| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | June |
| Marks | 8 |
| 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 with straightforward application of compound angle formulas and solving. Part (a) requires expanding cos(x-60°), collecting terms, and using R²=a²+b² with tan α=b/a. Part (b) is a routine substitution (x=2θ) followed by solving R cos(x-α)=2.5. While multi-step, it follows a well-practiced algorithm 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 |
|---|---|---|
| Answer | Mark | Guidance |
| \(\cos^{-1}\left(\frac{2.5}{R}\right)\) | B1 FT | SOI [55.0°]. Follow through their \(\sqrt{19}\) |
| Use correct method to find value of \(2\theta\) (not \(x\)) in interval. Allow sign error in moving \(\alpha\) to right side | M1 | \(2\theta = \cos^{-1}\left(\frac{2.5}{R}\right) + 23.41°\) or \(2\theta = 360° - \cos^{-1}\left(\frac{2.5}{R}\right) + 23.41°\) with \(R\) substituted |
| Obtain one correct answer e.g. \(39.2°\) | A1 | If working for M1 not seen then M1 implied by \(39.2°\) or \(164.2°\). Must be at least 1 d.p. |
| Obtain second correct answer e.g. \(164.2°\) and no others in the interval | A1 | Must be at least 1 d.p. Ignore answers outside the given interval |
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\cos^{-1}\left(\frac{2.5}{R}\right)$ | **B1 FT** | SOI [55.0°]. Follow through their $\sqrt{19}$ |
| Use correct method to find value of $2\theta$ (not $x$) in interval. Allow sign error in moving $\alpha$ to right side | **M1** | $2\theta = \cos^{-1}\left(\frac{2.5}{R}\right) + 23.41°$ or $2\theta = 360° - \cos^{-1}\left(\frac{2.5}{R}\right) + 23.41°$ with $R$ substituted |
| Obtain one correct answer e.g. $39.2°$ | **A1** | If working for M1 not seen then M1 implied by $39.2°$ or $164.2°$. Must be at least 1 d.p. |
| Obtain second correct answer e.g. $164.2°$ and no others in the interval | **A1** | Must be at least 1 d.p. Ignore answers outside the given interval |
---6
\begin{enumerate}[label=(\alph*)]
\item Express $3 \cos x + 2 \cos \left( x - 60 ^ { \circ } \right)$ in the form $R \cos ( x - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. State the exact value of $R$ and give $\alpha$ correct to 2 decimal places.
\item Hence solve the equation
$$3 \cos 2 \theta + 2 \cos \left( 2 \theta - 60 ^ { \circ } \right) = 2.5$$
for $0 ^ { \circ } < \theta < 180 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q6 [8]}}