| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.8 This is a Further Maths question requiring multiple applications of root relationships. Part (a) is routine, part (b) requires the identity α³+β³=(α+β)³-3αβ(α+β), and part (c) demands finding sum and product of transformed roots (α²/β + β²/α = (α³+β³)/αβ and α²/β · β²/α = αβ) then constructing the equation. The multi-step algebraic manipulation and transformation of roots elevates this above standard A-level. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks | Guidance |
|---|---|---|
| 1(a) | \(\alpha + \beta = 2, \alpha\beta = \frac{3}{8}\) | B1B1 |
| 1(b) | \(\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)\) or other appropriate formula | M1 |
| \(... = 8 - 3(\frac{1}{2})(2) = 6\) | m1A1 | m1 for substn of numerical values; A1 for result shown (AG) |
| 3 marks | ||
| 1(c) | Sum of roots = \(\frac{\alpha^3 + \beta^3}{\alpha\beta}\) | M1 |
| \(... = \frac{6}{\frac{3}{8}} = 18\) | A1F | ft wrong value for \(\alpha\beta\) |
| Product of roots = \(\alpha\beta = \frac{1}{8}\) | B1F | ditto |
| Equation is \(3x^2 - 54x + 1 = 0\) | A1F | Integer coeffs and "= 0" needed; ft wrong sum and/or product |
| 4 marks | ||
| Total for Q1 | 9 marks |
1(a) | $\alpha + \beta = 2, \alpha\beta = \frac{3}{8}$ | B1B1 | 2 marks |
1(b) | $\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$ or other appropriate formula | M1 | or other appropriate formula |
| $... = 8 - 3(\frac{1}{2})(2) = 6$ | m1A1 | m1 for substn of numerical values; A1 for result shown (AG) |
| | | 3 marks |
1(c) | Sum of roots = $\frac{\alpha^3 + \beta^3}{\alpha\beta}$ | M1 |
| $... = \frac{6}{\frac{3}{8}} = 18$ | A1F | ft wrong value for $\alpha\beta$ |
| Product of roots = $\alpha\beta = \frac{1}{8}$ | B1F | ditto |
| Equation is $3x^2 - 54x + 1 = 0$ | A1F | Integer coeffs and "= 0" needed; ft wrong sum and/or product |
| | | 4 marks |
| **Total for Q1** | **9 marks** |
---1 The quadratic equation
$$3 x ^ { 2 } - 6 x + 1 = 0$$
has roots $\alpha$ and $\beta$.
\begin{enumerate}[label=(\alph*)]
\item Write down the values of $\alpha + \beta$ and $\alpha \beta$.
\item Show that $\alpha ^ { 3 } + \beta ^ { 3 } = 6$.
\item Find a quadratic equation, with integer coefficients, which has roots $\frac { \alpha ^ { 2 } } { \beta }$ and $\frac { \beta ^ { 2 } } { \alpha }$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2010 Q1 [9]}}