| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.3 This is a standard harmonic form question requiring routine application of R cos(θ - α) = a cos θ + b sin θ formula, followed by straightforward equation solving. Part (i) uses R = √(a² + b²) and tan α = b/a, while part (ii) involves basic inverse trigonometry with one simple substitution (θ/2). All techniques 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 | Guidance |
|---|---|---|
| (i) Obtain or imply \(R = 4\) | B1 | |
| Use appropriate trigonometry to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 52.24\) or better from correct work | A1 | [3] |
| (ii) (a) State or imply \(\theta - \alpha = \cos^{-1}\left(-4 \div R\right)\) | M1 | |
| Obtain \(232.2\) or better | A1 | [2] |
| (ii) (b) Attempt at least one value using \(\cos^{-1}\left(3 \div R\right)\) | M1 | |
| Obtain one correct value e.g. \(\pm 41.41°\) | A1 | |
| Use \(\frac{1}{2}\theta - \alpha = \cos^{-1}\left(\frac{3}{R}\right)\) to find \(\theta\) | M1 | |
| Obtain \(21.7\) | A1 | [4] |
**(i)** Obtain or imply $R = 4$ | B1 |
Use appropriate trigonometry to find $\alpha$ | M1 |
Obtain $\alpha = 52.24$ or better from correct work | A1 | [3]
**(ii) (a)** State or imply $\theta - \alpha = \cos^{-1}\left(-4 \div R\right)$ | M1 |
Obtain $232.2$ or better | A1 | [2]
**(ii) (b)** Attempt at least one value using $\cos^{-1}\left(3 \div R\right)$ | M1 |
Obtain one correct value e.g. $\pm 41.41°$ | A1 |
Use $\frac{1}{2}\theta - \alpha = \cos^{-1}\left(\frac{3}{R}\right)$ to find $\theta$ | M1 |
Obtain $21.7$ | A1 | [4]8 (i) Express $( \sqrt { } 6 ) \cos \theta + ( \sqrt { } 10 ) \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. Give the value of $\alpha$ correct to 2 decimal places.\\
(ii) Hence, in each of the following cases, find the smallest positive angle $\theta$ which satisfies the equation
\begin{enumerate}[label=(\alph*)]
\item $( \sqrt { } 6 ) \cos \theta + ( \sqrt { } 10 ) \sin \theta = - 4$,
\item $( \sqrt { } 6 ) \cos \frac { 1 } { 2 } \theta + ( \sqrt { } 10 ) \sin \frac { 1 } { 2 } \theta = 3$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2010 Q8 [9]}}