| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2013 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Roots with given sum conditions |
| Difficulty | Standard +0.3 This is a standard FP2 question on symmetric functions of roots using Vieta's formulas. Part (a) requires direct recall of formulas, part (a)(ii) involves simple factorization, and part (b) uses the standard technique of expanding a square to relate symmetric functions. The reasoning in (b)(i) is straightforward (sum of squares cannot be negative for all real roots). While it requires multiple steps and some algebraic manipulation, these are well-practiced techniques with no novel insight required, making it slightly easier than average. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
The roots of the equation
$$z^3 - 5z^2 + kz - 4 = 0$$
are $\alpha$, $\beta$ and $\gamma$.
\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Write down the value of $\alpha + \beta + \gamma$ and the value of $\alpha\beta\gamma$. [2 marks]
\item Hence find the value of $\alpha^2\beta\gamma + \alpha\beta^2\gamma + \alpha\beta\gamma^2$. [2 marks]
\end{enumerate}
\item The value of $\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2$ is $-4$.
\begin{enumerate}[label=(\roman*)]
\item Explain why $\alpha$, $\beta$ and $\gamma$ cannot all be real. [1 mark]
\item By considering $(\alpha\beta + \beta\gamma + \gamma\alpha)^2$, find the possible values of $k$. [4 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2013 Q4 [9]}}