| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2023 |
| Session | January |
| 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 manipulation of roots through multiple transformations. Part (a) is routine Vieta's formulas, but parts (b) and (c) require algebraic manipulation of complex symmetric functions and forming a new quadratic from transformed roots—significantly more demanding than standard A-level work, though still a recognizable exercise type in F1. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
\begin{enumerate}
\item The quadratic equation
\end{enumerate}
$$4 x ^ { 2 } + 3 x + k = 0$$
where $k$ is an integer, has roots $\alpha$ and $\beta$\\
(a) Write down, in terms of $k$ where appropriate, the value of $\alpha + \beta$ and the value of $\alpha \beta$\\
(b) Determine, in simplest form in terms of $k$, the value of $\frac { \alpha } { \beta ^ { 2 } } + \frac { \beta } { \alpha ^ { 2 } }$\\
(c) Determine a quadratic equation which has roots
$$\frac { \alpha } { \beta ^ { 2 } } \text { and } \frac { \beta } { \alpha ^ { 2 } }$$
giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integer values in terms of $k$
\hfill \mbox{\textit{Edexcel F1 2023 Q5 [9]}}