| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.8 This is a Further Maths question requiring manipulation of symmetric functions of roots beyond basic sum/product. Part (a) needs expansion and substitution using Vieta's formulas. Part (b) requires forming a new equation from complex transformed roots (α/(α²+1)), which demands systematic application of sum and product of transformed roots—a non-trivial multi-step process typical of F1 but more demanding than standard root transformation questions. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
5.
$$f ( x ) = x ^ { 2 } - 6 x + 3$$
The equation $\mathrm { f } ( x ) = 0$ has roots $\alpha$ and $\beta$\\
Without solving the equation,
\begin{enumerate}[label=(\alph*)]
\item determine the value of
$$\left( \alpha ^ { 2 } + 1 \right) \left( \beta ^ { 2 } + 1 \right)$$
\item find a quadratic equation which has roots
$$\frac { \alpha } { \left( \alpha ^ { 2 } + 1 \right) } \text { and } \frac { \beta } { \left( \beta ^ { 2 } + 1 \right) }$$
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 2023 Q5 [10]}}