| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Session | Specimen |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.3 This is a standard Further Maths roots transformation question requiring Vieta's formulas and algebraic manipulation. Part (a) is direct recall, part (b) is routine simplification using α+β and αβ, and part (c) follows a well-practiced method of finding sum and product of transformed roots. While it requires more steps than basic A-level, it's a textbook exercise with no novel insight needed. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
\begin{enumerate}
\item The quadratic equation
\end{enumerate}
$$5 x ^ { 2 } - 4 x + 1 = 0$$
has roots $\alpha$ and $\beta$.\\
(a) Write down the value of $\alpha + \beta$ and the value of $\alpha \beta$.\\
(b) Show that $\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = \frac { 6 } { 5 }$\\
(c) Find a quadratic equation with integer coefficients, which has roots
$$\alpha + \frac { 1 } { \alpha } \text { and } \beta + \frac { 1 } { \beta }$$
\hfill \mbox{\textit{Edexcel F1 Q4 [12]}}