| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.3 This is a standard Further Maths question on complex roots with real coefficients. It requires knowing that complex roots come in conjugate pairs, then applying Vieta's formulas systematically. The calculations are straightforward (finding |α|², sum of roots, product of roots) with no conceptual surprises—slightly above average only because it's Further Maths content. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(2 + 3i\) | B1 | 1 mark |
| (b)(i) \(\alpha\beta = 13\) | B1 | 1 mark |
| (ii) \(\alpha\beta + \beta\gamma + \gamma\alpha = 25\) | M1 | M1A0 for -25 (no ft) |
| \(\gamma(\alpha + \beta) = 12\) | A1F | |
| \(\gamma = 3\) | A1F | 3 marks |
| (iii) \(p = -\sum\alpha = -7\) | M1 | M1 for a correct method for either \(p\) or \(q\) |
| \(q = -\alpha\beta\gamma = -39\) | A1F | A1F |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) Attempt at \((\overline{z} - 2 + 3i)(\overline{z} - 2 - 3i)\) | (M1) | |
| \(z^2 - 4z + 13\) | (A1) | |
| cubic is \((z^2 - 4z + 13)(\overline{z} - 3) \cdots \gamma = 3\) | (A1) | (3) |
| (iii) Multiply out or pick out coefficients | (M1) | |
| \(p = -7, q = -39\) | (A1, A1) | (3) |
**(a)** $2 + 3i$ | B1 | 1 mark
**(b)(i)** $\alpha\beta = 13$ | B1 | 1 mark
**(ii)** $\alpha\beta + \beta\gamma + \gamma\alpha = 25$ | M1 | M1A0 for -25 (no ft)
$\gamma(\alpha + \beta) = 12$ | A1F |
$\gamma = 3$ | A1F | 3 marks | ft error in $\alpha\beta$
**(iii)** $p = -\sum\alpha = -7$ | M1 | M1 for a correct method for either $p$ or $q$
$q = -\alpha\beta\gamma = -39$ | A1F | A1F | 3 marks | ft from previous errors; $p$ and $q$ must be real. For sign errors in $p$ and $q$ allow M1 but A0
**Alternative for (b)(ii) and (iii):**
**(ii)** Attempt at $(\overline{z} - 2 + 3i)(\overline{z} - 2 - 3i)$ | (M1) |
$z^2 - 4z + 13$ | (A1) |
cubic is $(z^2 - 4z + 13)(\overline{z} - 3) \cdots \gamma = 3$ | (A1) | (3)
**(iii)** Multiply out or pick out coefficients | (M1) |
$p = -7, q = -39$ | (A1, A1) | (3)
**Total: 8 marks**3 The cubic equation
$$z ^ { 3 } + p z ^ { 2 } + 25 z + q = 0$$
where $p$ and $q$ are real, has a root $\alpha = 2 - 3 \mathrm { i }$.
\begin{enumerate}[label=(\alph*)]
\item Write down another non-real root, $\beta$, of this equation.
\item Find:
\begin{enumerate}[label=(\roman*)]
\item the value of $\alpha \beta$;
\item the third root, $\gamma$, of the equation;
\item the values of $p$ and $q$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2009 Q3 [8]}}