2025 CJC P2 Q4

One of the roots of the equation \(2z^4 - 14z^3 + 33z^2 - 26z + p = 0\) where \(p\) text is a constant is \(3+i\).

1. Based on the above information only, a student claims that the equation has a root \(3-i\). State, with a reason, why the student's claim may not be true.[1]
2. Show that \(p=10\).[2]

For the rest of this question, do not use a calculator.

3. Find the roots of the equation \(2z^4 - 14z^3 + 33z^2 - 26z + 10 = 0\) and mark them clearly on a single labelled Argand diagram.[7]
4. The points of the Argand diagram in part (c) form the vertices of a quadrilateral. Identify the type of quadrilateral and determine its area.[2]