| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2008 |
| 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 resulting equation. Part (i) involves straightforward use of R = √(a² + b²) and tan α = b/a, while part (ii) requires basic inverse cosine and finding solutions in a given range. These are textbook exercises 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 |
|---|---|
| (i) State \(R = \sqrt{26}\) | B1 |
| Use trig formula to find \(a\) | M1 |
| Obtain \(a = 11.31°\) with no errors seen | A1 |
| [3] | |
| (ii) Carry out evaluation of \(\cos^{-1}\left(\frac{4}{\sqrt{26}}\right)\) (≈ 38.3288...°) | M1 |
| Obtain answer 27.0° | A1 |
| Carry out correct method for second answer | M1 |
| Obtain answer 310.4° and no others in the range [Ignore answers outside the given range.] | A1√ |
| [4] |
**(i)** State $R = \sqrt{26}$ | B1 |
Use trig formula to find $a$ | M1 |
Obtain $a = 11.31°$ with no errors seen | A1 |
| [3] |
**(ii)** Carry out evaluation of $\cos^{-1}\left(\frac{4}{\sqrt{26}}\right)$ (≈ 38.3288...°) | M1 |
Obtain answer 27.0° | A1 |
Carry out correct method for second answer | M1 |
Obtain answer 310.4° and no others in the range [Ignore answers outside the given range.] | A1√ |
| [4] |5 (i) Express $5 \cos \theta - \sin \theta$ in the form $R \cos ( \theta + \alpha )$, 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
$$5 \cos \theta - \sin \theta = 4$$
giving all solutions in the interval $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P2 2008 Q5 [7]}}