| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| 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 roots transformation question requiring routine application of sum/product formulas and algebraic manipulation. Part (a) is direct recall, part (b) is straightforward algebra, and part (c) follows a well-practiced method (finding sum and product of new roots, then constructing the equation). While it's Further Maths content, it's a textbook exercise with no novel insight required, making it slightly easier than average overall. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
| Answer | Marks |
|---|---|
| (a) \(\alpha + \beta = \frac{1}{2}\), \(\alpha\beta = 2\) | B1B1 |
| Answer | Marks | Guidance |
|---|---|---|
| (b) \(\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta}\) | M1 | |
| \(\ldots = \frac{1}{2} \div \frac{1}{4}\) | A1 | Convincingly shown (AG) |
| Answer | Marks | Guidance |
|---|---|---|
| (c) Sum of roots \(= 1\) | B1F | PI by term \(\pm x\); ft error(s) in (a) |
| Product of roots \(= \frac{16}{\alpha\beta} = 8\) | B1F | ft wrong value of \(\alpha\beta\) |
| Equation is \(x^2 - x + 8 = 0\) | B1F | ft wrong sum/product; "\(= 0\)" needed |
**(a)** $\alpha + \beta = \frac{1}{2}$, $\alpha\beta = 2$ | B1B1 |
**Total: 2 marks**
**(b)** $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta}$ | M1 |
$\ldots = \frac{1}{2} \div \frac{1}{4}$ | A1 | Convincingly shown (AG)
**Total: 2 marks**
**(c)** Sum of roots $= 1$ | B1F | PI by term $\pm x$; ft error(s) in (a)
Product of roots $= \frac{16}{\alpha\beta} = 8$ | B1F | ft wrong value of $\alpha\beta$
Equation is $x^2 - x + 8 = 0$ | B1F | ft wrong sum/product; "$= 0$" needed
**Total: 3 marks**
---4 The quadratic equation
$$2 x ^ { 2 } - x + 4 = 0$$
has roots $\alpha$ and $\beta$.
\begin{enumerate}[label=(\alph*)]
\item Write down the values of $\alpha + \beta$ and $\alpha \beta$.
\item Show that $\frac { 1 } { \alpha } + \frac { 1 } { \beta } = \frac { 1 } { 4 }$.
\item Find a quadratic equation with integer coefficients such that the roots of the equation are
$$\frac { 4 } { \alpha } \text { and } \frac { 4 } { \beta }$$
(3 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2007 Q4 [7]}}