| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Given factor, find all roots |
| Difficulty | Moderate -0.3 This is a structured multi-part question with clear guidance at each step. Part (i) is routine graph sketching, part (ii) is straightforward algebraic manipulation to form a cubic, and part (iii) explicitly gives one root and asks for standard factor theorem application followed by solving a quadratic. While it combines several techniques, each step is standard and well-signposted, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.02q Use intersection points: of graphs to solve equations |
\includegraphics{figure_12}
Fig. 12 shows the graph of $y = \frac{-1}{x - 3}$.
\begin{enumerate}[label=(\roman*)]
\item Draw accurately, on the copy of Fig. 12, the graph of $y = x^2 - 4x + 1$ for $-1 < x < 5$. Use your graph to estimate the coordinates of the intersections of $y = \frac{-1}{x - 3}$ and $y = x^2 - 4x + 1$. [5]
\item Show algebraically that, where the curves intersect, $x^3 - 7x^2 + 13x - 4 = 0$. [3]
\item Use the fact that $x = 4$ is a root of $x^3 - 7x^2 + 13x - 4 = 0$ to find a quadratic factor of $x^3 - 7x^2 + 13x - 4$. Hence find the exact values of the other two roots of this equation. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C1 2012 Q12 [13]}}