| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2024 |
| 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 systematic application of Vieta's formulas across multiple parts, culminating in finding a quadratic with complex transformed roots (α - 1/β² and β - 1/α²). Part (c) requires algebraic manipulation of symmetric functions and careful bookkeeping, going beyond standard A-level transformations like α+1, 1/α. The multi-step nature and algebraic complexity place it moderately above average difficulty. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
\begin{enumerate}
\item The quadratic equation
\end{enumerate}
$$2 x ^ { 2 } - 3 x + 7 = 0$$
has roots $\alpha$ and $\beta$\\
Without solving the equation,\\
(a) write down the value of $( \alpha + \beta )$ and the value of $\alpha \beta$\\
(b) determine the value of $\alpha ^ { 2 } + \beta ^ { 2 }$\\
(c) find a quadratic equation which has roots
$$\left( \alpha - \frac { 1 } { \beta ^ { 2 } } \right) \text { and } \left( \beta - \frac { 1 } { \alpha ^ { 2 } } \right)$$
giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers to be determined.
\hfill \mbox{\textit{Edexcel F1 2024 Q5 [9]}}