| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | November |
| Marks | 10 |
| 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 cos(θ + α) = R cos θ cos α - R sin θ sin α, solving a basic trigonometric equation, and finding max/min values using the range of the harmonic form. All three parts follow textbook 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) etc |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(R = 1.3\) or \(\dfrac{10}{3}\) | B1 | Not \(\sqrt{1.69}\) |
| Use appropriate trigonometry to find \(\alpha\) | M1 | AWRT \(\pm1.18\) rads, AWRT \(\pm0.391\) rads, AWRT \(\pm67.4°\), AWRT \(\pm22.6°\) |
| Obtain 67.38 with no errors seen | A1 | AWRT |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out correct method to find one value of \(\theta\) between 0 and 360 | M1 | |
| Obtain 240.6 (or 344.6) | A1 | |
| Carry out correct method to find second value of \(\theta\) between 0 and 360 | M1 | Must be using either degrees throughout or radians throughout for M marks |
| Obtain 344.6 (or 240.6) | A1 | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Recognise expression as \([3 - 2R\cos(\theta + \alpha)]^2\) | M1 | |
| Obtain \([3 - 2\times(-1.3)]^2\) and hence 31.36 or 31.4 | A1 | |
| Obtain \([3 - 2\times1.3]^2\) and hence 0.16 | A1 | |
| Total: 3 |
## Question 8:
### Part 8(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $R = 1.3$ or $\dfrac{10}{3}$ | B1 | Not $\sqrt{1.69}$ |
| Use appropriate trigonometry to find $\alpha$ | M1 | AWRT $\pm1.18$ rads, AWRT $\pm0.391$ rads, AWRT $\pm67.4°$, AWRT $\pm22.6°$ |
| Obtain 67.38 with no errors seen | A1 | AWRT |
| **Total: 3** | | |
### Part 8(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out correct method to find one value of $\theta$ between 0 and 360 | M1 | |
| Obtain 240.6 (or 344.6) | A1 | |
| Carry out correct method to find second value of $\theta$ between 0 and 360 | M1 | Must be using either degrees throughout or radians throughout for M marks |
| Obtain 344.6 (or 240.6) | A1 | |
| **Total: 4** | | |
### Part 8(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Recognise expression as $[3 - 2R\cos(\theta + \alpha)]^2$ | M1 | |
| Obtain $[3 - 2\times(-1.3)]^2$ and hence 31.36 or 31.4 | A1 | |
| Obtain $[3 - 2\times1.3]^2$ and hence 0.16 | A1 | |
| **Total: 3** | | |8 (i) Express $0.5 \cos \theta - 1.2 \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving the value of $\alpha$ correct to 2 decimal places.\\
(ii) Hence solve the equation $0.5 \cos \theta - 1.2 \sin \theta = 0.8$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.\\
(iii) Determine the greatest and least possible values of $( 3 - \cos \theta + 2.4 \sin \theta ) ^ { 2 }$ as $\theta$ varies.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE P2 2019 Q8 [10]}}