| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 13 |
| 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 knowledge of polynomial theory, complex conjugate roots, and transformations. Parts (a)-(b) involve standard Vieta's formulas and algebraic manipulation (routine for FP2). Part (c) requires recognizing complex conjugate pairs and computing products/sums with complex numbers. Part (d) requires forming a new polynomial from reciprocal roots. While systematic, it demands multiple techniques and careful algebraic work across several steps, making it moderately challenging but still within standard FP2 territory. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.05a Roots and coefficients: symmetric functions |
4 The cubic equation
$$z ^ { 3 } + p z + q = 0$$
has roots $\alpha , \beta$ and $\gamma$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the value of $\alpha + \beta + \gamma$.
\item Express $\alpha \beta \gamma$ in terms of $q$.
\end{enumerate}\item Show that
$$\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 3 \alpha \beta \gamma$$
\item Given that $\alpha = 4 + 7 \mathrm { i }$ and that $p$ and $q$ are real, find the values of:
\begin{enumerate}[label=(\roman*)]
\item $\beta$ and $\gamma$;
\item $p$ and $q$.
\end{enumerate}\item Find a cubic equation with integer coefficients which has roots $\frac { 1 } { \alpha } , \frac { 1 } { \beta }$ and $\frac { 1 } { \gamma }$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2012 Q4 [13]}}