| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Factored form to roots |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question requiring coefficient comparison to find a and b, then solving two quadratic factors for complex roots, and plotting on an Argand diagram. All steps are routine applications of standard techniques with no novel insight required, making it slightly easier than average. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02k Argand diagrams: geometric interpretation |
5. Given that
$$z ^ { 4 } - 6 z ^ { 3 } + 34 z ^ { 2 } - 54 z + 225 \equiv \left( z ^ { 2 } + 9 \right) \left( z ^ { 2 } + a z + b \right)$$
where $a$ and $b$ are real numbers,
\begin{enumerate}[label=(\alph*)]
\item find the value of $a$ and the value of $b$.
\item Hence find the exact roots of the equation
$$z ^ { 4 } - 6 z ^ { 3 } + 34 z ^ { 2 } - 54 z + 225 = 0$$
\item Show your roots on a single Argand diagram.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2018 Q5 [8]}}