground of a passenger on a Ferris wheel by the equation
$$H = 12 - 10 \cos ( 30 t ) ^ { \circ } + 3 \sin ( 30 t ) ^ { \circ }$$
where the height of the passenger above the ground is \(H\) metres at time \(t\) minutes after the wheel starts turning.
\includegraphics[max width=\textwidth, alt={}, center]{03548211-79cb-4629-b6ca-aa9dfcc77a33-23_419_567_516_1160} - Calculate
- the maximum value of \(H\) predicted by this model,
- the value of \(t\) when this maximum first occurs.
Give each answer to 2 decimal places.
- Calculate the value of \(t\) when the passenger is 18 m above the ground for the first time. Give your answer to 2 decimal places.
- Determine the time taken for the Ferris wheel to complete two revolutions.