| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Moderate -0.3 This is a standard FP1 question on roots of polynomials using Vieta's formulas and symmetric functions. Part (a) involves routine application of sum/product of roots and algebraic manipulation to find a new equation. Part (b) uses Vieta's formulas with a constraint on the roots. All techniques are textbook exercises with no novel insight required, making it slightly easier than average A-level difficulty despite being Further Maths content. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.05b Transform equations: substitution for new roots |
\begin{enumerate}[label=(\alph*)]
\item The quadratic equation $x^2 - 2x + 4 = 0$ has roots $\alpha$ and $\beta$.
\begin{enumerate}[label=(\roman*)]
\item Write down the values of $\alpha + \beta$ and $\alpha\beta$. [2]
\item Show that $\alpha^2 + \beta^2 = -4$. [2]
\item Hence find a quadratic equation which has roots $\alpha^2$ and $\beta^2$. [3]
\end{enumerate}
\item The cubic equation $x^3 - 12x^2 + ax - 48 = 0$ has roots $p$, $2p$ and $3p$.
\begin{enumerate}[label=(\roman*)]
\item Find the value of $p$. [2]
\item Hence find the value of $a$. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 Q8 [11]}}