| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.8 This is a standard Further Maths roots transformation question requiring Vieta's formulas, the identity for sum of cubes, and systematic algebraic manipulation to find the new equation. Part (c) requires finding sum and product of transformed roots through multi-step algebra, which is moderately challenging but follows established techniques taught in FP1. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
The equation
$$2x^2 + 3x - 6 = 0$$
has roots $a$ and $b$.
**(a) Write down the value of $a + b$ and the value of $ab$. (2 marks)**
**(b) Hence show that $a^3 + b^3 = -\frac{135}{8}$. (3 marks)**
**(c) Find a quadratic equation, with integer coefficients, whose roots are $\frac{a}{b^2} + \frac{b}{a^2}$. (6 marks)**6 The equation
$$2 x ^ { 2 } + 3 x - 6 = 0$$
has roots $\alpha$ and $\beta$.
\begin{enumerate}[label=(\alph*)]
\item Write down the value of $\alpha + \beta$ and the value of $\alpha \beta$.
\item Hence show that $\alpha ^ { 3 } + \beta ^ { 3 } = - \frac { 135 } { 8 }$.
\item Find a quadratic equation, with integer coefficients, whose roots are $\alpha + \frac { \alpha } { \beta ^ { 2 } }$ and $\beta + \frac { \beta } { \alpha ^ { 2 } }$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2013 Q6 [11]}}