| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| 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 Vieta's formulas, forming expressions for α²+β² and α²β², then solving the resulting quadratic. While it involves multiple steps and is from FP1, the techniques are routine and well-practiced, making it slightly easier than average for Further Maths content but slightly harder than typical A-level Core questions. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| 1(a) \(\alpha + \beta = 6, \alpha\beta = 1\) | B1B1 | 2 marks |
| 1(b) Sum of new roots \(= 6^2 - 2(18) = 0\) | M1A1F | ft wrong value(s) in (a) |
| Product \(= 18^2 = 324\) | B1F | ditto |
| Equation \(x^2 + 324 = 0\) | A1F | '= 0' needed here; ft wrong value(s) for sum/product |
| 1(c) \(\alpha^2\) and \(\beta^2\) are \(\pm 18i\) | B1 | 1 mark |
**1(a)** $\alpha + \beta = 6, \alpha\beta = 1$ | B1B1 | 2 marks
**1(b)** Sum of new roots $= 6^2 - 2(18) = 0$ | M1A1F | ft wrong value(s) in (a)
Product $= 18^2 = 324$ | B1F | ditto
Equation $x^2 + 324 = 0$ | A1F | '= 0' needed here; ft wrong value(s) for sum/product | 4 marks
**1(c)** $\alpha^2$ and $\beta^2$ are $\pm 18i$ | B1 | 1 mark
**Total: 7 marks**
---1 The quadratic equation $x ^ { 2 } - 6 x + 18 = 0$ has roots $\alpha$ and $\beta$.
\begin{enumerate}[label=(\alph*)]
\item Write down the values of $\alpha + \beta$ and $\alpha \beta$.
\item Find a quadratic equation, with integer coefficients, which has roots $\alpha ^ { 2 }$ and $\beta ^ { 2 }$.
\item Hence find the values of $\alpha ^ { 2 }$ and $\beta ^ { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2011 Q1 [7]}}