| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Equation with nonlinearly transformed roots |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring systematic application of Vieta's formulas and algebraic manipulation to find symmetric functions of transformed roots. While the techniques are standard for FP2 (using sum of roots squared identity, expressing new symmetric functions in terms of old), the multi-step nature, algebraic complexity, and requirement to construct a new equation from transformed roots places it moderately above average difficulty. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
The roots of the equation $z^3 + 2z^2 + 3z - 4 = 0$ are $\alpha$, $\beta$ and $\gamma$.
(a)(i) Write down the value of $\alpha + \beta + \gamma$ and the value of $\alpha\beta + \beta\gamma + \gamma\alpha$.
[2 marks]
(ii) Hence show that $\alpha^2 + \beta^2 + \gamma^2 = -2$.
[2 marks]
(b) Find the value of:
(i) $(\alpha + \beta)(\beta + \gamma) + (\beta + \gamma)(\gamma + \alpha) + (\gamma + \alpha)(\alpha + \beta)$
[3 marks]
(ii) $(\alpha + \beta)(\beta + \gamma)(\gamma + \alpha)$
[4 marks]
(c) Find a cubic equation whose roots are $\alpha + \beta$, $\beta + \gamma$ and $\gamma + \alpha$.
[3 marks]4 The roots of the equation
$$z ^ { 3 } + 2 z ^ { 2 } + 3 z - 4 = 0$$
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 $\alpha \beta + \beta \gamma + \gamma \alpha$.
\item Hence show that $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 2$.
\end{enumerate}\item Find the value of:
\begin{enumerate}[label=(\roman*)]
\item $( \alpha + \beta ) ( \beta + \gamma ) + ( \beta + \gamma ) ( \gamma + \alpha ) + ( \gamma + \alpha ) ( \alpha + \beta )$;
\item $( \alpha + \beta ) ( \beta + \gamma ) ( \gamma + \alpha )$.
\end{enumerate}\item Find a cubic equation whose roots are $\alpha + \beta , \beta + \gamma$ and $\gamma + \alpha$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2014 Q4 [14]}}