| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Roots with given sum conditions |
| Difficulty | Challenging +1.2 This is a structured Further Maths question on complex roots and symmetric functions that guides students through each step. While it involves complex numbers and requires knowledge of Vieta's formulas, the multi-part scaffolding makes it more accessible than it initially appears. Part (a) uses a standard identity, parts (b-c) apply direct formulas, and the final part involves straightforward quadratic solving. The complexity is moderate for FP2 level but the extensive guidance keeps it below average difficulty for Further Maths material. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots4.05a Roots and coefficients: symmetric functions |
7 The numbers $\alpha , \beta$ and $\gamma$ satisfy the equations
$$\begin{aligned}
& \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 10 - 12 \mathrm { i } \\
& \alpha \beta + \beta \gamma + \gamma \alpha = 5 + 6 \mathrm { i }
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\alpha + \beta + \gamma = 0$.
\item The numbers $\alpha , \beta$ and $\gamma$ are also the roots of the equation
$$z ^ { 3 } + p z ^ { 2 } + q z + r = 0$$
Write down the value of $p$ and the value of $q$.
\item It is also given that $\alpha = 3 \mathrm { i }$.
\begin{enumerate}[label=(\roman*)]
\item Find the value of $r$.
\item Show that $\beta$ and $\gamma$ are the roots of the equation
$$z ^ { 2 } + 3 \mathrm { i } z - 4 + 6 \mathrm { i } = 0$$
\item Given that $\beta$ is real, find the values of $\beta$ and $\gamma$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2012 Q7 [12]}}