| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| 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 harmonic form question requiring routine application of the R cos(x + α) formula with R = √(a² + b²) and tan α = b/a, followed by solving a straightforward equation. The only minor complication is the √2 and √5 coefficients and the double angle in part (b), but the method is entirely procedural with no problem-solving insight required. |
| 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 \(R = \sqrt{7}\) | B1 | |
| Use trig formulae to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 57.688°\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Evaluate \(\cos^{-1}\!\left(\frac{1}{\sqrt{7}}\right)\) to at least 3 d.p. (\(67.792°\)) | B1 FT | FT is on their \(R\) |
| Use correct method to find a value of \(\theta\) in the interval | M1 | |
| Obtain answer, e.g. \(5.1°\) | A1 | |
| Obtain second answer, e.g. \(117.3°\), only | A1 |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $R = \sqrt{7}$ | B1 | |
| Use trig formulae to find $\alpha$ | M1 | |
| Obtain $\alpha = 57.688°$ | A1 | |
---
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Evaluate $\cos^{-1}\!\left(\frac{1}{\sqrt{7}}\right)$ to at least 3 d.p. ($67.792°$) | B1 FT | FT is on their $R$ |
| Use correct method to find a value of $\theta$ in the interval | M1 | |
| Obtain answer, e.g. $5.1°$ | A1 | |
| Obtain second answer, e.g. $117.3°$, only | A1 | |
---5
\begin{enumerate}[label=(\alph*)]
\item Express $\sqrt { 2 } \cos x - \sqrt { 5 } \sin x$ in the form $R \cos ( x + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. Give the exact value of $R$ and the value of $\alpha$ correct to 3 decimal places.
\item Hence solve the equation $\sqrt { 2 } \cos 2 \theta - \sqrt { 5 } \sin 2 \theta = 1$, for $0 ^ { \circ } < \theta < 180 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q5 [7]}}