| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Moderate -0.3 This is a standard Further Maths roots of polynomials question requiring routine application of Vieta's formulas and transformation techniques. Part (a) is direct recall, part (b) uses the standard identity α²+β²=(α+β)²-2αβ, and part (c) follows a well-practiced method for finding equations with transformed roots. While it requires multiple steps and is harder than typical A-level questions, it's a textbook exercise with no novel problem-solving required. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
2. The quadratic equation
$$5 x ^ { 2 } - 4 x + 2 = 0$$
has roots $\alpha$ and $\beta$.
\begin{enumerate}[label=(\alph*)]
\item Write down the value of $\alpha + \beta$ and the value of $\alpha \beta$.
\item Find the value of $\alpha ^ { 2 } + \beta ^ { 2 }$.
\item Find a quadratic equation which has roots
$$\frac { 1 } { \alpha ^ { 2 } } \text { and } \frac { 1 } { \beta ^ { 2 } }$$
giving your answer in the form $p x ^ { 2 } + q x + r = 0$, where $p , q$ and $r$ are integers.\\
\includegraphics[max width=\textwidth, alt={}, center]{4da2bb2c-a51b-493c-a9f2-f4ff008a3aac-07_70_51_2663_1896}
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2014 Q2 [8]}}