| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Equation with nonlinearly transformed roots |
| Difficulty | Standard +0.3 This is a standard FP1 roots of equations question testing Vieta's formulas and symmetric functions. Parts (a) and (b) are routine applications of sum/product formulas and algebraic manipulation. Part (c) requires finding sum and product of transformed roots, which is a textbook technique for this module. While it has multiple steps and requires careful algebra, it follows predictable patterns without requiring novel insight. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
The quadratic equation
$$2x^2 + 8x + 1 = 0$$
has roots $\alpha$ and $\beta$.
\begin{enumerate}[label=(\alph*)]
\item Write down the value of $\alpha + \beta$ and the value of $\alpha\beta$. [2 marks]
\item
\begin{enumerate}[label=(\roman*)]
\item Find the value of $\alpha^2 + \beta^2$. [2 marks]
\item Hence, or otherwise, show that $\alpha^4 + \beta^4 = \frac{449}{2}$. [2 marks]
\end{enumerate}
\item Find a quadratic equation, with integer coefficients, which has roots
$$2\alpha^4 + \frac{1}{\beta^2} \text{ and } 2\beta^4 + \frac{1}{\alpha^2}$$
[5 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2014 Q2 [11]}}