| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2009 |
| Session | November |
| Marks | 7 |
| 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 cos(x - α) formula (finding R = 5 using Pythagoras, α using tan⁻¹(4/3)) followed by solving a straightforward equation. While it involves multiple steps, both parts follow textbook procedures 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 |
|---|---|
| State answer \(R = 5\) | B1 |
| Use trig formula to find \(a\) | M1 |
| Obtain \(a = 53.13°\) | A1 |
| Answer | Marks |
|---|---|
| Evaluate \(\cos^{-1}(4.5/5) \approx 25.84°\) | M1 |
| Obtain answer \(79.0°\) | A1 |
| Carry out correct method for second answer | M1 |
| Obtain answer \(27.3°\) and no others in the given range | A1 |
**(i)**
| State answer $R = 5$ | B1 |
| Use trig formula to find $a$ | M1 |
| Obtain $a = 53.13°$ | A1 |
**Total: [3]**
**(ii)**
| Evaluate $\cos^{-1}(4.5/5) \approx 25.84°$ | M1 |
| Obtain answer $79.0°$ | A1 |
| Carry out correct method for second answer | M1 |
| Obtain answer $27.3°$ and no others in the given range | A1 |
[Treat the giving of answers in radians as a misread. Ignore answers outside the given range.]
**Total: [4]**
---6 (i) Express $3 \cos x + 4 \sin x$ in the form $R \cos ( x - \alpha )$, 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
$$3 \cos x + 4 \sin x = 4.5$$
giving all solutions in the interval $0 ^ { \circ } < x < 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P2 2009 Q6 [7]}}