| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.3 This is a standard two-part harmonic form question requiring routine application of R sin(θ - α) = R sin θ cos α - R cos θ sin α, followed by solving a straightforward equation. The technique is well-practiced in P3/C3 courses 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) etc |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply \(R = 25\) | B1 | |
| Use correct trigonometric formula to find \(\alpha\) | M1 | |
| Obtain \(16.26°\) | A1 | with no errors seen [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Evaluate \(\sin^{-1}\frac{17}{R}\) \(( = 42.84...°)\) | M1 | |
| Obtain answer \(59.1°\) | A1 | [2] |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply $R = 25$ | B1 | |
| Use correct trigonometric formula to find $\alpha$ | M1 | |
| Obtain $16.26°$ | A1 | **with no errors seen** [3] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Evaluate $\sin^{-1}\frac{17}{R}$ $( = 42.84...°)$ | M1 | |
| Obtain answer $59.1°$ | A1 | [2] |
---2 (i) Express $24 \sin \theta - 7 \cos \theta$ 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.\\
(ii) Hence find the smallest positive value of $\theta$ satisfying the equation
$$24 \sin \theta - 7 \cos \theta = 17$$
\hfill \mbox{\textit{CAIE P3 2012 Q2 [5]}}