Solve the equation \(x ^ { 2 } + 4 x + 20 = 0\), giving your answers in the form \(c + d \mathrm { i }\), where \(c\) and \(d\) are integers.
The roots of the quadratic equation
$$z ^ { 2 } + ( 4 + i + q i ) z + 20 = 0$$
are \(w\) and \(w ^ { * }\).
In the case where \(q\) is real, explain why \(q\) must be - 1 .
In the case where \(w = p + 2 \mathrm { i }\), where \(p\) is real, find the possible values of \(q\). [0pt]
[5 marks]
\(8 \quad\) The matrix \(\mathbf { A }\) is defined by \(\mathbf { A } = \left[ \begin{array} { l l } 2 & 0 0 & 1 \end{array} \right]\).
Find the matrix \(\mathbf { A } ^ { 2 }\).
Describe fully the single geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\).
Given that the matrix \(\mathbf { B }\) represents a reflection in the line \(x + \sqrt { 3 } y = 0\), find the matrix \(\mathbf { B }\), giving the exact values of any trigonometric expressions.
Hence find the coordinates of the point \(P\) which is mapped onto \(( 0 , - 4 )\) under the transformation represented by \(\mathbf { A } ^ { 2 }\) followed by a reflection in the line \(x + \sqrt { 3 } y = 0\). [0pt]
[6 marks]
\(9 \quad\) A curve \(C\) has equation \(y = \frac { x - 1 } { ( x - 2 ) ( 2 x - 1 ) }\).
The line \(L\) has equation \(y = \frac { 1 } { 2 } ( x - 1 )\).
Write down the equations of the asymptotes of \(C\).
By forming and solving a suitable cubic equation, find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\).
Given that \(C\) has no stationary points, sketch \(C\) and \(L\) on the same axes.
Hence solve the inequality \(\frac { x - 1 } { ( x - 2 ) ( 2 x - 1 ) } \geqslant \frac { 1 } { 2 } ( x - 1 )\).