AQA
Further Paper 2
2022
June
Q13
16 marks
Challenging +1.2
13
- The matrix A represents a reflection in the line \(y = m x\), where \(m\) is a constant.
Show that \(\mathbf { A } = \left( \frac { 1 } { m ^ { 2 } + 1 } \right) \left[ \begin{array} { c c } 1 - m ^ { 2 } & 2 m \\ 2 m & m ^ { 2 } - 1 \end{array} \right]\)
You may use the result in the formulae booklet.
13
- \(\quad\) The matrix \(\mathbf { B }\) is defined as \(\mathbf { B } = \left[ \begin{array} { l l } 3 & 0 \\ 0 & 3 \end{array} \right]\)
Show that \(( \mathbf { B A } ) ^ { 2 } = k \mathbf { I }\)
where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix and \(k\) is an integer.
13 - (i) The diagram below shows a point \(P\) and the line \(y = m x\)
Draw four lines on the diagram to demonstrate the result proved in part (b).
Label as \(P ^ { \prime }\) the image of \(P\) under the transformation represented by (BA) \({ } ^ { 2 }\)
\includegraphics[max width=\textwidth, alt={}, center]{74b8239a-1f46-45e7-ad20-2dce7bf4baf6-20_579_1068_584_488}
13 (c) (ii) Explain how your completed diagram shows the result proved in part (b).
13 - The matrix \(\mathbf { C }\) is defined as \(\mathbf { C } = \left[ \begin{array} { c c } \frac { 12 } { 5 } & \frac { 9 } { 5 } \\ \frac { 9 } { 5 } & - \frac { 12 } { 5 } \end{array} \right]\)
Find the value of \(m\) such that \(\mathbf { C } = \mathbf { B A }\)
Fully justify your answer.
[0pt]
[4 marks]