| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Reflections |
| Difficulty | Standard +0.3 This question tests understanding of curve properties and transformations at a basic level. Part (a) requires reading values from a graph, part (b) is a standard reflection in the y-axis, and part (c) asks for recognition that a cubic with only one real root cannot be factored into three linear factors. All parts are routine applications of A-level concepts with no problem-solving or novel insight required. |
| Spec | 1.02h Express solutions: using 'and', 'or', set and interval notation1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x < 1\) | B1 | Any notation. Allow \(0.4 < x < 1\) |
| \(2 < x < 3\) | B1 | Any notation. Lose only one B1 if \(\leq\) seen instead of \(<\) |
\(\{x: x<1\} \cup \{x: 2| B1f |
ft their (i) dep at least two ranges; allow \(\leq\) |
|
| [3] | SC: (MR '>') \(1 |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt reflect in \(y\)-axis | M1 | Approx correct shape, orientation & location |
| Through \((-3, 0.5)\), \((-2, 0.5)\), \((-1, 0.5)\), \((-0.5, -1.5)\) | A1 | Allow \(\pm 2\)mm. SC. All four points plotted \(\pm 2\)mm: B1 |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Curve intersects \(x\)-axis only once | B1 | oe |
| [1] |
# Question 3:
## Part (a):
| $x < 1$ | B1 | Any notation. Allow $0.4 < x < 1$ |
|---|---|---|
| $2 < x < 3$ | B1 | Any notation. Lose only one B1 if $\leq$ seen instead of $<$ |
| $\{x: x<1\} \cup \{x: 2<x<3\}$ | B1f | ft their (i) dep at least two ranges; allow $\leq$ |
| | [3] | SC: (MR '>') $1<x<2$; $x>3$ B0B1 $\{x: 1<x<2\}\cup\{x: x>3\}$ B1 |
## Part (b):
| Attempt reflect in $y$-axis | M1 | Approx correct shape, orientation & location |
|---|---|---|
| Through $(-3, 0.5)$, $(-2, 0.5)$, $(-1, 0.5)$, $(-0.5, -1.5)$ | A1 | Allow $\pm 2$mm. SC. All four points plotted $\pm 2$mm: B1 |
| | [2] | |
## Part (c):
| Curve intersects $x$-axis only once | B1 | oe |
|---|---|---|
| | [1] | |
---3 The diagram shows the curve $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x )$ is a cubic polynomial in $x$. This diagram is repeated in the Printed Answer Booklet.\\
\includegraphics[max width=\textwidth, alt={}, center]{d44919ed-806d-48c0-9726-c5fd67764504-03_896_1467_1382_244}
\begin{enumerate}[label=(\alph*)]
\item State the values of $x$ for which $\mathrm { f } ( x ) < \frac { 1 } { 2 }$, giving your answer in set notation.
\item On the diagram in the Printed Answer Booklet, draw the graph of $y = \mathrm { f } ( - x )$.
\item Explain how you can tell that $\mathrm { f } ( x )$ cannot be expressed as the product of three real linear factors.
\end{enumerate}
\hfill \mbox{\textit{OCR PURE Q3 [6]}}