| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| 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 transformation of roots. Part (a) is routine recall, parts (b)(i)-(ii) require standard algebraic manipulation with symmetric functions, but part (c) demands finding sum and product of asymmetric transformed roots (α³-β and β³-α), requiring careful algebraic work and insight into how to express these in terms of elementary symmetric functions. The extended multi-step nature and the non-standard root transformation elevate this above typical A-level questions. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
\begin{enumerate}
\item The quadratic equation
\end{enumerate}
$$2 x ^ { 2 } - 3 x + 5 = 0$$
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\\
(i) $\alpha ^ { 2 } + \beta ^ { 2 }$\\
(ii) $\alpha ^ { 3 } + \beta ^ { 3 }$\\
(c) find a quadratic equation which has roots
$$\left( \alpha ^ { 3 } - \beta \right) \text { and } \left( \beta ^ { 3 } - \alpha \right)$$
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 2022 Q5 [10]}}