| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Sketch transformations from algebraic function |
| Difficulty | Moderate -0.8 This is a straightforward question testing basic understanding of trigonometric transformations. Part (i) requires simple recall that cos(2x) has period 180°. Part (ii) involves identifying standard transformations (horizontal stretch factor 1/2 and vertical translation +1). Part (iii) is routine curve sketching with clear amplitude and period. All parts are textbook-standard with no problem-solving or novel insight required, making this easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05f Trigonometric function graphs: symmetries and periodicities |
| Answer | Marks | Guidance |
|---|---|---|
| \(0 \le x \le 180°\) or \(\pi\) | B1 | condone \(0 \le x \le 180°\) or \(\pi\) |
| Answer | Marks | Guidance |
|---|---|---|
| One-way stretch in x-direction, scale factor \(\frac{1}{2}\), translation in y-direction through \(\begin{pmatrix}0\\1\end{pmatrix}\) | M1, A1, M1, A1 | [4] Either way round... condone 'squeeze', 'contract' for M1; stretch used and s.f \(\frac{1}{2}\); condone 'move', 'shift', etc. for M1; 'translation' used, +1 unit; \(\begin{pmatrix}0\\1\end{pmatrix}\) only is M1 A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Correct shape, touching x-axis at \(-90°, 90°\) | M1, B1 | Correct domain; \((0, 2)\) marked or indicated (i.e. amplitude is 2) |
## Part (i)
$0 \le x \le 180°$ or $\pi$ | B1 | condone $0 \le x \le 180°$ or $\pi$
## Part (ii)
One-way stretch in x-direction, scale factor $\frac{1}{2}$, translation in y-direction through $\begin{pmatrix}0\\1\end{pmatrix}$ | M1, A1, M1, A1 | [4] Either way round... condone 'squeeze', 'contract' for M1; stretch used and s.f $\frac{1}{2}$; condone 'move', 'shift', etc. for M1; 'translation' used, +1 unit; $\begin{pmatrix}0\\1\end{pmatrix}$ only is M1 A0
## Part (iii)
Correct shape, touching x-axis at $-90°, 90°$ | M1, B1 | Correct domain; $(0, 2)$ marked or indicated (i.e. amplitude is 2) | A1 | [3]
---5 (i) State the period of the function $\mathrm { f } ( x ) = 1 + \cos 2 x$, where $x$ is in degrees.\\
(ii) State a sequence of two geometrical transformations which maps the curve $y = \cos x$ onto the curve $y = \mathrm { f } ( x )$.\\
(iii) Sketch the graph of $y = \mathrm { f } ( x )$ for $- 180 ^ { \circ } < x < 180 ^ { \circ }$.
\hfill \mbox{\textit{OCR MEI C3 2009 Q5 [8]}}