10
\end{array} \right) + \lambda \left( \begin{array} { r }
6
c
- 1
\end{array} \right)$$
where \(a , b\) and \(c\) are constants and \(\lambda\) is a parameter.
Given that the point \(A\) lies on the line \(l\),
- find the value of \(a\).
Given also that the vector \(\overrightarrow { A B }\) is perpendicular to \(l\),
- find the values of \(b\) and \(c\),
- find the distance \(A B\).
The image of the point \(B\) after reflection in the line \(l\) is the point \(B ^ { \prime }\).
- Find the position vector of the point \(B ^ { \prime }\).
7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_542_1164_251_477}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of the curve \(C\) with parametric equations
$$x = 27 \sec ^ { 3 } t , y = 3 \tan t , \quad 0 \leqslant t \leqslant \frac { \pi } { 3 }$$ - Find the gradient of the curve \(C\) at the point where \(t = \frac { \pi } { 6 }\)
- Show that the cartesian equation of \(C\) may be written in the form
$$y = \left( x ^ { \frac { 2 } { 3 } } - 9 \right) ^ { \frac { 1 } { 2 } } , \quad a \leqslant x \leqslant b$$
stating the values of \(a\) and \(b\).
(3)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_581_1173_1628_475}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
The finite region \(R\) which is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 125\) is shown shaded in Figure 3. This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution. - Use calculus to find the exact value of the volume of the solid of revolution.
8. In an experiment testing solid rocket fuel, some fuel is burned and the waste products are collected. Throughout the experiment the sum of the masses of the unburned fuel and waste products remains constant.
Let \(x\) be the mass of waste products, in kg , at time \(t\) minutes after the start of the experiment. It is known that at time \(t\) minutes, the rate of increase of the mass of waste products, in kg per minute, is \(k\) times the mass of unburned fuel remaining, where \(k\) is a positive constant.
The differential equation connecting \(x\) and \(t\) may be written in the form
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( M - x ) , \text { where } M \text { is a constant. }$$
- Explain, in the context of the problem, what \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) and \(M\) represent.
Given that initially the mass of waste products is zero,
- solve the differential equation, expressing \(x\) in terms of \(k , M\) and \(t\).
Given also that \(x = \frac { 1 } { 2 } M\) when \(t = \ln 4\),
- find the value of \(x\) when \(t = \ln 9\), expressing \(x\) in terms of \(M\), in its simplest form.