| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2007 |
| Session | November |
| Marks | 7 |
| 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 the R sin(θ - α) formula (finding R = √(64+225) = 17 and α = arctan(15/8)), then solving a straightforward trigonometric equation. While it involves multiple steps, the techniques are well-practiced and follow a predictable pattern with no novel insight required, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| State answer \(R = 17\), allow \(\sqrt{289}\) | B1 | |
| Use trig formula to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 61.93°\), (1.08 radians) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Carry out evaluation of \(\sin^{-1}(14/17) \approx 55.44°\), or equivalent | M1 | |
| Obtain answer \(117.4°\), (2.06 radians) | A1 | |
| Carry out correct method for second answer | M1 | |
| Obtain answer \(186.5°\) and no others in the range (3.255 radians) | A1√ | [Ignore answers outside the given range.] [4] |
**(i)**
State answer $R = 17$, allow $\sqrt{289}$ | B1 |
Use trig formula to find $\alpha$ | M1 |
Obtain $\alpha = 61.93°$, (1.08 radians) | A1 | [3]
**(ii)**
Carry out evaluation of $\sin^{-1}(14/17) \approx 55.44°$, or equivalent | M1 |
Obtain answer $117.4°$, (2.06 radians) | A1 |
Carry out correct method for second answer | M1 |
Obtain answer $186.5°$ and no others in the range (3.255 radians) | A1√ | [Ignore answers outside the given range.] [4]6 (i) Express $8 \sin \theta - 15 \cos \theta$ in the form $R \sin ( \theta - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving the exact value of $R$ and the value of $\alpha$ correct to 2 decimal places.\\
(ii) Hence solve the equation
$$8 \sin \theta - 15 \cos \theta = 14$$
giving all solutions in the interval $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.\\
\hfill \mbox{\textit{CAIE P2 2007 Q6 [7]}}