| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Parametric polynomials with root conditions |
| Difficulty | Standard +0.3 This is a standard Further Pure complex roots question testing well-rehearsed techniques: complex conjugate roots theorem, Vieta's formulas, and basic complex arithmetic. The multi-part structure guides students through the solution methodically with no novel insight required—slightly easier than average A-level difficulty. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) \(1 - 2i\) | B1 | 1 mark |
| (a)(ii) \((\alpha\beta = 1 + 4 =) 5\) | B1 | 1 mark |
| (b) \(\sum\alpha\beta = \frac{17}{3}\) | B1 | PI by next line |
| \(\alpha\gamma + \beta\gamma + \text{"their"} 5 = \text{"their"} \frac{17}{3}\) | M1 | FT "their" \(\alpha\beta\) and \(\sum\alpha\beta\) values |
| \(\Rightarrow \gamma = \frac{1}{3}\) | A1 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| (c) \(\alpha + \beta + \gamma = \frac{-p}{3}\), \(\alpha\beta\gamma = \frac{-q}{3}\) | M1 | Either of these expressions correct |
| \(p = -7\) | A1 | PI by correct \(p\) or \(q\) |
| \(q = -5\) | A1 | 3 marks |
**(a)(i)** $1 - 2i$ | B1 | **1 mark**
**(a)(ii)** $(\alpha\beta = 1 + 4 =) 5$ | B1 | **1 mark**
**(b)** $\sum\alpha\beta = \frac{17}{3}$ | B1 | PI by next line
$\alpha\gamma + \beta\gamma + \text{"their"} 5 = \text{"their"} \frac{17}{3}$ | M1 | FT "their" $\alpha\beta$ and $\sum\alpha\beta$ values
$\Rightarrow \gamma = \frac{1}{3}$ | A1 | **3 marks**
**Alternative:** $z^3 + \frac{p}{3}z^2 + \frac{17}{3}z + \frac{q}{3}$; quadratic factor $z^2 - 2z + 5$ B1; $(z^2 - 2z + 5)(z - \gamma)$ comparing coefficient of $z$: $5 + 2\gamma = \frac{17}{3}$ M1; $\Rightarrow \gamma = \frac{1}{3}$ A1 (3)
**(c)** $\alpha + \beta + \gamma = \frac{-p}{3}$, $\alpha\beta\gamma = \frac{-q}{3}$ | M1 | Either of these expressions correct
$p = -7$ | A1 | PI by correct $p$ or $q$
$q = -5$ | A1 | **3 marks**
**Alternative:** comparing coefficients either $-5y = \frac{q}{3}$ or $-\gamma - 2 = \frac{p}{3}$ M1; $p = -7$ A1; $q = -5$ A1 (3)
**Total: 8 marks**
**Notes:**
- Allow M1 for $5 + 2\gamma = \frac{17}{3}$ if $\sum\alpha\beta$ not seen
- **(c) Example:** $\alpha + \beta + \gamma = -p$; $\alpha + \beta + \gamma = 2 + \frac{1}{3} - \frac{7}{3}$ $\Rightarrow p = -7$ Award M1 A1 assuming first statement was meant as candidate's "reminder" for signs but "wiggly underline" incorrect statement
- **(c) Example:** $\gamma = \frac{4}{3}$; $\alpha + \beta + \gamma = \frac{10}{3}$ $\Rightarrow p = -10$ Award M1 (implied) A0
- **Alternative:** substituting $z = 1 + 2i$ or $1 - 2i$ leading to correct simultaneous equations $3p - q + 16 = 0$ and $4p + 28 = 0$ M1 then $p = -7$ A1; $q = -5$ A1
---The cubic equation $3z^3 + pz^2 + 17z + q = 0$, where $p$ and $q$ are real, has a root $\alpha = 1 + 2\mathrm{i}$.
\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Write down the value of another non-real root, $\beta$, of this equation. [1 mark]
\item Hence find the value of $\alpha\beta$. [1 mark]
\end{enumerate}
\item Find the value of the third root, $\gamma$, of this equation. [3 marks]
\item Find the values of $p$ and $q$. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2016 Q2 [8]}}