3 Solve the equation
$$\cos \theta + 3 \cos 2 \theta = 2$$
giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
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Question 3:
Answer Marks
Guidance
Answer/Working Mark
Guidance
Use correct \(\cos 2A\) formula, or equivalent pair of correct formulas, to obtain an equation in \(\cos\theta\) M1
Obtain 3-term quadratic \(6\cos^2\theta + \cos\theta - 5 = 0\), or equivalent A1
Attempt to solve quadratic and reach \(\theta = \cos^{-1}(a)\) M1
Obtain answer \(33.6°\) (or \(33.5°\)) or \(0.586\) (or \(0.585\)) radians A1
Obtain answer \(180°\) or \(\pi\) (or \(3.14\)) radians and no others in range A1
Notes: The answer \(\theta = 180°\) found by inspection can earn B1. Ignore answers outside the given range.Total: [5]
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## Question 3:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use correct $\cos 2A$ formula, or equivalent pair of correct formulas, to obtain an equation in $\cos\theta$ | M1 | |
| Obtain 3-term quadratic $6\cos^2\theta + \cos\theta - 5 = 0$, or equivalent | A1 | |
| Attempt to solve quadratic and reach $\theta = \cos^{-1}(a)$ | M1 | |
| Obtain answer $33.6°$ (or $33.5°$) or $0.586$ (or $0.585$) radians | A1 | |
| Obtain answer $180°$ or $\pi$ (or $3.14$) radians and no others in range | A1 | |
**Notes:** The answer $\theta = 180°$ found by inspection can earn B1. Ignore answers outside the given range.
**Total: [5]**
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3 Solve the equation
$$\cos \theta + 3 \cos 2 \theta = 2$$
giving all solutions in the interval $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P3 2003 Q3 [5]}}