| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.8 This is a multi-part FP2 question requiring complex number manipulation, polynomial theory (sum/product of roots), and solving simultaneous equations with complex coefficients. Part (a) is routine, but parts (b)(i)-(iii) require systematic application of Vieta's formulas and the conjugate root theorem, with careful algebraic manipulation across multiple steps. More demanding than standard A-level but typical for Further Maths. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots |
3
\begin{enumerate}[label=(\alph*)]
\item Show that $( 1 + \mathrm { i } ) ^ { 3 } = 2 \mathrm { i } - 2$.
\item The cubic equation
$$z ^ { 3 } - ( 5 + \mathrm { i } ) z ^ { 2 } + ( 9 + 4 \mathrm { i } ) z + k ( 1 + \mathrm { i } ) = 0$$
where $k$ is a real constant, has roots $\alpha , \beta$ and $\gamma$.\\
It is given that $\alpha = 1 + \mathrm { i }$.
\begin{enumerate}[label=(\roman*)]
\item Find the value of $k$.
\item Show that $\beta + \gamma = 4$.
\item Find the values of $\beta$ and $\gamma$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2011 Q3 [11]}}