| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.8 This Further Pure F1 question requires multiple sophisticated techniques: using Vieta's formulas, factoring α³+β³ using sum of cubes, then finding sum and product of transformed roots α²/β and β²/α through algebraic manipulation. While systematic, it demands careful algebraic reasoning across multiple steps beyond standard A-level, placing it moderately above average difficulty. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
6. It is given that $\alpha$ and $\beta$ are roots of the equation $3 x ^ { 2 } + 5 x - 1 = 0$
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $\alpha ^ { 3 } + \beta ^ { 3 }$
\item Find a quadratic equation which has roots $\frac { \alpha ^ { 2 } } { \beta }$ and $\frac { \beta ^ { 2 } } { \alpha }$, giving your answer in the form $a x ^ { 2 } + b x + c = 0$, where $a$, $b$ and $c$ are integers.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2014 Q6 [8]}}