| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 7 |
| 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 requiring straightforward application of Vieta's formulas and algebraic manipulation. Part (a) is direct recall, part (b) requires factoring α²β + αβ² = αβ(α+β), and part (c) uses sum and product of roots formulae with values already computed. While it's Further Maths content, the techniques are routine and mechanical with no novel insight required. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks |
|---|---|
| \(\alpha + \beta = 5\), \(\alpha\beta = -2\) | B1, B1 (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\alpha^2\beta + \alpha\beta^2 = \alpha\beta(\alpha + \beta) = -10\) | M1A1ft (2 marks) | ft wrong values |
| Answer | Marks | Guidance |
|---|---|---|
| \((\alpha^2\beta)(\alpha\beta^2) = (\alpha\beta)^3 = -8\) | M1A1ft | ft wrong values |
| Equation is \(x^2 + 10x - 8 = 0\) | A1ft (3 marks) | Dep on both M1s; ft wrong values; Condone omission of "\(= 0\)" |
## Question 1:
**Part (a)**
$\alpha + \beta = 5$, $\alpha\beta = -2$ | B1, B1 (2 marks) |
**Part (b)**
$\alpha^2\beta + \alpha\beta^2 = \alpha\beta(\alpha + \beta) = -10$ | M1A1ft (2 marks) | ft wrong values
**Part (c)**
$(\alpha^2\beta)(\alpha\beta^2) = (\alpha\beta)^3 = -8$ | M1A1ft | ft wrong values
Equation is $x^2 + 10x - 8 = 0$ | A1ft (3 marks) | Dep on both M1s; ft wrong values; Condone omission of "$= 0$"
---1 The equation
$$x ^ { 2 } - 5 x - 2 = 0$$
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 $\alpha ^ { 2 } \beta + \alpha \beta ^ { 2 }$.
\item Find a quadratic equation which has roots
$$\alpha ^ { 2 } \beta \quad \text { and } \quad \alpha \beta ^ { 2 }$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2005 Q1 [7]}}