14
2
\end{array} \right) ,
$$
where \(a\) is a constant and \(s\) and \(t\) are scalar parameters.\\
Given that the two lines intersect,
- find the position vector of their point of intersection,
- find the value of \(a\).
Given also that \(\theta\) is the acute angle between the lines,
- find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.\\
8. A small town had a population of 9000 in the year 2001.
In a model, it is assumed that the population of the town, \(P\), at time \(t\) years after 2001 satisfies the differential equation
$$\frac { \mathrm { d } P } { \mathrm {~d} t } = 0.05 P \mathrm { e } ^ { - 0.05 t }$$
- Show that, according to the model, the population of the town in 2011 will be 13300 to 3 significant figures.
- Find the value which the population of the town will approach in the long term, according to the model.\\
9. A curve has parametric equations
$$x = t ( t - 1 ) , \quad y = \frac { 4 t } { 1 - t } , \quad t \neq 1$$
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
The point \(P\) on the curve has parameter \(t = - 1\).
- Show that the tangent to the curve at \(P\) has the equation
$$x + 3 y + 4 = 0$$
The tangent to the curve at \(P\) meets the curve again at the point \(Q\).
- Find the coordinates of \(Q\).