| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Finding polynomial from root properties |
| Difficulty | Standard +0.8 This is a substantial FP2 question requiring systematic application of symmetric function theory and Newton's identities. Part (a) uses the standard algebraic identity relating power sums to elementary symmetric functions. Parts (b-c) build the polynomial from Vieta's formulas. Part (d) requires understanding of discriminants or complex conjugate pairs. Part (e) involves solving a cubic and working with complex roots. While the techniques are syllabus-standard for Further Maths, the multi-step nature, algebraic manipulation demands, and need to work with complex numbers place this moderately above average difficulty. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| - Use of \(\left(\sum\alpha\right)^2 = \sum\alpha^2 + 2\sum\alpha\beta\) | M1 | |
| - \(1 = -5 + 2\sum\alpha\beta\) | A1 | |
| - \(\sum\alpha\beta = 3\) | A1 | 3 marks; AG |
| Answer | Marks | Guidance |
|---|---|---|
| - \(1(-5-3) = -23 - 3\alpha\beta\gamma\) | M1 | For use of identity |
| - \(\alpha\beta\gamma = -5\) | A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| - \(z^3 - z^2 + 3z + 5 = 0\) | M1 | |
| - | A1F | 2 marks; For correct signs and "= 0" |
| Answer | Marks | Guidance |
|---|---|---|
| - \(\alpha^2 + \beta^2 + \gamma^2 < 0 \Rightarrow \text{non real roots}\) | B1 | |
| - Coefficients real \(\therefore\) conjugate pair | B1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| - \(f(-1) = 0 \Rightarrow z + 1\) is a factor | M1A1 | |
| - \((z+1)(z^2 - 2z + 5) = 0\) | A1 | |
| - \(z = -1, 1 \pm 2i\) | A1 | 4 marks |
**(a)**
- Use of $\left(\sum\alpha\right)^2 = \sum\alpha^2 + 2\sum\alpha\beta$ | M1 |
- $1 = -5 + 2\sum\alpha\beta$ | A1 |
- $\sum\alpha\beta = 3$ | A1 | 3 marks; AG
- **Total: 3 marks**
**(b)**
- $1(-5-3) = -23 - 3\alpha\beta\gamma$ | M1 | For use of identity
- $\alpha\beta\gamma = -5$ | A1 | 2 marks
- **Total: 2 marks**
**(c)**
- $z^3 - z^2 + 3z + 5 = 0$ | M1 |
- | A1F | 2 marks; For correct signs and "= 0"
- **Total: 2 marks**
**(d)**
- $\alpha^2 + \beta^2 + \gamma^2 < 0 \Rightarrow \text{non real roots}$ | B1 |
- Coefficients real $\therefore$ conjugate pair | B1 | 2 marks
- **Total: 2 marks**
**(e)**
- $f(-1) = 0 \Rightarrow z + 1$ is a factor | M1A1 |
- $(z+1)(z^2 - 2z + 5) = 0$ | A1 |
- $z = -1, 1 \pm 2i$ | A1 | 4 marks
- **Total: 4 marks**
**Question 4 Total: 13 marks**
---4 It is given that $\alpha , \beta$ and $\gamma$ satisfy the equations
$$\begin{aligned}
& \alpha + \beta + \gamma = 1 \\
& \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 5 \\
& \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 23
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\alpha \beta + \beta \gamma + \gamma \alpha = 3$.
\item Use the identity
$$( \alpha + \beta + \gamma ) \left( \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } - \alpha \beta - \beta \gamma - \gamma \alpha \right) = \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } - 3 \alpha \beta \gamma$$
to find the value of $\alpha \beta \gamma$.
\item Write down a cubic equation, with integer coefficients, whose roots are $\alpha , \beta$ and $\gamma$.
\item Explain why this cubic equation has two non-real roots.
\item Given that $\alpha$ is real, find the values of $\alpha , \beta$ and $\gamma$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2009 Q4 [13]}}