| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2010 |
| 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 = √5 and α ≈ 26.57°) followed by solving a straightforward equation using inverse sine. While it involves multiple steps, the techniques are well-practiced and follow a predictable template 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 |
|---|---|---|
| State \(R = \sqrt{5}\) | B1 | |
| Use trig formula to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 26.57°\) with no errors seen | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Carry out evaluation of \(\sin\left(\frac{\pm 0.4}{\sqrt{5}}\right)\) (≈ ±10.3048°) | M1 | |
| Obtain answer 16.3° | A1 | |
| Carry out correct method for second answer | M1 | |
| Obtain answer 216.9° and no others in the range | A1 | [4] |
**(i)**
| State $R = \sqrt{5}$ | B1 |
| Use trig formula to find $\alpha$ | M1 |
| Obtain $\alpha = 26.57°$ with no errors seen | A1 | [3] |
**(ii)**
| Carry out evaluation of $\sin\left(\frac{\pm 0.4}{\sqrt{5}}\right)$ (≈ ±10.3048°) | M1 |
| Obtain answer 16.3° | A1 |
| Carry out correct method for second answer | M1 |
| Obtain answer 216.9° and no others in the range | A1 | [4] |6 (i) Express $2 \sin \theta - \cos \theta$ in the form $R \sin ( \theta - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving the exact value of $R$ and the value of $\alpha$ correct to 2 decimal places.\\
(ii) Hence solve the equation
$$2 \sin \theta - \cos \theta = - 0.4$$
giving all solutions in the interval $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P2 2010 Q6 [7]}}