| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2020 |
| 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 question on roots of polynomials requiring systematic application of Vieta's formulas and algebraic manipulation. Part (a) is direct recall, parts (b)(i)-(ii) use standard identities (α²+β² from (α+β)²-2αβ, and α³+β³ from factorization), and part (c) requires finding sum and product of transformed roots using known values. While it involves multiple steps and careful algebra, all techniques are routine for FM students with no novel problem-solving required. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
2\\
2. The quadratic equation
$$5 x ^ { 2 } - 2 x + 3 = 0$$
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 determine, giving each answer as a simplified fraction, the value of
\begin{enumerate}[label=(\roman*)]
\item $\alpha ^ { 2 } + \beta ^ { 2 }$
\item $\alpha ^ { 3 } + \beta ^ { 3 }$
\end{enumerate}\item determine a quadratic equation that has roots
$$\left( \alpha + \beta ^ { 2 } \right) \text { and } \left( \beta + \alpha ^ { 2 } \right)$$
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 2020 Q2 [9]}}