| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2012 |
| 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 routine steps: finding R and α using Pythagorean theorem and inverse tan, solving a transformed equation, and identifying the maximum value. All techniques are textbook exercises requiring no novel insight, 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 |
|---|---|---|
| (i) State or imply \(R = 15\) | B1 | |
| Use appropriate formula to find \(a\) | M1 | |
| Obtain \(53.13°\) | A1 | [3] |
| (ii) Attempt to find at least one value of \(\theta - a\) | M1 | |
| Obtain one correct value \(68.6°\) of \(\theta\) | A1 | |
| Carry out correct method to find second answer | M1 | |
| Obtain \(217.7°\) and no others in range | A1 | [4] |
| (iii) State \(15\), following their value of \(R\) from part (i) | B1✓ | [1] |
**(i)** State or imply $R = 15$ | B1 |
Use appropriate formula to find $a$ | M1 |
Obtain $53.13°$ | A1 | [3]
**(ii)** Attempt to find at least one value of $\theta - a$ | M1 |
Obtain one correct value $68.6°$ of $\theta$ | A1 |
Carry out correct method to find second answer | M1 |
Obtain $217.7°$ and no others in range | A1 | [4]
**(iii)** State $15$, following their value of $R$ from part (i) | B1✓ | [1]
---4 (i) Express $9 \sin \theta - 12 \cos \theta$ in the form $R \sin ( \theta - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. Give the value of $\alpha$ correct to 2 decimal places.
Hence\\
(ii) solve the equation $9 \sin \theta - 12 \cos \theta = 4$ for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$,\\
(iii) state the largest value of $k$ for which the equation $9 \sin \theta - 12 \cos \theta = k$ has any solutions.
\hfill \mbox{\textit{CAIE P2 2012 Q4 [8]}}