| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2008 |
| Session | November |
| Marks | 8 |
| 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 R-formula (finding R and α using Pythagoras and tan), followed by solving a transformed trigonometric equation with double angle substitution. The techniques are well-practiced at A-level, though the double angle adds a minor complication in finding all solutions in the given range. |
| 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 or imply at any stage answer \(R = 13\) | B1 | |
| Use trig formula to find \(a\) | M1 | |
| Obtain \(a = 67.38°\) with no errors seen | A1 | [3] |
| [Do not allow radians in this part. If the only trig error is a sign error in \(\sin(x + \alpha)\) give M1A0.] | ||
| (ii) Evaluate \(\sin^{-1}\left(\frac{11}{13}\right)\) correctly to at least 1 d.p. \((57.79577\ldots°)\) | B1√ | |
| Carry out an appropriate method to find a value of \(2\theta\) in \(0° < 2\theta < 360°\) | M1 | |
| Obtain an answer for \(\theta\) in the given range, e.g. \(\theta = 27.4°\) | A1 | |
| Use an appropriate method to find another value of \(2\theta\) in the above range | M1 | |
| Obtain second angle, e.g. \(\theta = 175.2°\) and no others in the given range | A1 | [5] |
| [Ignore answers outside the given range.] | ||
| [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).] |
**(i)** State or imply at any stage answer $R = 13$ | B1 |
Use trig formula to find $a$ | M1 |
Obtain $a = 67.38°$ with no errors seen | A1 | [3] |
[Do not allow radians in this part. If the only trig error is a sign error in $\sin(x + \alpha)$ give M1A0.] |
**(ii)** Evaluate $\sin^{-1}\left(\frac{11}{13}\right)$ correctly to at least 1 d.p. $(57.79577\ldots°)$ | B1√ |
Carry out an appropriate method to find a value of $2\theta$ in $0° < 2\theta < 360°$ | M1 |
Obtain an answer for $\theta$ in the given range, e.g. $\theta = 27.4°$ | A1 |
Use an appropriate method to find another value of $2\theta$ in the above range | M1 |
Obtain second angle, e.g. $\theta = 175.2°$ and no others in the given range | A1 | [5] |
[Ignore answers outside the given range.] |
[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 $5 \sin x + 12 \cos x$ in the form $R \sin ( x + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving the value of $\alpha$ correct to 2 decimal places.\\
(ii) Hence solve the equation
$$5 \sin 2 \theta + 12 \cos 2 \theta = 11$$
giving all solutions in the interval $0 ^ { \circ } < \theta < 180 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P3 2008 Q6 [8]}}