| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.3 This is a standard Further Maths question on transformed roots requiring Vieta's formulas and algebraic manipulation. Part (a) uses the identity α²+β²=(α+β)²-2αβ, while part (b) requires finding sum and product of reciprocal ratios using known relationships. Though it's Further Maths content, the techniques are routine and well-practiced, making it slightly easier than average overall. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
3. It is given that $\alpha$ and $\beta$ are roots of the equation
$$2 x ^ { 2 } - 7 x + 4 = 0$$
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $\alpha ^ { 2 } + \beta ^ { 2 }$
\item Find a quadratic equation which has roots $\frac { \alpha } { \beta }$ and $\frac { \beta } { \alpha }$, giving your answer in the form $a x ^ { 2 } + b x + c = 0$, where $a , b$ and $c$ are integers.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2015 Q3 [6]}}