| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | Specimen |
| Marks | 9 |
| 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 expansion of cos(x+30°), combining terms into Rcos(x+α) form, then solving a straightforward equation. While it involves multiple steps, the techniques are routine for P3 level with no novel insight required—slightly easier than average due to its predictable structure. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use \(\cos(A+B)\) formula to obtain expression in terms of \(\cos x\) and \(\sin x\) | M1 | |
| Collect terms and reach \(\frac{\cos x}{\sqrt{2}} - \frac{3}{\sqrt{2}}\sin x\), or equivalent | A1 | |
| Obtain \(R = 2.236\) | A1 | |
| Use trig formula to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 71.57°\) with no errors seen | A1 | |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Evaluate \(\cos^{-1}\left(\frac{2}{2.236}\right)\) to at least 1 dp | B1FT | |
| Carry out appropriate method to find a value of \(x\) in the interval \(0° < x < 360°\) | M1 | |
| Obtain answer, e.g. \(x = 315°\) | A1 | |
| Obtain second answer, e.g. \(261.9°\) and no others in the given interval | A1 | |
| Total | 4 |
# Question 7:
## Part 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use $\cos(A+B)$ formula to obtain expression in terms of $\cos x$ and $\sin x$ | M1 | |
| Collect terms and reach $\frac{\cos x}{\sqrt{2}} - \frac{3}{\sqrt{2}}\sin x$, or equivalent | A1 | |
| Obtain $R = 2.236$ | A1 | |
| Use trig formula to find $\alpha$ | M1 | |
| Obtain $\alpha = 71.57°$ with no errors seen | A1 | |
| **Total** | **5** | |
## Part 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Evaluate $\cos^{-1}\left(\frac{2}{2.236}\right)$ to at least 1 dp | B1FT | |
| Carry out appropriate method to find a value of $x$ in the interval $0° < x < 360°$ | M1 | |
| Obtain answer, e.g. $x = 315°$ | A1 | |
| Obtain second answer, e.g. $261.9°$ and no others in the given interval | A1 | |
| **Total** | **4** | |
---7 (a) By first expanding $\cos \left( x + 45 ^ { \circ } \right)$, express $\cos \left( x + 45 ^ { \circ } \right) - \sqrt { 2 } \sin x$ in the form $R \cos ( x + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. Give the value of $R$ correct to 4 significant figures and the value of $\alpha$ correct to 2 decimal places.\\
(b) Hence solve the equation
$$\cos \left( x + 45 ^ { \circ } \right) - \sqrt { 2 } \sin x = 2$$
for $0 ^ { \circ } < x < 360 ^ { \circ }$.\\
\hfill \mbox{\textit{CAIE P3 2020 Q7 [9]}}