| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.2 This is a standard harmonic form question requiring routine application of the R-formula (finding r and α using Pythagoras and tan), reading off max/min values directly, and solving a straightforward equation. All steps are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(r\sin(\theta+\alpha) \equiv r\sin\theta\cos\alpha + r\cos\theta\sin\alpha\) | ||
| \(\Rightarrow \begin{cases} r\cos\alpha = 3 \\ r\sin\alpha = 4 \end{cases}\) | M1 | |
| \(\Rightarrow r = 5\) | A1 | |
| \(\alpha = \arctan\frac{4}{3} = 53.13\ldots°\) | A1 | |
| Total: 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(-5 \leq f(\theta) \leq 5\) | B1 | |
| Total: 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1 = 5\sin(\theta + 53.1\ldots)\) | ||
| \(\Rightarrow \theta + 53.1\ldots = \arcsin 0.2\) | M1 | |
| or \(= 180° - \arcsin 0.2\) | ||
| \(\Rightarrow \theta = 11.53\ldots - 53.1\ldots\) | ||
| or \(= 180° - 11.53\ldots - 53.1\ldots\) | ||
| \(\Rightarrow \theta = 115.33\ldots = 115.3°\) to 1 d.p. | A1 | |
| Total: 2 marks |
## Question 6(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $r\sin(\theta+\alpha) \equiv r\sin\theta\cos\alpha + r\cos\theta\sin\alpha$ | | |
| $\Rightarrow \begin{cases} r\cos\alpha = 3 \\ r\sin\alpha = 4 \end{cases}$ | M1 | |
| $\Rightarrow r = 5$ | A1 | |
| $\alpha = \arctan\frac{4}{3} = 53.13\ldots°$ | A1 | |
| **Total: 3 marks** | | |
---
## Question 6(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $-5 \leq f(\theta) \leq 5$ | B1 | |
| **Total: 1 mark** | | |
---
## Question 6(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1 = 5\sin(\theta + 53.1\ldots)$ | | |
| $\Rightarrow \theta + 53.1\ldots = \arcsin 0.2$ | M1 | |
| or $= 180° - \arcsin 0.2$ | | |
| $\Rightarrow \theta = 11.53\ldots - 53.1\ldots$ | | |
| or $= 180° - 11.53\ldots - 53.1\ldots$ | | |
| $\Rightarrow \theta = 115.33\ldots = 115.3°$ to 1 d.p. | A1 | |
| **Total: 2 marks** | | |
---6 The function $\mathrm { f } ( \theta ) = 3 \sin \theta + 4 \cos \theta$ is to be expressed in the form $r \sin ( \theta + \alpha )$ where $r > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.\\
(i) Find the values of $r$ and $\alpha$.\\
(ii) Write down the maximum and minimum value of $\mathrm { f } ( \theta )$.\\
(iii) Solve the equation $\mathrm { f } ( \theta ) = 1$ for $0 ^ { \circ } \leq \theta \leq 180 ^ { \circ }$.
\hfill \mbox{\textit{OCR MEI C4 Q6 [6]}}