| Exam Board | OCR |
|---|---|
| Module | H240/03 (Pure Mathematics and Mechanics) |
| Year | 2023 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Multiple transformations in sequence |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard A-level techniques: differentiation to find a constant, using the factor theorem, and applying a translation. Part (a) involves routine calculus and algebra, while part (b) requires careful algebraic expansion but follows a standard procedure. The question is slightly easier than average because each step is clearly signposted and uses well-practiced methods with no novel problem-solving required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02w Graph transformations: simple transformations of f(x)1.07i Differentiate x^n: for rational n and sums |
The cubic polynomial $\text{f}(x)$ is defined by $\text{f}(x) = x^3 + px + q$, where $p$ and $q$ are constants.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Given that $\text{f}'(2) = 13$, find the value of $p$. [2]
\item Given also that $(x - 2)$ is a factor of $\text{f}(x)$, find the value of $q$. [2]
\end{enumerate}
The curve $y = \text{f}(x)$ is translated by the vector $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$.
\item Using the values from part (a), determine the equation of the curve after it has been translated. Give your answer in the form $y = x^3 + ax^2 + bx + c$, where $a$, $b$ and $c$ are integers to be found. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/03 2023 Q3 [8]}}