| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.3 This is a standard Further Maths roots of polynomials question requiring systematic application of Vieta's formulas and algebraic manipulation. Part (a) is direct recall, part (b) is routine manipulation, and part (c) requires finding sum and product of transformed roots using established techniques. While it involves multiple steps, the methods are well-practiced in F1 and require no novel insight—slightly easier than average A-level difficulty overall. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
The quadratic equation
$$2 x ^ { 2 } - x + 3 = 0$$
has roots $\alpha$ and $\beta$.\\
Without solving the equation,
\begin{enumerate}[label=(\alph*)]
\item write down the value of $( \alpha + \beta )$ and the value of $\alpha \beta$
\item find the value of $\frac { 1 } { \alpha } + \frac { 1 } { \beta }$
\item find a quadratic equation which has roots
$$\left( 2 \alpha - \frac { 1 } { \beta } \right) \text { and } \left( 2 \beta - \frac { 1 } { \alpha } \right)$$
giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2017 Q2 [7]}}