| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| 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) = R cos θ cos α - R sin θ sin α, followed by straightforward equation solving and identifying max/min values. While it involves multiple parts and some algebraic manipulation with √5, it follows a well-practiced template 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/Working | Mark | Guidance |
| State or imply \(R=3\) | B1 | |
| Use appropriate formula to find \(\alpha\) | M1 | |
| Obtain \(41.81°\) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt to find one correct value of \(\theta+\alpha\) | M1 | |
| Obtain one correct value (\(30.7\) or \(245.6\)) of \(\theta\) | A1 | |
| Carry out correct method to find second answer | M1 | |
| Obtain second correct answer and no others in range | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State greatest value is \(13\), following their value of \(R\) | B1 | |
| State least value is \(7\), following their value of \(R\) | B1 | [2] |
## Question 6:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply $R=3$ | B1 | |
| Use appropriate formula to find $\alpha$ | M1 | |
| Obtain $41.81°$ | A1 | [3] |
### Part (ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt to find one correct value of $\theta+\alpha$ | M1 | |
| Obtain one correct value ($30.7$ or $245.6$) of $\theta$ | A1 | |
| Carry out correct method to find second answer | M1 | |
| Obtain second correct answer and no others in range | A1 | [4] |
### Part (ii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State greatest value is $13$, following their value of $R$ | B1 | |
| State least value is $7$, following their value of $R$ | B1 | [2] |
---6 (i) Express $( \sqrt { } 5 ) \cos \theta - 2 \sin \theta$ 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
\begin{enumerate}[label=(\alph*)]
\item solve the equation $( \sqrt { } 5 ) \cos \theta - 2 \sin \theta = 0.9$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$,
\item state the greatest and least values of
$$10 + ( \sqrt { } 5 ) \cos \theta - 2 \sin \theta$$
as $\theta$ varies.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2015 Q6 [9]}}