| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2002 |
| 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 harmonic form question requiring routine application of R sin(θ - α) transformation, solving a basic trigonometric equation, and finding a maximum value. All three parts follow textbook procedures with no novel insight required, making it slightly easier than average for A-level. |
| 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 |
|---|---|---|
| Content | Mark | Guidance |
| (i) State or imply at any stage that \(R = 5\) | B1 | |
| Use trig formula to find \(\alpha\) | M1 | |
| Obtain answer \(\alpha = 36.87°\) | A1 | Max 3 marks |
| (ii) EITHER: Carry out, or indicate need for, calculation of \(\sin^{-1}(\frac{3}{5})\) | M1 | |
| Obtain answer \(60.4°\) (or \(60.5°\)) | A1 | |
| Carry out correct method for second root i.e. \(180° - 23.578° + 36.870°\) | M1 | |
| Obtain answer \(193.3°\) and no others in range | A1 | |
| OR: Obtain a three-term quadratic equation in \(\sin\theta\) or \(\cos\theta\) | M1 | |
| Solve a two- or three-term quadratic and calculate an angle | M1 | |
| Obtain answer \(60.4°\) (or \(60.5°\)) | A1 | |
| Obtain answer \(193.3°\) and no others in range | A1 | Max 4 marks |
| (iii) State greatest value is \(1\) | B1✓ | Treat work in radians as a misread, scoring a maximum of 7. The angles are 0.644, 1.06 and 3.37. |
| Content | Mark | Guidance |
|---------|------|----------|
| **(i)** State or imply at any stage that $R = 5$ | B1 | |
| Use trig formula to find $\alpha$ | M1 | |
| Obtain answer $\alpha = 36.87°$ | A1 | Max 3 marks |
| **(ii)** **EITHER:** Carry out, or indicate need for, calculation of $\sin^{-1}(\frac{3}{5})$ | M1 | |
| Obtain answer $60.4°$ (or $60.5°$) | A1 | |
| Carry out correct method for second root i.e. $180° - 23.578° + 36.870°$ | M1 | |
| Obtain answer $193.3°$ and no others in range | A1 | |
| **OR:** Obtain a three-term quadratic equation in $\sin\theta$ or $\cos\theta$ | M1 | |
| Solve a two- or three-term quadratic and calculate an angle | M1 | |
| Obtain answer $60.4°$ (or $60.5°$) | A1 | |
| Obtain answer $193.3°$ and no others in range | A1 | Max 4 marks |
| **(iii)** State greatest value is $1$ | B1✓ | Treat work in radians as a misread, scoring a maximum of 7. The angles are 0.644, 1.06 and 3.37. |Max 1 mark |
---5 (i) Express $4 \sin \theta - 3 \cos \theta$ in the form $R \sin ( \theta - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, stating the value of $\alpha$ correct to 2 decimal places.
Hence\\
(ii) solve the equation
$$4 \sin \theta - 3 \cos \theta = 2$$
giving all values of $\theta$ such that $0 ^ { \circ } < \theta < 360 ^ { \circ }$,\\
(iii) write down the greatest value of $\frac { 1 } { 4 \sin \theta - 3 \cos \theta + 6 }$.
\hfill \mbox{\textit{CAIE P3 2002 Q5 [8]}}