| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Moderate -0.3 This is a standard two-part harmonic form question requiring routine application of the R cos(θ + α) formula and solving a resulting trigonometric equation. While it involves multiple steps (finding R and α, then solving), the techniques are well-practiced C3 material with no novel problem-solving 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 | Mark | Guidance |
| Obtain \(R = \sqrt{17}\) or 4.12 or 4.1 | B1 | or greater accuracy |
| Attempt recognisable process for finding \(\alpha\) | M1 | allow for sin/cos confusion |
| Obtain \(\alpha = 14\) | A1 | 3 or greater accuracy \(14.036\ldots\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt to find at least one value of \(\theta + \alpha\) | M1 | |
| Obtain or imply value 61 | A1\(\sqrt{}\) | following \(R\) value; or value rounding to 61 |
| Obtain 46.9 | A1 | allow \(\pm 0.1\); allow greater accuracy |
| Show correct process for obtaining second angle | M1 | |
| Obtain \(-75\) | A1 | 5 allow \(\pm 0.1\); allow greater accuracy; max of 4/5 if extra angles between \(-180\) and \(180\) |
# Question 5:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain $R = \sqrt{17}$ or 4.12 or 4.1 | B1 | or greater accuracy |
| Attempt recognisable process for finding $\alpha$ | M1 | allow for sin/cos confusion |
| Obtain $\alpha = 14$ | A1 | **3** or greater accuracy $14.036\ldots$ |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt to find at least one value of $\theta + \alpha$ | M1 | |
| Obtain or imply value 61 | A1$\sqrt{}$ | following $R$ value; or value rounding to 61 |
| Obtain 46.9 | A1 | allow $\pm 0.1$; allow greater accuracy |
| Show correct process for obtaining second angle | M1 | |
| Obtain $-75$ | A1 | **5** allow $\pm 0.1$; allow greater accuracy; max of 4/5 if extra angles between $-180$ and $180$ |
---5 (i) Express $4 \cos \theta - \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.\\
(ii) Hence solve the equation $4 \cos \theta - \sin \theta = 2$, giving all solutions for which $- 180 ^ { \circ } < \theta < 180 ^ { \circ }$.
\hfill \mbox{\textit{OCR C3 2007 Q5 [8]}}