12 \end{array} \right) + \mu \left( \begin{array} { r } 5
- 3
4 \end{array} \right) \text { where } \mu \text { is a scalar parameter. }$$ The point \(A\) does not lie on \(l _ { 2 }\). Given that the vector \(\overrightarrow { O A }\) is parallel to the line \(l _ { 2 }\) and \(| \overrightarrow { O A } | = \sqrt { 2 }\) units, calculate the possible position vectors of the point \(A\).
[0pt] [BLANK PAGE] Q7.
The simultaneous equations $$\begin{aligned} & 2 x - y = 1
& 3 x + k y = b \end{aligned}$$ are represented by the matrix equation \(\mathbf { M } \binom { x } { y } = \binom { 1 } { b }\).

1. State the value of \(k\) for which \(\mathbf { M } ^ { - 1 }\) does not exist and find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\) when \(\mathbf { M } ^ { - 1 }\) exists.
   Use \(\mathbf { M } ^ { - 1 }\) to solve the simultaneous equations when \(k = 5\) and \(b = 21\).
2. The two equations can be interpreted as representing two lines in the \(x - y\) plane. Describe the relationship between these two lines
   (A) when \(k = 5\) and \(b = 21\),
   (B) when \(k = - \frac { 3 } { 2 }\) and \(b = 1\),
   (C) when \(k = - \frac { 3 } { 2 }\) and \(b = \frac { 3 } { 2 }\).
   [0pt] [BLANK PAGE] Q8.
   You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 0.8 & 0.6
   0.6 & - 0.8 \end{array} \right)\).
3. Calculate \(\mathbf { M } ^ { 2 }\). You are now given that the matrix \(\mathbf { M }\) represents a reflection in a line through the origin.
4. Explain how your answer to part (i) relates to this information.
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5. By investigating the invariant points of the reflection, find the equation of the mirror line. [3]
6. Describe fully the transformation represented by the matrix \(\mathbf { P } = \left( \begin{array} { r r } 0.8 & - 0.6
   0.6 & 0.8 \end{array} \right)\).
7. A composite transformation is formed by the transformation represented by \(\mathbf { P }\) followed by the transformation represented by \(\mathbf { M }\). Find the single matrix that represents this composite transformation.
8. The composite transformation described in part (v) is equivalent to a single reflection. What is the equation of the mirror line of this reflection?
   [0pt] [BLANK PAGE] Q9.
   (a) Two points, \(A\) and \(B\), on an Argand diagram are represented by the complex numbers \(2 + 3 \mathrm { i }\) and \(- 4 - 5 \mathrm { i }\) respectively. Given that the points \(A\) and \(B\) are at the ends of a diameter of a circle \(C _ { 1 }\), express the equation of \(C _ { 1 }\) in the form \(\left| z - z _ { 0 } \right| = k\).
   (4 marks)
   (b) A second circle, \(C _ { 2 }\), is represented on the Argand diagram by the equation \(| z - 5 + 4 i | = 4\). Sketch on one Argand diagram both \(C _ { 1 }\) and \(C _ { 2 }\).
   (3 marks)
   (c) The points representing the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) lie on \(C _ { 1 }\) and \(C _ { 2 }\) respectively and are such that \(\left| z _ { 1 } - z _ { 2 } \right|\) has its maximum value. Find this maximum value, giving your answer in the form \(a + b \sqrt { 5 }\).
   (5 marks)
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