| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | January |
| Marks | 8 |
| 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) uses the identity α²+β²=(α+β)²-2αβ, and part (c) requires finding sum and product of transformed roots using known values. While it involves multiple steps, the techniques are routine for FM students with no novel insight required. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
5. The quadratic equation
$$4 x ^ { 2 } + 3 x + 1 = 0$$
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 $\left( \alpha ^ { 2 } + \beta ^ { 2 } \right)$.
\item Find a quadratic equation which has roots
$$( 4 \alpha - \beta ) \text { and } ( 4 \beta - \alpha )$$
giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers to be determined.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2015 Q5 [8]}}