| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Integration using harmonic form |
| Difficulty | Standard +0.3 This is a standard harmonic form question with routine applications. Part (i) requires straightforward use of R cos(θ-α) expansion and inverse tan. Part (ii)(a) is a direct solve using the harmonic form. Part (ii)(b) involves a standard integration technique (sec² substitution) that follows mechanically once the harmonic form is established. 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) etc1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply \(R = 5\) | B1 | |
| Use relevant trigonometry to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 0.6435\) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Carry out appropriate method to find one value in given range | M1 | |
| Obtain \(1.80\) | A1 | |
| Carry out appropriate method to find second value in given range | M1 | |
| Obtain \(5.77\) and no other value | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Express integrand as \(k\sec^2(\theta - \alpha)\) for any constant \(k\) | M1 | |
| Integrate to obtain result \(k\tan(\theta - \alpha)\) | A1 | |
| Obtain correct answer \(2\tan(\theta - 0.6435)\) | A1 | [3] |
## Question 9(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply $R = 5$ | B1 | |
| Use relevant trigonometry to find $\alpha$ | M1 | |
| Obtain $\alpha = 0.6435$ | A1 | [3] |
## Question 9(ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Carry out appropriate method to find one value in given range | M1 | |
| Obtain $1.80$ | A1 | |
| Carry out appropriate method to find second value in given range | M1 | |
| Obtain $5.77$ and no other value | A1 | [4] |
## Question 9(ii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Express integrand as $k\sec^2(\theta - \alpha)$ for any constant $k$ | M1 | |
| Integrate to obtain result $k\tan(\theta - \alpha)$ | A1 | |
| Obtain correct answer $2\tan(\theta - 0.6435)$ | A1 | [3] |
---9 (i) Express $4 \cos \theta + 3 \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$. Give the value of $\alpha$ correct to 4 decimal places.\\
(ii) Hence
\begin{enumerate}[label=(\alph*)]
\item solve the equation $4 \cos \theta + 3 \sin \theta = 2$ for $0 < \theta < 2 \pi$,
\item find $\int \frac { 50 } { ( 4 \cos \theta + 3 \sin \theta ) ^ { 2 } } \mathrm {~d} \theta$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2013 Q9 [10]}}