| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Solve reciprocal trig equation |
| Difficulty | Standard +0.8 This question requires converting a cosecant-cotangent equation to harmonic form through algebraic manipulation (multiplying by sin x, recognizing the resulting form), then solving the resulting equation. While the technique is standard for P3, the presence of surds, the non-obvious initial manipulation, and the need to handle the restricted domain carefully make this moderately challenging compared to typical A-level questions. |
| 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 |
|---|---|---|
| 6(i) | Rearrange in the form 3sinx−cosx= 2 | B1 |
| State R = 2 | B1 | |
| Use trig formulae to obtain α | M1 | |
| Obtain α = 30° with no errors seen | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks |
|---|---|
| 6(ii) | 2 |
| Answer | Marks |
|---|---|
| | B1ft |
| Carry out a correct method to find a value of x in the given interval | M1 |
| Obtain answer x = 75° | A1 |
| Answer | Marks |
|---|---|
| [Treat answers in radians as a misread. Ignore answers outside the given interval.] | A1ft |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 6:
--- 6(i) ---
6(i) | Rearrange in the form 3sinx−cosx= 2 | B1
State R = 2 | B1
Use trig formulae to obtain α | M1
Obtain α = 30° with no errors seen | A1
4
Question | Answer | Marks | Guidance
--- 6(ii) ---
6(ii) | 2
Evaluate sin−1
R
| B1ft
Carry out a correct method to find a value of x in the given interval | M1
Obtain answer x = 75° | A1
Obtain a second answer e.g. x = 165° and no others
[Treat answers in radians as a misread. Ignore answers outside the given interval.] | A1ft
4
Question | Answer | Marks | Guidance\begin{enumerate}[label=(\roman*)]
\item Show that the equation $(\sqrt{2})\cos ec x + \cot x = \sqrt{3}$ can be expressed in the form $R\sin(x - \alpha) = \sqrt{2}$, where $R > 0$ and $0° < \alpha < 90°$. [4]
\item Hence solve the equation $(\sqrt{2})\cos ec x + \cot x = \sqrt{3}$, for $0° < x < 180°$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q6 [8]}}