| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2021 |
| Session | June |
| Marks | 4 |
| Topic | Roots of polynomials |
| Type | Quadratic with transformed roots |
| Difficulty | Standard +0.3 Part (a) is direct application of sum/product of roots formulas. Part (b) requires algebraic manipulation to find the sum and product of transformed roots, which is a standard Further Maths technique but involves more steps than typical A-level questions. The series question is routine application of standard formulae. Overall slightly easier than average for Further Maths content. |
| Spec | 4.05a Roots and coefficients: symmetric functions4.06a Summation formulae: sum of r, r^2, r^3 |
1 In this question you must show detailed reasoning.\\
The quadratic equation $x ^ { 2 } - 2 x + 5 = 0$ has roots $\alpha$ and $\beta$.
\begin{enumerate}[label=(\alph*)]
\item Write down the values of $\alpha + \beta$ and $\alpha \beta$.
\item Hence find a quadratic equation with roots $\alpha + \frac { 1 } { \beta }$ and $\beta + \frac { 1 } { \alpha }$.
Using the formulae for $\sum _ { r = 1 } ^ { n } r$ and $\sum _ { r = 1 } ^ { n } r ^ { 2 }$, show that $\sum _ { r = 1 } ^ { 10 } r ( 3 r - 2 ) = 1045$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2021 Q1 [4]}}