10
3
\end{array} \right) + \lambda \left( \begin{array} { c }
2
- 2
1
\end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }
5
2
4
\end{array} \right) + \mu \left( \begin{array} { c }
3
1
- 2
\end{array} \right)
\end{aligned}$$
\(P\) is the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
- Find the position vector of \(P\).
- Find, correct to 1 decimal place, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\). \(Q\) is a point on \(l _ { 1 }\) which is 12 metres away from \(P . R\) is the point on \(l _ { 2 }\) such that \(Q R\) is perpendicular to \(l _ { 1 }\).
- Determine the length \(Q R\).
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8. In this question you must show detailed reasoning.
The equation \(\mathrm { f } ( x ) = 0\), where \(\mathrm { f } ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 26 x + 169\), has a root \(x = 2 + 3 \mathrm { i }\). - Express \(\mathrm { f } ( x )\) as a product of two quadratic factors.
- Hence write down all the roots of the equation \(\mathrm { f } ( x ) = 0\).
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9. \(O\) is the origin of a coordinate system whose units are cm .
The points \(A , B , C\) and \(D\) have coordinates \(( 1,0 ) , ( 1,4 ) , ( 6,9 )\) and \(( 0,9 )\) respectively.
The arc \(B C\) is part of the curve with equation \(x ^ { 2 } + ( y - 10 ) ^ { 2 } = 37\).
The closed shape \(O A B C D\) is formed, in turn, from the line segments \(O A\) and \(A B\), the arc \(B C\) and the line segments \(C D\) and \(D O\) (see diagram).
A funnel can be modelled by rotating \(O A B C D\) by \(2 \pi\) radians about the \(y\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{7126440a-3ac4-4dc5-b39f-09d7d6b5c87b-18_663_1166_541_210}
Find the volume of the funnel according to the model.
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10. A transformation is equivalent to a shear parallel to the \(x\)-axis followed by a shear parallel to the \(y\)-axis and is represented by the matrix \(\left( \begin{array} { c c } 1 & s
t & 0 \end{array} \right)\).
Find in terms of \(s\) the matrices which represent each of the shears.
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