| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.3 This is a multi-part question involving standard C3 techniques: transformations of graphs (vertical stretch and translation), expressing trigonometric functions in R-cos form using standard identities, and solving trigonometric equations. While it requires multiple steps, each part follows routine procedures taught in C3 with no novel problem-solving required. The R-cos conversion is a textbook exercise, and finding A from given stationary points is straightforward substitution. Slightly easier than average due to the scaffolded structure. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks |
|---|---|
| Maximum at \((-45, 15)\), minimum at \((135, -1)\) | B3 |
| Answer | Marks |
|---|---|
| \(\therefore f(x) = A + 4 \cos (x + 45)°\) | M1, A1, A1 |
| Answer | Marks |
|---|---|
| \(3\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 93.6, 176.4\) (1dp) | M1, A1, A2 | (11) |
## (i)
Maximum at $(-45, 15)$, minimum at $(135, -1)$ | B3 |
## (ii)
$2\sqrt{2} \cos x - 2\sqrt{2} \sin x = R \cos x \cos \alpha - R \sin x \sin \alpha$
$R \cos \alpha = 2\sqrt{2}$, $R \sin \alpha = 2\sqrt{2}$
$\therefore R = \sqrt{8+8} = 4$
$\tan \alpha = 1$, $\alpha = 45$
$\therefore f(x) = A + 4 \cos (x + 45)°$ | M1, A1, A1 |
## (iii)
$3$ | B1 |
## (iv)
$3 + 4 \cos (x + 45) = 0$
$\cos (x + 45) = -\frac{3}{4}$
$x + 45 = 180 - 41.4, 180 + 41.4 = 138.6, 221.4$
$x = 93.6, 176.4$ (1dp) | M1, A1, A2 | (11)
---\includegraphics{figure_7}
The diagram shows the curve $y = \text{f}(x)$ which has a maximum point at $(-45, 7)$ and a minimum point at $(135, -1)$.
\begin{enumerate}[label=(\roman*)]
\item Showing the coordinates of any stationary points, sketch the curve with equation $y = 1 + 2\text{f}(x)$. [3]
\end{enumerate}
Given that
$$\text{f}(x) = A + 2\sqrt{2} \cos x° - 2\sqrt{2} \sin x°, \quad x \in \mathbb{R}, \quad -180 \leq x \leq 180,$$
where $A$ is a constant,
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item show that f$(x)$ can be expressed in the form
$$\text{f}(x) = A + R \cos (x + \alpha)°,$$
where $R > 0$ and $0 < \alpha < 90$, [3]
\item state the value of $A$, [1]
\item find, to 1 decimal place, the $x$-coordinates of the points where the curve $y = \text{f}(x)$ crosses the $x$-axis. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q7 [11]}}