6 Let \(\mathrm { f } ( x ) = \frac { 5 a } { ( 2 x - a ) ( 3 a - x ) }\), where \(a\) is a positive constant.
- Express \(\mathrm { f } ( x )\) in partial fractions.
- Hence show that \(\int _ { a } ^ { 2 a } \mathrm { f } ( x ) \mathrm { d } x = \ln 6\).
\(7 \quad\) Two lines have equations \(\mathbf { r } = \left( \begin{array} { l } 1
3
2 \end{array} \right) + s \left( \begin{array} { r } 2
- 1
3 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { l } 2
1
4 \end{array} \right) + t \left( \begin{array} { r } 1
- 1
4 \end{array} \right)\). - Show that the lines are skew.
- Find the acute angle between the directions of the two lines.