| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.3 This is a structured Further Maths question on roots of polynomials with complex numbers, but it's highly scaffolded with clear steps. Part (a) uses standard sum/product of roots formulas and basic algebraic manipulation. Parts (b) and (c) apply these to transformed roots with explicit guidance. While it involves complex numbers and is from FP1, the question requires only routine application of formulas with no novel insight or problem-solving—making it slightly easier than average even for Further Maths. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| \(\alpha + \beta = 4, \alpha\beta = 13\) | B1B1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(\alpha' + \beta' = (\alpha + \beta)^2 - 2\alpha\beta\); \(\ldots = 4^2 - 26 = -10\) | M1, A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| The square of a real number is positive (or zero); The sum of two such squares is positive (or zero) | E1, E1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \((\alpha + i) + (\beta + i) = 4 + 2i\) | B1F | 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| \((\alpha + i)(\beta + i) = 12 + 4i\) | M1A1F | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Correct coeff. of \(x\) or constant term; \(x^2 - (4+2i)x + (12+4i) = 0\) | M1, A1F | 2 marks |
**Part (a)(i)**
| $\alpha + \beta = 4, \alpha\beta = 13$ | B1B1 | 2 marks | — |
**Part (a)(ii)**
| $\alpha' + \beta' = (\alpha + \beta)^2 - 2\alpha\beta$; $\ldots = 4^2 - 26 = -10$ | M1, A1 | 2 marks | convincingly shown (AG) |
**Part (a)(iii)**
| The square of a real number is positive (or zero); The sum of two such squares is positive (or zero) | E1, E1 | 2 marks | — |
**Part (b)(i)**
| $(\alpha + i) + (\beta + i) = 4 + 2i$ | B1F | 1 mark | ft wrong value in (a)(i) |
**Part (b)(ii)**
| $(\alpha + i)(\beta + i) = 12 + 4i$ | M1A1F | 2 marks | ditto |
**Part (b)(c)**
| Correct coeff. of $x$ or constant term; $x^2 - (4+2i)x + (12+4i) = 0$ | M1, A1F | 2 marks | Using c's answers in (b); ft wrong answers in (b) |
**Total: 11 marks**
---6 The equation
$$x ^ { 2 } - 4 x + 13 = 0$$
has roots $\alpha$ and $\beta$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the values of $\alpha + \beta$ and $\alpha \beta$.
\item Deduce that $\alpha ^ { 2 } + \beta ^ { 2 } = - 10$.
\item Explain why the statement $\alpha ^ { 2 } + \beta ^ { 2 } = - 10$ implies that $\alpha$ and $\beta$ cannot both be real.
\end{enumerate}\item Find in the form $p + \mathrm { i } q$ the values of:
\begin{enumerate}[label=(\roman*)]
\item $( \alpha + \mathrm { i } ) + ( \beta + \mathrm { i } )$;
\item $( \alpha + \mathrm { i } ) ( \beta + \mathrm { i } )$.
\end{enumerate}\item Hence find a quadratic equation with roots $( \alpha + \mathrm { i } )$ and $( \beta + \mathrm { i } )$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2005 Q6 [11]}}