| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | March |
| 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 R cos(θ + α) = R cos α cos θ - R sin α sin θ, followed by solving a transformed equation. The method is well-practiced in P3/C3 courses with no novel insight required, though the double angle (2x) in part (ii) adds a minor complication in finding solutions in the given range. Slightly easier than average due to its predictable structure. |
| 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 |
|---|---|---|
| Answer | Mark | Guidance |
| State \(R = 17\) | B1 | |
| Use trig formula to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 61.93°\) with no errors seen | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Evaluate \(\cos^{-1}(4/17)\) to at least 1 d.p. (\(76.39°\) to 2 d.p.) | B1\(\checkmark\) | Follow through mark |
| Use a correct method to find a value of \(x\) in the interval \(0° < x < 180°\) | M1 | |
| Obtain answer, e.g. \(x = 7.2°\) | A1 | |
| Obtain second answer, e.g. \(x = 110.8°\) and no others | A1 | Ignore answers outside given interval; treat radians as misread |
## Question 4(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $R = 17$ | B1 | |
| Use trig formula to find $\alpha$ | M1 | |
| Obtain $\alpha = 61.93°$ with no errors seen | A1 | |
**Total: 3**
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## Question 4(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Evaluate $\cos^{-1}(4/17)$ to at least 1 d.p. ($76.39°$ to 2 d.p.) | B1$\checkmark$ | Follow through mark |
| Use a correct method to find a value of $x$ in the interval $0° < x < 180°$ | M1 | |
| Obtain answer, e.g. $x = 7.2°$ | A1 | |
| Obtain second answer, e.g. $x = 110.8°$ and no others | A1 | Ignore answers outside given interval; treat radians as misread |
**Total: 4**
---4 (i) Express $8 \cos \theta - 15 \sin \theta$ in the form $R \cos ( \theta + \alpha )$, 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
$$8 \cos 2 x - 15 \sin 2 x = 4$$
for $0 ^ { \circ } < x < 180 ^ { \circ }$.\\
\hfill \mbox{\textit{CAIE P3 2017 Q4 [7]}}