| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Quadratic from one complex root |
| Difficulty | Moderate -0.8 This is a standard Further Maths question testing the fundamental property that complex roots of real polynomials come in conjugate pairs, followed by routine application of Vieta's formulas or expansion. Requires only recall of key facts and basic algebraic manipulation with no problem-solving insight needed. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Other root is \(2-3i\) | B1 | |
| Total | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Sum of roots \(= 4\), so \(b = -4\) | B1F, B1F | ft error in (a); ft wrong value for sum |
| Product is \(13\), so \(c = 13\) | B1, B1F | ft wrong value for product |
| Alternative: Substituting \(2+3i\) into equation | M1 | |
| Equating R and I parts | m1 | |
| \(12+3b=0\), so \(b=-4\) | A1 | |
| \(-5+2b+c=0\), so \(c=13\) | A1F | ft wrong value for \(b\) |
| Total | 5 |
## Question 2(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Other root is $2-3i$ | B1 | |
| **Total** | **1** | |
## Question 2(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Sum of roots $= 4$, so $b = -4$ | B1F, B1F | ft error in (a); ft wrong value for sum |
| Product is $13$, so $c = 13$ | B1, B1F | ft wrong value for product |
| **Alternative:** Substituting $2+3i$ into equation | M1 | |
| Equating R and I parts | m1 | |
| $12+3b=0$, so $b=-4$ | A1 | |
| $-5+2b+c=0$, so $c=13$ | A1F | ft wrong value for $b$ |
| **Total** | **5** | |
---2 The complex number $2 + 3 \mathrm { i }$ is a root of the quadratic equation
$$x ^ { 2 } + b x + c = 0$$
where $b$ and $c$ are real numbers.
\begin{enumerate}[label=(\alph*)]
\item Write down the other root of this equation.
\item Find the values of $b$ and $c$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2009 Q2 [5]}}