| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question on complex roots requiring direct substitution to find k, then using Vieta's formulas or polynomial division to find the third root. The steps are mechanical with no novel insight needed—slightly easier than average even for FP2 material. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02i Quadratic equations: with complex roots |
3 The cubic equation
$$z ^ { 3 } + 2 ( 1 - \mathrm { i } ) z ^ { 2 } + 32 ( 1 + \mathrm { i } ) = 0$$
has roots $\alpha , \beta$ and $\gamma$.
\begin{enumerate}[label=(\alph*)]
\item It is given that $\alpha$ is of the form $k \mathrm { i }$, where $k$ is real. By substituting $z = k \mathrm { i }$ into the equation, show that $k = 4$.
\item Given that $\beta = - 4$, find the value of $\gamma$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2007 Q3 [7]}}