| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.3 This is a standard Further Maths transformation of roots question requiring systematic application of Vieta's formulas. Part (a) uses the identity (α+β)²-2αβ divided by (αβ)², while part (b) scales the result from (a). The techniques are well-practiced in F1, though slightly above average A-level difficulty due to the algebraic manipulation required. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
7. It is given that $\alpha$ and $\beta$ are roots of the equation $5 x ^ { 2 } - 4 x + 3 = 0$
Without solving the quadratic equation,
\begin{enumerate}[label=(\alph*)]
\item find the exact value of $\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }$
\item find a quadratic equation which has roots $\frac { 3 } { \alpha ^ { 2 } }$ and $\frac { 3 } { \beta ^ { 2 } }$\\
giving your answer in the form $a x ^ { 2 } + b x + c = 0$, where $a$, $b$ and $c$ are integers to be found.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2018 Q7 [9]}}