| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.3 This is a standard Further Maths question on transformed roots using Vieta's formulas. Parts (a)-(c) involve routine algebraic manipulation of sum and product of roots, while part (d) requires forming a new equation—a common FP1 exercise. The techniques are well-practiced and straightforward, making it slightly easier than an average A-level question overall. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\alpha + \beta = -1, \alpha\beta = 5\) | B1B1 | 2 marks |
| (b) \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\) ... \(= 1 - 10 = -9\) | M1, A1F | 2 marks |
| (c) \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta}\) ... \(= -\frac{9}{5}\) | M1, A1 | 2 marks |
| (d) Product of new roots is 1; Eqn is \(5x^2 + 9x + 5 = 0\) | B1, B1F | 2 marks |
**(a)** $\alpha + \beta = -1, \alpha\beta = 5$ | B1B1 | 2 marks |
**(b)** $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$ ... $= 1 - 10 = -9$ | M1, A1F | 2 marks | with numbers substituted; ft sign error(s) in (a)
**(c)** $\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta}$ ... $= -\frac{9}{5}$ | M1, A1 | 2 marks | AG: A0 if $\alpha + \beta = 1$ used
**(d)** Product of new roots is 1; Eqn is $5x^2 + 9x + 5 = 0$ | B1, B1F | 2 marks | PI by constant term 1 or 5; ft wrong value for product
**Total: 8 marks**1 The equation
$$x ^ { 2 } + x + 5 = 0$$
has roots $\alpha$ and $\beta$.
\begin{enumerate}[label=(\alph*)]
\item Write down the values of $\alpha + \beta$ and $\alpha \beta$.
\item Find the value of $\alpha ^ { 2 } + \beta ^ { 2 }$.
\item Show that $\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = - \frac { 9 } { 5 }$.
\item Find a quadratic equation, with integer coefficients, which has roots $\frac { \alpha } { \beta }$ and $\frac { \beta } { \alpha }$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2008 Q1 [8]}}