| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2018 |
| Session | December |
| Marks | 5 |
| Topic | Curve Sketching |
| Type | Basic factored form sketching |
| Difficulty | Moderate -0.3 This question tests understanding of repeated roots and polynomial factorization. Part (a) requires recognizing that 'exactly two roots' means one must be repeated, which is conceptually straightforward. Part (b) involves writing two factorizations (either (x-2)²(x-3) or (x-2)(x-3)²) and expanding, which is routine algebraic manipulation. While it requires some insight about multiplicity, the execution is mechanical and well within standard A-level expectations. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials |
$\text{f}(x)$ is a cubic polynomial in which the coefficient of $x^3$ is 1. The equation $\text{f}(x) = 0$ has exactly two roots.
\begin{enumerate}[label=(\alph*)]
\item Sketch a possible graph of $y = \text{f}(x)$. [2]
\end{enumerate}
It is now given that the two roots are $x = 2$ and $x = 3$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find, in expanded form, the two possible polynomials $\text{f}(x)$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2018 Q2 [5]}}