| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| Session | November |
| 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 sin(θ + α) formula (finding R = 17, α ≈ 61.93°) 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 |
|---|---|---|
| (i) State or imply \(R = 17\). Use appropriate formula to find \(\alpha\). Obtain \(61.93\) | B1 M1 A1 | [3] |
| (ii) Attempt to find at least one value of \(\theta + \alpha\). Obtain one correct value of \(\theta(97.4\) or \(318.7)\). Carry out correct method to find second answer. Obtain second correct value and no others between 0 and 360 | M1 A1 M1 A1 | [4] |
(i) State or imply $R = 17$. Use appropriate formula to find $\alpha$. Obtain $61.93$ | B1 M1 A1 | [3]
(ii) Attempt to find at least one value of $\theta + \alpha$. Obtain one correct value of $\theta(97.4$ or $318.7)$. Carry out correct method to find second answer. Obtain second correct value and no others between 0 and 360 | M1 A1 M1 A1 | [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 2015 Q3 [7]}}