3. \(f ( x ) = \frac { 1 + 14 x } { ( 1 - x ) ( 1 + 2 x ) } , \quad | x | < \frac { 1 } { 2 } .\)
- Express \(\mathrm { f } ( x )\) in partial fractions.
(3) - Hence find the exact value of \(\int _ { \frac { 1 } { 6 } } ^ { \frac { 1 } { 3 } } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in the form \(\ln p\), where \(p\) is rational.
(5) - Use the binomial theorem to expand \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying each term.
(5)
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- continued
- The line \(l _ { 1 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { c } 11
5
6 \end{array} \right) + \lambda \left( \begin{array} { l } 4
2
4 \end{array} \right)\), where \(\lambda\) is a parameter.
The line \(l _ { 2 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { c } 24
4
13 \end{array} \right) + \mu \left( \begin{array} { l } 7
1
5 \end{array} \right)\), where \(\mu\) is a parameter. - Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
- Find the coordinates of their point of intersection.
Given that \(\theta\) is the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\),
- find the value of \(\cos \theta\). Give your answer in the form \(k \sqrt { } 3\), where \(k\) is a simplified fraction.