| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Roots with given sum conditions |
| Difficulty | Standard +0.8 This FP2 question requires systematic application of Vieta's formulas with complex coefficients, then solving a system involving sum and product of roots. Part (c) adds complexity by requiring solution of a quadratic with complex coefficients where one root is purely imaginary. While methodical, it demands careful algebraic manipulation with complex numbers and multiple interconnected steps beyond standard A-level, typical of Further Maths material. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| 3(a)(i) \(\alpha\beta = -18 + 12i\) | B1 | 1 mark; accept \(-(18 - 12i)\) |
| 3(a)(ii) \(\alpha + \beta + \gamma = 0\) | B1 | 1 mark |
| 3(b)(i) \(\alpha = -2\) | B1F | 1 mark |
| 3(b)(ii) \(\beta\gamma = \frac{\alpha\beta}{\alpha} = 9 - 6i\) | M1, A1F | 2 marks; ft sign errors in (a) or (b)(i) or slips such as miscopy |
| 3(b)(iii) \(q = \sum \alpha\beta = \alpha(\beta + \gamma) + \beta\gamma = -2 \times 2 + 9 - 6i = 5 - 6i\) | M1, A1F, A1F | 3 marks; ft incorrect \(\beta\gamma\) or \(\alpha\) |
| 3(c) \(\beta = ki, \gamma = 2 - ki\) | B1 | |
| \(ki(2-ki) = 9 - 6i\) | M1 | |
| \(2k = -6\) \((k^2 = 9)\) \(k = -3\) | m1 | imaginary parts |
| \(\beta = -3i, \gamma = 2 + 3i\) | A1 | 4 marks (12 marks for question) |
**3(a)(i)** $\alpha\beta = -18 + 12i$ | B1 | 1 mark; accept $-(18 - 12i)$
**3(a)(ii)** $\alpha + \beta + \gamma = 0$ | B1 | 1 mark
**3(b)(i)** $\alpha = -2$ | B1F | 1 mark
**3(b)(ii)** $\beta\gamma = \frac{\alpha\beta}{\alpha} = 9 - 6i$ | M1, A1F | 2 marks; ft sign errors in (a) or (b)(i) or slips such as miscopy
**3(b)(iii)** $q = \sum \alpha\beta = \alpha(\beta + \gamma) + \beta\gamma = -2 \times 2 + 9 - 6i = 5 - 6i$ | M1, A1F, A1F | 3 marks; ft incorrect $\beta\gamma$ or $\alpha$
**3(c)** $\beta = ki, \gamma = 2 - ki$ | B1
$ki(2-ki) = 9 - 6i$ | M1
$2k = -6$ $(k^2 = 9)$ $k = -3$ | m1 | imaginary parts
$\beta = -3i, \gamma = 2 + 3i$ | A1 | 4 marks (12 marks for question)3 The cubic equation
$$z ^ { 3 } + q z + ( 18 - 12 i ) = 0$$
where $q$ is a complex number, has roots $\alpha , \beta$ and $\gamma$.
\begin{enumerate}[label=(\alph*)]
\item Write down the value of:
\begin{enumerate}[label=(\roman*)]
\item $\alpha \beta \gamma$;
\item $\alpha + \beta + \gamma$.
\end{enumerate}\item Given that $\beta + \gamma = 2$, find the value of:
\begin{enumerate}[label=(\roman*)]
\item $\alpha$;
\item $\quad \beta \gamma$;
\item $q$.
\end{enumerate}\item Given that $\beta$ is of the form $k \mathrm { i }$, where $k$ is real, find $\beta$ and $\gamma$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2008 Q3 [12]}}