| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.8 This is a Further Maths question requiring systematic application of Vieta's formulas and algebraic manipulation to find sums/products of transformed roots. Part (b)(ii) requires forming a new quadratic from complex transformed roots (α³-β and β³-α), which demands careful algebraic work across multiple steps. While the techniques are standard for FM, the length and algebraic complexity elevate it above typical questions. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
3. The quadratic equation
$$x ^ { 2 } - 2 x + 3 = 0$$
has roots $\alpha$ and $\beta$.\\
Without solving the equation,
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item write down the value of $( \alpha + \beta )$ and the value of $\alpha \beta$
\item show that $\alpha ^ { 2 } + \beta ^ { 2 } = - 2$
\item find the value of $\alpha ^ { 3 } + \beta ^ { 3 }$
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item show that $\alpha ^ { 4 } + \beta ^ { 4 } = \left( \alpha ^ { 2 } + \beta ^ { 2 } \right) ^ { 2 } - 2 ( \alpha \beta ) ^ { 2 }$
\item find a quadratic equation which has roots
$$\text { ( } \alpha ^ { 3 } - \beta \text { ) and ( } \beta ^ { 3 } - \alpha \text { ) }$$
giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2016 Q3 [11]}}