| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Moderate -0.3 This is a standard harmonic form question with routine application of R-formula (finding R and α using Pythagorean identity and arctan) followed by solving a double-angle equation. While it requires multiple steps and careful angle work, it follows a well-practiced algorithm 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 |
|---|---|
| 5 | Where a candidate has misread a number in the question and used that value consistently throughout, provided that number does not alter the difficulty or |
| Answer | Marks | Guidance |
|---|---|---|
| 5(a) | State R= 11 | B1 |
| Use trig formulae to find α | M1 | |
| Obtain α=37.09° | A1 |
| Answer | Marks |
|---|---|
| 5(b) | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| 11 | B1 FT | The FT is on R. |
| Use correct method to find a value of θ in the interval | M1 | |
| Obtain answer, e.g. 62.7° | A1 | |
| Use a correct method to obtain a second answer | M1 | |
| Obtain second answer, e.g. 170.2°, and no other in the interval | A1 | Ignore answers outside the given interval. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 5:
5 | Where a candidate has misread a number in the question and used that value consistently throughout, provided that number does not alter the difficulty or
the method required, award all marks earned and deduct just 1 mark for the misread.
--- 5(a) ---
5(a) | State R= 11 | B1
Use trig formulae to find α | M1
Obtain α=37.09° | A1
3
--- 5(b) ---
5(b) | 1
−1
Evaluate sin to at least 2 dp (17.5484°)
11 | B1 FT | The FT is on R.
Use correct method to find a value of θ in the interval | M1
Obtain answer, e.g. 62.7° | A1
Use a correct method to obtain a second answer | M1
Obtain second answer, e.g. 170.2°, and no other in the interval | A1 | Ignore answers outside the given interval.
5
Question | Answer | Marks | Guidance\begin{enumerate}[label=(\alph*)]
\item Express $\sqrt{7} \sin x + 2 \cos x$ in the form $R \sin(x + \alpha)$, where $R > 0$ and $0° < \alpha < 90°$. State the exact value of $R$ and give $\alpha$ correct to 2 decimal places. [3]
\item Hence solve the equation $\sqrt{7} \sin 2\theta + 2 \cos 2\theta = 1$, for $0° < \theta < 180°$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q5 [8]}}