| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | November |
| 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 two-part harmonic form question requiring routine application of the R cos(x + α) formula, followed by solving a straightforward equation. The techniques are well-practiced at A-level, though the double angle in part (b) adds a minor complication. Slightly easier than average due to its predictable structure and standard method. |
| 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 | Marks | Guidance |
| State \(R = \sqrt{17}\) | B1 | Allow if working from an incorrect expansion but not from decimals |
| Use correct trig formulae to find \(\alpha\) (correct expansion and correct expression for trig ratio for \(\alpha\)) | M1 | NB: \(\cos\alpha = 4\) and \(\sin\alpha = 1\) scores M0A0. M0 for incorrect expansion of \(\cos(x-\alpha)\). M1 for correct expression for trig ratio for \(\alpha\) and no errors seen |
| Obtain \(\alpha = 14.04°\) | A1 | 2 d.p. required. Allow M1A1 for correct answer with no working shown. \(\tan^{-1}\!\left(-\frac{1}{4}\right)\) is awarded M0A0. \(180° - \tan^{-1}\!\left(-\frac{1}{4}\right)\) is awarded M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Evaluate \(\cos^{-1}\!\left(\frac{3}{\sqrt{17}}\right)\) to at least 1 d.p. \((43.31\ 38...°)\) | B1 FT | FT *their* \(R\). Accept awrt \(43.3°\) or awrt \(316.7°\). Can be implied by subsequent working |
| Use correct method to find a value of \(x\) in the interval | M1 | Must be working with \(2x\) and *their* \(\alpha\) |
| Obtain answer, e.g. \(14.6°\) | A1 | Accept overspecified answers but they need to be correct. \((14.6388...\) and \(151.3249...)\) |
| Use a correct method to find a second answer in the interval | M1 | Must be working with \(2x\), *their* \(\alpha\) and \(360°\) − *their* \(43.3\) |
| Obtain second answer in the interval, e.g. \(151.3°\), and no other in the interval | A1 | Ignore answers outside the given interval. Treat answers in radians \((0.255...\) and \(2.64...)\) as a misread |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $R = \sqrt{17}$ | B1 | Allow if working from an incorrect expansion but not from decimals |
| Use correct trig formulae to find $\alpha$ (correct expansion and correct expression for trig ratio for $\alpha$) | M1 | NB: $\cos\alpha = 4$ and $\sin\alpha = 1$ scores M0A0. M0 for incorrect expansion of $\cos(x-\alpha)$. M1 for correct expression for trig ratio for $\alpha$ and no errors seen |
| Obtain $\alpha = 14.04°$ | A1 | 2 d.p. required. Allow M1A1 for correct answer with no working shown. $\tan^{-1}\!\left(-\frac{1}{4}\right)$ is awarded M0A0. $180° - \tan^{-1}\!\left(-\frac{1}{4}\right)$ is awarded M1 |
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## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Evaluate $\cos^{-1}\!\left(\frac{3}{\sqrt{17}}\right)$ to at least 1 d.p. $(43.31\ 38...°)$ | B1 FT | FT *their* $R$. Accept awrt $43.3°$ or awrt $316.7°$. Can be implied by subsequent working |
| Use correct method to find a value of $x$ in the interval | M1 | Must be working with $2x$ and *their* $\alpha$ |
| Obtain answer, e.g. $14.6°$ | A1 | Accept overspecified answers but they need to be correct. $(14.6388...$ and $151.3249...)$ |
| Use a correct method to find a second answer in the interval | M1 | Must be working with $2x$, *their* $\alpha$ and $360°$ − *their* $43.3$ |
| Obtain second answer in the interval, e.g. $151.3°$, and no other in the interval | A1 | Ignore answers outside the given interval. Treat answers in radians $(0.255...$ and $2.64...)$ as a misread |
---4
\begin{enumerate}[label=(\alph*)]
\item Express $4 \cos x - \sin x$ in the form $R \cos ( x + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. State the exact value of $R$ and give $\alpha$ correct to 2 decimal places.
\item Hence solve the equation $4 \cos 2 x - \sin 2 x = 3$ for $0 ^ { \circ } < x < 180 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q4 [8]}}