| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Sum of powers of roots |
| Difficulty | Standard +0.8 This is a substantial FP2 question on roots of polynomials requiring systematic application of Newton's identities and power sum techniques. While part (a) involves standard Vieta's formulas and routine manipulations, part (b) requires working with fourth powers to find k=2, then computing fifth powers—this demands careful algebraic manipulation across multiple steps. The question is more challenging than typical A-level pure maths but represents standard Further Maths fare rather than requiring novel insight. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
The cubic equation
$$z^3 - 2z^2 + k = 0 \quad (k \neq 0)$$
has roots $\alpha$, $\beta$ and $\gamma$.
\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Write down the values of $\alpha + \beta + \gamma$ and $\alpha\beta + \beta\gamma + \gamma\alpha$. [2 marks]
\item Show that $\alpha^2 + \beta^2 + \gamma^2 = 4$. [2 marks]
\item Explain why $\alpha^3 - 2\alpha^2 + k = 0$. [1 mark]
\item Show that $\alpha^3 + \beta^3 + \gamma^3 = 8 - 3k$. [2 marks]
\end{enumerate}
\item Given that $\alpha^4 + \beta^4 + \gamma^4 = 0$:
\begin{enumerate}[label=(\roman*)]
\item show that $k = 2$; [4 marks]
\item find the value of $\alpha^5 + \beta^5 + \gamma^5$. [3 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2011 Q4 [14]}}