| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2011 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Moderate -0.3 This is a standard harmonic form question requiring routine application of R cos(θ + α) = R cos α cos θ - R sin α sin θ, solving a trigonometric equation using the result, and recognizing that the minimum of the related expression is simply -3R. All steps are 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 \(R = \sqrt{34}\) | B1 | |
| Use trig formula to find \(a\) | M1 | |
| Obtain \(\alpha = 30.96°\) with no errors seen | A1 | [3] |
| (ii) Carry out evaluation of \(\cos\left(\frac{4}{R}\right)\) (\(\approx 46.6861°\) or \(313.3139°\)) | M1 | |
| Obtain answer \(15.7°\) | A1 | |
| Carry out correct method for second answer | M1 | |
| Obtain answer \(282.3°\) or \(282.4°\) and no others in the range | A1 | [4] |
| (iii) State \(-3\sqrt{34}\) (\(= -3R\)) | B1√ | [1] |
**(i)** State $R = \sqrt{34}$ | B1 |
Use trig formula to find $a$ | M1 |
Obtain $\alpha = 30.96°$ with no errors seen | A1 | [3] |
**(ii)** Carry out evaluation of $\cos\left(\frac{4}{R}\right)$ ($\approx 46.6861°$ or $313.3139°$) | M1 |
Obtain answer $15.7°$ | A1 |
Carry out correct method for second answer | M1 |
Obtain answer $282.3°$ or $282.4°$ and no others in the range | A1 | [4] |
**(iii)** State $-3\sqrt{34}$ ($= -3R$) | B1√ | [1] |8 (i) Express $5 \cos \theta - 3 \sin \theta$ 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
$$5 \cos \theta - 3 \sin \theta = 4$$
giving all solutions in the interval $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.\\
(iii) Write down the least value of $15 \cos \theta - 9 \sin \theta$ as $\theta$ varies.
\hfill \mbox{\textit{CAIE P2 2011 Q8 [8]}}