9 The matrix \(\mathbf { M }\) represents the transformation T and is given by
$$\mathbf { M } = \left[ \begin{array} { c c }
3 p + 1 & 12
p + 2 & p ^ { 2 } - 3
\end{array} \right]$$
9
- In the case when \(p = 0\) show that the image of the point \(( 4,5 )\) under T is the point \(( 64 , - 7 )\)
9 - In the case when \(p = - 2\) find the gradient of the line of invariant points under \(T\)
9 - Show that \(p = 3\) is the only real value of \(p\) for which \(\mathbf { M }\) is singular.
The curve \(C\) has equation
$$y = \frac { 3 x ^ { 2 } + m x + p } { x ^ { 2 } + p x + m }$$
where \(m\) and \(p\) are integers.
The vertical asymptotes of \(C\) are \(x = - 4\) and \(x = - 1\)
The curve \(C\) is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{b37e2ee7-1cde-4d75-895a-381b32f4e95a-12_867_1102_733_463}