| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2005 |
| 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 cos(θ + α) formula (R = √(144+25) = 13, α from tan α = 5/12), followed by solving a straightforward trigonometric equation. While it involves multiple steps, both parts follow well-practiced 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 | Guidance |
|---|---|---|
| (i) State answer \(R = 13\) | B1 | |
| Use trig formula to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 22.62°\) | A1 | 3 |
| (ii) Carry out evaluation of \(\cos^{-1}(\frac{39}{715.4}°)\) (≈ 39.715...°) | M1 | |
| Obtain answer 17.1° | A1 | |
| Carry out correct method for second answer | M1 | |
| Obtain answer 297.7° and no others in the range [Ignore answers outside the given range] | A1/* | 4 |
(i) State answer $R = 13$ | B1 |
Use trig formula to find $\alpha$ | M1 |
Obtain $\alpha = 22.62°$ | A1 | 3
(ii) Carry out evaluation of $\cos^{-1}(\frac{39}{715.4}°)$ (≈ 39.715...°) | M1 |
Obtain answer 17.1° | A1 |
Carry out correct method for second answer | M1 |
Obtain answer 297.7° and no others in the range [Ignore answers outside the given range] | A1/* | 43 (i) Express $12 \cos \theta - 5 \sin \theta$ in the form $R \cos ( \theta + \alpha )$, 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
$$12 \cos \theta - 5 \sin \theta = 10$$
giving all solutions in the interval $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P2 2005 Q3 [7]}}