| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Session | Specimen |
| Marks | 7 |
| 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 R sin(θ + α) = a sin θ + b cos θ formula (R = √(64+225) = 17, tan α = 15/8), followed by solving a straightforward trigonometric equation. While it involves multiple steps, both parts follow 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 | Marks | Guidance |
| State or imply \(R = 17\) | B1 | |
| Use appropriate formula to find \(\alpha\) | M1 | |
| Obtain \(61.93\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt to find at least one value of \(\theta + \alpha\) | M1 | |
| Obtain one correct value of \(\theta\) (\(97.4\) or \(318.7\)) | A1 | |
| Carry out correct method to find second answer | M1 | |
| Obtain second correct value and no others between \(0\) and \(360\) | A1 | |
| Total: 4 |
## Question 3(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $R = 17$ | B1 | |
| Use appropriate formula to find $\alpha$ | M1 | |
| Obtain $61.93$ | A1 | |
| **Total: 3** | | |
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## Question 3(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt to find at least one value of $\theta + \alpha$ | M1 | |
| Obtain one correct value of $\theta$ ($97.4$ or $318.7$) | A1 | |
| Carry out correct method to find second answer | M1 | |
| Obtain second correct value and no others between $0$ and $360$ | A1 | |
| **Total: 4** | | |
---3 (i) Express $8 \sin \theta + 15 \cos \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. Give the value of $\alpha$ correct to 2 decimal places.\\
(ii) Hence solve the equation
$$8 \sin \theta + 15 \cos \theta = 6$$
for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.\\
\hfill \mbox{\textit{CAIE P2 Q3 [7]}}