| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2017 |
| Session | June |
| 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 the R cos(θ - α) formula and solving a basic trigonometric equation. The technique is well-practiced at A-level, with straightforward calculations (R = 3, α = arctan(√5/2)) and standard angle-finding in part (ii). Slightly easier than average due to its predictable structure and lack of complicating features. |
| 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 \(R = 3\) | B1 | Allow marks for (i) if seen in (ii) |
| Use appropriate trigonometric formula to find \(\alpha\) | M1 | |
| Obtain 48.19 with no errors seen | A1 | |
| Total: | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Carry out evaluation of \(\cos^{-1}\frac{1}{3}\ (= 70.528...)\) | M1 | M1 for \(\cos^{-1}\left(\frac{1}{R}\right)\) |
| Obtain correct answer 118.7 | A1 | |
| Carry out correct method to find second answer | M1 | |
| Obtain 337.7 and no others between 0 and 360 | A1 | |
| Total: | 4 |
## Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $R = 3$ | B1 | Allow marks for (i) if seen in (ii) |
| Use appropriate trigonometric formula to find $\alpha$ | M1 | |
| Obtain 48.19 with no errors seen | A1 | |
| **Total:** | **3** | |
## Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Carry out evaluation of $\cos^{-1}\frac{1}{3}\ (= 70.528...)$ | M1 | M1 for $\cos^{-1}\left(\frac{1}{R}\right)$ |
| Obtain correct answer 118.7 | A1 | |
| Carry out correct method to find second answer | M1 | |
| Obtain 337.7 and no others between 0 and 360 | A1 | |
| **Total:** | **4** | |5 (i) Express $2 \cos \theta + ( \sqrt { } 5 ) \sin \theta$ in the form $R \cos ( \theta - \alpha )$ where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving the value of $\alpha$ correct to 2 decimal places.\\
(ii) Hence solve the equation $2 \cos \theta + ( \sqrt { } 5 ) \sin \theta = 1$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.\\
\hfill \mbox{\textit{CAIE P2 2017 Q5 [7]}}