| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then solve equation |
| Difficulty | Standard +0.3 Part (a) is a standard bookwork proof using addition and double angle formulae with clear guidance. Part (b) requires substituting the proven identity and using cos 2θ = 2cos²θ - 1, leading to a factorisable cubic in cos θ. While multi-step, this follows predictable patterns for P3 trigonometric equations with no novel insight required, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct expansion for \(\cos(2\theta + \theta)\) | *M1 | |
| Use correct double angle formulae to express \(\cos 3\theta\) in terms of \(\cos\theta\) and \(\sin\theta\) | DM1 | |
| Show sufficient working to confirm \(\cos 3\theta \equiv 4\cos^3\theta - 3\cos\theta\) | A1 | AG |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the identity and correct double angle formula to obtain an equation in \(\cos\theta\) only. Must come from using all three terms in the given equation | *M1 | e.g. \(4\cos^3\theta - 3\cos\theta + \cos\theta(2\cos^2\theta - 1) = \cos^2\theta\); \(6\cos^3\theta - \cos^2\theta - 4\cos\theta = 0\); or \(6\cos^2\theta - \cos\theta - 4 = 0\) |
| Obtain \(\theta = 90°\) | B1 | Allow if \(\cos\theta\) obtained correctly as a factor of their expression (even if error in quadratic factor). Can follow M0 |
| Solve a 3-term quadratic in \(\cos\theta\) to obtain a value of \(\theta\) | DM1 | |
| Obtain one value e.g. \(25.3°\) | A1 | Accept awrt \(25.3°\) |
| Obtain a second value e.g. \(137.5°\) and no extras in range | A1 | Accept awrt \(137.5°\). Ignore values outside the range. Mark solutions in radians as a misread \((0.442, 1.57, 2.40)\) |
| Total | 5 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct expansion for $\cos(2\theta + \theta)$ | *M1 | |
| Use correct double angle formulae to express $\cos 3\theta$ in terms of $\cos\theta$ and $\sin\theta$ | DM1 | |
| Show sufficient working to confirm $\cos 3\theta \equiv 4\cos^3\theta - 3\cos\theta$ | A1 | AG |
| **Total** | **3** | |
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## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the identity and correct double angle formula to obtain an equation in $\cos\theta$ only. Must come from using all three terms in the given equation | *M1 | e.g. $4\cos^3\theta - 3\cos\theta + \cos\theta(2\cos^2\theta - 1) = \cos^2\theta$; $6\cos^3\theta - \cos^2\theta - 4\cos\theta = 0$; or $6\cos^2\theta - \cos\theta - 4 = 0$ |
| Obtain $\theta = 90°$ | B1 | Allow if $\cos\theta$ obtained correctly as a factor of their expression (even if error in quadratic factor). Can follow M0 |
| Solve a 3-term quadratic in $\cos\theta$ to obtain a value of $\theta$ | DM1 | |
| Obtain one value e.g. $25.3°$ | A1 | Accept awrt $25.3°$ |
| Obtain a second value e.g. $137.5°$ and no extras in range | A1 | Accept awrt $137.5°$. Ignore values outside the range. Mark solutions in radians as a misread $(0.442, 1.57, 2.40)$ |
| **Total** | **5** | |7
\begin{enumerate}[label=(\alph*)]
\item By expressing $3 \theta$ as $2 \theta + \theta$, prove the identity $\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta$.
\item Hence solve the equation
$$\cos 3 \theta + \cos \theta \cos 2 \theta = \cos ^ { 2 } \theta$$
for $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q7 [8]}}