| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Given factor, find all roots |
| Difficulty | Moderate -0.3 This is a standard C1 multi-part question combining graph sketching, algebraic manipulation, and factorization. Part (i) is routine plotting, part (ii) is straightforward equation manipulation (equating two functions), and part (iii) involves standard factor theorem application and checking rationality of roots using the quadratic formula. All techniques are core C1 skills with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.02q Use intersection points: of graphs to solve equations |
Answer part (i) of this question on the insert provided.
The insert shows the graph of $y = \frac{1}{x}$.
\begin{enumerate}[label=(\roman*)]
\item On the insert, on the same axes, plot the graph of $y = x^2 - 5x + 5$ for $0 \leq x \leq 5$. [4]
\item Show algebraically that the $x$-coordinates of the points of intersection of the curves $y = \frac{1}{x}$ and $y = x^2 - 5x + 5$ satisfy the equation $x^3 - 5x^2 + 5x - 1 = 0$. [2]
\item Given that $x = 1$ at one of the points of intersection of the curves, factorise $x^3 - 5x^2 + 5x - 1$ into a linear and a quadratic factor.
Show that only one of the three roots of $x^3 - 5x^2 + 5x - 1 = 0$ is rational. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C1 Q2 [11]}}