| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.8 This is a Further Maths question requiring systematic use of Vieta's formulas and algebraic manipulation to find sums and products of transformed roots. Part (a) is standard (using α+β and αβ to find power sums), but part (b) requires finding both sum and product of (α²+β) and (β²+α), which involves non-trivial algebraic manipulation and is more demanding than typical A-level questions. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
9. The quadratic equation
$$2 x ^ { 2 } + 4 x - 3 = 0$$
has roots $\alpha$ and $\beta$.\\
Without solving the quadratic equation,
\begin{enumerate}[label=(\alph*)]
\item find the exact value of
\begin{enumerate}[label=(\roman*)]
\item $\alpha ^ { 2 } + \beta ^ { 2 }$
\item $\alpha ^ { 3 } + \beta ^ { 3 }$
\end{enumerate}\item Find a quadratic equation which has roots ( $\alpha ^ { 2 } + \beta$ ) and ( $\beta ^ { 2 } + \alpha$ ), giving your answer in the form $a x ^ { 2 } + b x + c = 0$, where $a , b$ and $c$ are integers.
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\includegraphics[max width=\textwidth, alt={}, center]{0b7ef4a1-51bf-4f0c-908a-7caf26a144dc-27_99_332_2622_1466}
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2016 Q9 [9]}}