8
5
- 2 \end{array} \right) + \mu \left( \begin{array} { r } 3
4
- 5 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant. The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\).

1. Find the coordinates of \(A\).
2. Find the value of the constant \(p\).
3. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in degrees to 2 decimal places. The point \(B\) lies on \(l _ { 2 }\) where \(\mu = 1\)
4. Find the shortest distance from the point \(B\) to the line \(l _ { 1 }\), giving your answer to 3 significant figures.
   1. A curve \(C\) has parametric equations
   $$x = 4 t + 3 , \quad y = 4 t + 8 + \frac { 5 } { 2 t } , \quad t \neq 0$$
5. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point on \(C\) where \(t = 2\), giving your answer as a fraction in its simplest form.
6. Show that the cartesian equation of the curve \(C\) can be written in the form $$y = \frac { x ^ { 2 } + a x + b } { x - 3 } , \quad x \neq 3$$ where \(a\) and \(b\) are integers to be determined.
   6. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{89d4a7a5-3f4f-4d16-b14e-a27243cedd78-11_666_993_244_392} \captionsetup{labelformat=empty} \caption{Diagram not to scale}
   \end{figure} Figure 2 Figure 2 shows a sketch of the curve with equation \(y = \sqrt { ( 3 - x ) ( x + 1 ) } , 0 \leqslant x \leqslant 3\)
   The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis, and the \(y\)-axis.
7. Use the substitution \(x = 1 + 2 \sin \theta\) to show that $$\int _ { 0 } ^ { 3 } \sqrt { ( 3 - x ) ( x + 1 ) } d x = k \int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \cos ^ { 2 } \theta d \theta$$ where \(k\) is a constant to be determined.
8. Hence find, by integration, the exact area of \(R\). 7. (a) Express \(\frac { 2 } { P ( P - 2 ) }\) in partial fractions. A team of biologists is studying a population of a particular species of animal. The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 2 } P ( P - 2 ) \cos 2 t , t \geqslant 0$$ where \(P\) is the population in thousands, and \(t\) is the time measured in years since the start of the study. Given that \(P = 3\) when \(t = 0\),
9. solve this differential equation to show that $$P = \frac { 6 } { 3 - \mathrm { e } ^ { \frac { 1 } { 2 } \sin 2 t } }$$
10. find the time taken for the population to reach 4000 for the first time. Give your answer in years to 3 significant figures.
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{89d4a7a5-3f4f-4d16-b14e-a27243cedd78-15_696_1418_287_262} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = 3 ^ { x }$$ The point \(P\) lies on \(C\) and has coordinates \(( 2,9 )\).
    The line \(l\) is a tangent to \(C\) at \(P\). The line \(l\) cuts the \(x\)-axis at the point \(Q\).
11. Find the exact value of the \(x\) coordinate of \(Q\). The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(l\). This region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
12. Use integration to find the exact value of the volume of the solid generated. Give your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are exact constants.
    [0pt] [You may assume the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.]