| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2021 |
| Session | October |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring systematic application of Vieta's formulas and algebraic manipulation to find sums and products of transformed roots. Part (c) requires finding a new quadratic equation with complex transformed roots (1/(α²+β) and 1/(β²+α)), demanding careful algebraic work across multiple steps. While the techniques are standard for FM students, the length and computational complexity elevate this above average difficulty. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
3. The quadratic equation
$$2 x ^ { 2 } - 5 x + 7 = 0$$
has roots $\alpha$ and $\beta$
Without solving the equation,
\begin{enumerate}[label=(\alph*)]
\item write down the value of $( \alpha + \beta )$ and the value of $\alpha \beta$
\item determine, giving each answer as a simplified fraction, the value of
\begin{enumerate}[label=(\roman*)]
\item $\alpha ^ { 2 } + \beta ^ { 2 }$
\item $\alpha ^ { 3 } + \beta ^ { 3 }$
\end{enumerate}\item find a quadratic equation that has roots
$$\frac { 1 } { \alpha ^ { 2 } + \beta } \text { and } \frac { 1 } { \beta ^ { 2 } + \alpha }$$
giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers to be determined.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2021 Q3 [9]}}