| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Moderate -0.3 This is a standard two-part harmonic form question requiring routine application of the R sin(θ + α) formula and solving a transformed trigonometric equation. Part (i) involves straightforward use of R = √(a² + b²) and tan α = b/a, while part (ii) requires solving R sin(θ + α) = 1 within a given range. Both parts are textbook exercises with no novel insight required, making this 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 \(R = \sqrt{13}\) | B1 | |
| Use trig formula to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 33.69°\) with no errors seen | A1 | [3] |
| (ii) Evaluate \(\sin^{-1}(1/\sqrt{13})\) to at least 1 d.p. (\(16.10°\) to 2 d.p.) | B1✱ | |
| Carry out an appropriate method to find a value of \(\theta\) in the interval \(0° < \theta < 180°\) | M1 | |
| Obtain answer \(\theta = 130.2°\) and no other in the given interval | A1 | [3] |
**(i)** State $R = \sqrt{13}$ | B1 |
Use trig formula to find $\alpha$ | M1 |
Obtain $\alpha = 33.69°$ with no errors seen | A1 | [3]
**(ii)** Evaluate $\sin^{-1}(1/\sqrt{13})$ to at least 1 d.p. ($16.10°$ to 2 d.p.) | B1✱ |
Carry out an appropriate method to find a value of $\theta$ in the interval $0° < \theta < 180°$ | M1 |
Obtain answer $\theta = 130.2°$ and no other in the given interval | A1 | [3]
[Ignore answers outside the given interval.]
[Treat answers in radians as a misread and deduct A1 from the marks for the angles.]4 (i) Express $3 \sin \theta + 2 \cos \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, stating the exact value of $R$ and giving the value of $\alpha$ correct to 2 decimal places.\\
(ii) Hence solve the equation
$$3 \sin \theta + 2 \cos \theta = 1$$
for $0 ^ { \circ } < \theta < 180 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P3 2015 Q4 [6]}}