| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2003 |
| Session | November |
| 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 harmonic form question requiring routine application of the R cos(θ - α) formula with exact values. Part (i) involves straightforward calculation (R = 2, α = π/3), and part (ii) requires solving a simple trigonometric equation with one solution given. While it requires multiple steps and exact value manipulation, it follows a well-practiced textbook procedure with no novel insight needed, 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 |
|---|---|
| State answer \(R = 2\) | B1 |
| Use trig formula to find \(\alpha\) | M1 |
| Obtain answer \(\alpha = \frac{1}{3}\pi\) | A1 |
| Answer | Marks |
|---|---|
| Carry out, or indicate need for, evaluation of \(\cos^{-1}(\sqrt{2}/2)\) | M1* |
| Obtain, or verify, the solution \(\theta = \frac{7}{12}\pi\) | A1 |
| Attempt correct method for the other solution in range i.e. \(-\cos^{-1}(\sqrt{2}/2) + \alpha\) | M1(dep*) |
| Obtain solution \(\theta = \frac{1}{12}\pi\) : [M1A0 for \(\frac{25\pi}{12}\)] | A1 |
### (i)
State answer $R = 2$ | B1 |
Use trig formula to find $\alpha$ | M1 |
Obtain answer $\alpha = \frac{1}{3}\pi$ | A1 |
**Total: [3]**
### (ii)
Carry out, or indicate need for, evaluation of $\cos^{-1}(\sqrt{2}/2)$ | M1* |
Obtain, or verify, the solution $\theta = \frac{7}{12}\pi$ | A1 |
Attempt correct method for the other solution in range i.e. $-\cos^{-1}(\sqrt{2}/2) + \alpha$ | M1(dep*) |
Obtain solution $\theta = \frac{1}{12}\pi$ : [M1A0 for $\frac{25\pi}{12}$] | A1 |
**Total: [4]**\begin{enumerate}[label=(\roman*)]
\item Express $\cos \theta + (\sqrt{3}) \sin \theta$ in the form $R \cos(\theta - \alpha)$, where $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$, giving the exact value of $\alpha$. [3]
\item Hence show that one solution of the equation
$$\cos \theta + (\sqrt{3}) \sin \theta = \sqrt{2}$$
is $\theta = \frac{7}{12}\pi$, and find the other solution in the interval $0 < \theta < 2\pi$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2003 Q4 [7]}}