| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.3 This is a standard two-part harmonic form question requiring routine application of the R cos(θ - α) formula (finding R and α using Pythagorean theorem and arctan), followed by solving a straightforward trigonometric equation. While it involves multiple steps, the techniques are well-practiced and require no novel insight, 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 |
| (i) State or imply \(R = 17\) | B1 | |
| Use correct trigonometric formula to find \(\alpha\) | M1 | |
| Obtain \(61.93°\) with no errors seen | A1 | [3] |
| (ii) Evaluate \(\cos^{-1}\frac{12}{R}\) \((= 45.099°)\) | M1 | |
| Obtain answer \(107.0°\) | A1 | |
| Carry out correct method for second answer | M1 | |
| Obtain answer \(16.8°\) and no others between \(0°\) and \(360°\) | A1 | [4] |
## Question 3:
| Answer/Working | Mark | Guidance |
|---|---|---|
| **(i)** State or imply $R = 17$ | B1 | |
| Use correct trigonometric formula to find $\alpha$ | M1 | |
| Obtain $61.93°$ with no errors seen | A1 | [3] |
| **(ii)** Evaluate $\cos^{-1}\frac{12}{R}$ $(= 45.099°)$ | M1 | |
| Obtain answer $107.0°$ | A1 | |
| Carry out correct method for second answer | M1 | |
| Obtain answer $16.8°$ and no others between $0°$ and $360°$ | A1 | [4] |
---3 (i) Express $8 \cos \theta + 15 \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. Give the value of $\alpha$ correct to 2 decimal places.\\
(ii) Hence solve the equation $8 \cos \theta + 15 \sin \theta = 12$, giving all solutions in the interval $0 ^ { \circ } < \theta < 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P3 2011 Q3 [7]}}