| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2006 |
| 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 (finding R = √(7² + 24²) = 25 and α = arctan(24/7)), followed by solving a straightforward trigonometric equation. While it involves multiple steps and careful arithmetic, it follows a well-practiced textbook procedure 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 answer \(R = 25\) | B1 | |
| Use trig formula to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 73.74°\) | A1 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Carry out evaluation of \(\cos^{-1}(15/25)\) (\(= 53.1301...\)°) | M1 | |
| Obtain answer \(126.9°\) | A1 | |
| Carry out correct method for second answer | M1 | |
| Obtain answer \(20.6°\) and no others in the range [Ignore answers outside the given range.] | A1♦ | 4 |
**(i)**
| State answer $R = 25$ | B1 |
| Use trig formula to find $\alpha$ | M1 |
| Obtain $\alpha = 73.74°$ | A1 | 3 |
**(ii)**
| Carry out evaluation of $\cos^{-1}(15/25)$ ($= 53.1301...$°) | M1 |
| Obtain answer $126.9°$ | A1 |
| Carry out correct method for second answer | M1 |
| Obtain answer $20.6°$ and no others in the range [Ignore answers outside the given range.] | A1♦ | 4 |\begin{enumerate}[label=(\roman*)]
\item Express $7\cos \theta + 24\sin \theta$ in the form $R\cos(\theta - \alpha)$, where $R > 0$ and $0° < \alpha < 90°$, giving the exact value of $R$ and the value of $\alpha$ correct to 2 decimal places. [3]
\item Hence solve the equation
$$7\cos \theta + 24\sin \theta = 15,$$
giving all solutions in the interval $0° \leqslant \theta \leqslant 360°$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2006 Q4 [7]}}