| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| 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 question requiring straightforward application of sum/product formulas and root transformation. Part (a) is direct recall, part (b) uses the identity α²+β²=(α+β)²-2αβ, and part (c) applies standard transformation techniques. While it's Further Maths content, the execution is mechanical with no novel insight required, making it slightly easier than average overall. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
| Answer | Marks |
|---|---|
| \(\alpha + \beta = -\frac{1}{2}\) | B1 |
| \(\alpha\beta = -4\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta\) | M1 | Use of correct identity |
| \(= (-\frac{1}{2})^2 - 2(-4) = \frac{1}{4} + 8 = \frac{33}{4}\) | A1 |
| Answer | Marks |
|---|---|
| Sum of new roots \(= 4\alpha^2 + 4\beta^2 = 4(\alpha^2+\beta^2) = 4 \times \frac{33}{4} = 33\) | M1 |
| Product of new roots \(= 4\alpha^2 \times 4\beta^2 = 16(\alpha\beta)^2 = 16 \times 16 = 256\) | M1 |
| \(x^2 - 33x + 256 = 0\) | A1 |
# Question 1:
## Part (a)
| $\alpha + \beta = -\frac{1}{2}$ | B1 | |
| $\alpha\beta = -4$ | B1 | |
## Part (b)
| $\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta$ | M1 | Use of correct identity |
| $= (-\frac{1}{2})^2 - 2(-4) = \frac{1}{4} + 8 = \frac{33}{4}$ | A1 | |
## Part (c)
| Sum of new roots $= 4\alpha^2 + 4\beta^2 = 4(\alpha^2+\beta^2) = 4 \times \frac{33}{4} = 33$ | M1 | |
| Product of new roots $= 4\alpha^2 \times 4\beta^2 = 16(\alpha\beta)^2 = 16 \times 16 = 256$ | M1 | |
| $x^2 - 33x + 256 = 0$ | A1 | |
---1 The equation
$$2 x ^ { 2 } + x - 8 = 0$$
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 Find a quadratic equation which has roots $4 \alpha ^ { 2 }$ and $4 \beta ^ { 2 }$. Give your answer in the form $x ^ { 2 } + p x + q = 0$, where $p$ and $q$ are integers.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2009 Q1 [7]}}