| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | November |
| Marks | 8 |
| 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 = √10 and α using tan⁻¹(3)) followed by a straightforward equation solve. While it requires multiple steps and careful angle work, it follows a well-practiced textbook procedure with no novel insight needed, 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 |
|---|---|
| Use trig formulae to find \(a\) | M1 |
| Obtain \(a = 71.57°\) with no errors seen | A1 |
| Answer | Marks |
|---|---|
| [3] |
| Answer | Marks |
|---|---|
| Allow 50.7° here. Carry out an appropriate method to find a value of \(2\theta\) in \(0° < 2\theta < 180°\) | M1 |
| Obtain an answer for \(\theta\) in the given range, e.g. \(\theta = 61.2°\) | A1 |
| Use an appropriate method and find another value of \(2\theta\) in the above range | M1 |
| Obtain second angle, e.g. \(\theta = 10.4°\), and no others in the given range | A1 |
| [Ignore answers outside the given range.] | [5] |
**(i) State or imply $R = \sqrt{10}$**
Use trig formulae to find $a$ | M1 |
Obtain $a = 71.57°$ with no errors seen | A1 |
[Do not allow radians in this part. If the only trig error is a sign error in $\cos(x - a)$ give M1A0]
| [3]
**(ii) Evaluate $\cos^{-1}(2/\sqrt{10})$ correctly to at least 1 d.p. (50.7684...°)**
Allow 50.7° here. Carry out an appropriate method to find a value of $2\theta$ in $0° < 2\theta < 180°$ | M1 |
Obtain an answer for $\theta$ in the given range, e.g. $\theta = 61.2°$ | A1 |
Use an appropriate method and find another value of $2\theta$ in the above range | M1 |
Obtain second angle, e.g. $\theta = 10.4°$, and no others in the given range | A1 |
[Ignore answers outside the given range.] | [5]
[Treat answers in radians as a misread and deduct A1 from the answers for the angles.]
[SR: The use of correct trig formulae to obtain a 3-term quadratic in $\tan \theta$, $\sin 2\theta$, $\cos 2\theta$ or $\tan 2\theta$ earns M1; then A1 for a correct quadratic, M1 for obtaining a value of $\theta$ in the given range, and A1 + A1 for the two correct answers (candidates who square must reject the spurious roots to get the final A1).]6 (i) Express $\cos x + 3 \sin x$ in the form $R \cos ( x - \alpha )$, 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 $\cos 2 \theta + 3 \sin 2 \theta = 2$, for $0 ^ { \circ } < \theta < 90 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P3 2011 Q6 [8]}}