Hence verify that \(\sqrt { 5 } - \mathrm { i }\) is a root of the equation
$$( 2 + \mathrm { i } \sqrt { 5 } ) z = 3 z ^ { * }$$
where \(z ^ { * }\) is the conjugate of \(z\).
The quadratic equation
$$x ^ { 2 } + p x + q = 0$$
in which the coefficients \(p\) and \(q\) are real, has a complex root \(\sqrt { 5 } - \mathrm { i }\).
Write down the other root of the equation.
Find the sum and product of the two roots of the equation.
5 (a) (i) Calculate $( 2 + \mathrm { i } \sqrt { 5 } ) ( \sqrt { 5 } - \mathrm { i } )$.\\
(ii) Hence verify that $\sqrt { 5 } - \mathrm { i }$ is a root of the equation
$$( 2 + \mathrm { i } \sqrt { 5 } ) z = 3 z ^ { * }$$
where $z ^ { * }$ is the conjugate of $z$.\\
(b) The quadratic equation
$$x ^ { 2 } + p x + q = 0$$
in which the coefficients $p$ and $q$ are real, has a complex root $\sqrt { 5 } - \mathrm { i }$.\\
(i) Write down the other root of the equation.\\
(ii) Find the sum and product of the two roots of the equation.\\
(iii) Hence state the values of $p$ and $q$.
\hfill \mbox{\textit{AQA FP1 Q5}}