13
- 3
- 8
\end{array} \right) + t \left( \begin{array} { r }
- 5
3
4
\end{array} \right)$$
The point \(P\) has position vector \(\left( \begin{array} { r } - 7 \\ 2 \\ 7 \end{array} \right)\) ．\\
The point \(P ^ { \prime }\) is the reflection of \(P\) in \(L\) ．\\
（a）Find the position vector of \(P ^ { \prime }\) ．\\
（b）Show that the point \(A\) with position vector \(\left( \begin{array} { r } - 7 \\ 9 \\ 8 \end{array} \right)\) lies on \(L\) ．\\
（c）Show that angle \(P A P ^ { \prime } = 120 ^ { \circ }\) ．
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12d3f92f-8464-4ba1-93a2-c7b841e3d3de-5_483_1367_1263_347}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
The point \(B\) lies on \(L\) and \(A P B P ^ { \prime }\) forms a kite as shown in Figure 3.\\
The area of the kite is \(50 \sqrt { } 3\)\\
（d）Find the position vector of the point \(B\) ．\\
（e）Show that angle \(B P A = 90 ^ { \circ }\) ．
The circle \(C\) passes through the points \(A , P , P ^ { \prime }\) and \(B\) ．\\
（f）Find the position vector of the centre of \(C\) ．\\
7.\\
\includegraphics[max width=\textwidth, alt={}, center]{12d3f92f-8464-4ba1-93a2-c7b841e3d3de-6_675_1145_237_459}
\section*{Figure 4}
（a）Figure 4 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) ，where
$$\mathrm { f } ( x ) = \frac { x ^ { 2 } - 5 } { 3 - x } , \quad x \in \mathbb { R } , x \neq 3$$
The curve has a minimum at the point \(A\) ，with \(x\)－coordinate \(\alpha\) ，and a maximum at the point \(B\) ， with \(x\)－coordinate \(\beta\) ．
Find the value of \(\alpha\) ，the value of \(\beta\) and the \(y\)－coordinates of the points \(A\) and \(B\) ．\\
(b) The functions g and h are defined as follows
$$\begin{array} { l l }
\mathrm { g } : x \rightarrow x + p & x \in \mathbb { R }
\mathrm {~h} : x \rightarrow | x | & x \in \mathbb { R }
\end{array}$$
where \(p\) is a constant.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12d3f92f-8464-4ba1-93a2-c7b841e3d3de-7_673_1338_591_367}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
Figure 5 shows a sketch of the curve with equation \(y = \mathrm { h } ( \mathrm { fg } ( x ) + q ) , x \in \mathbb { R } , x \neq 0\), where \(q\) is a constant. The curve is symmetric about the \(y\)-axis and has minimum points at \(C\) and \(D\).
- Find the value of \(p\) and the value of \(q\).
- Write down the coordinates of \(D\).\\
(c) The function \(m\) is given by
$$\mathrm { m } ( x ) = \frac { x ^ { 2 } - 5 } { 3 - x } , \quad x \in \mathbb { R } , x \leqslant \alpha$$
where \(\alpha\) is the \(x\)-coordinate of \(A\) as found in part (a).
- Find \(\mathrm { m } ^ { - 1 }\)
- Write down the domain of \(\mathrm { m } ^ { - 1 }\)
- Find the value of \(t\) such that \(\mathrm { m } ( t ) = \mathrm { m } ^ { - 1 } ( t )\)