8 Solve the equation
$$\sec ^ { 2 } \theta = 4 , \quad 0 \leqslant \theta \leqslant \pi ,$$
giving your answers in terms of \(\pi\).
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Question 8:
\(\sec^2\theta = 4\)
Answer Marks
Guidance
Answer Mark
Guidance
\(\Rightarrow \frac{1}{\cos^2\theta} = 4\) M1
\(\sec\theta = 1/\cos\theta\) used
\(\Rightarrow \cos^2\theta = \frac{1}{4}\) \(\Rightarrow \cos\theta = \frac{1}{2}\) or \(-\frac{1}{2}\) M1
\(\pm\frac{1}{2}\)
\(\Rightarrow \theta = \pi/3, 2\pi/3\) A1 A1
Allow unsupported answers
OR \(\sec^2\theta = 1 + \tan^2\theta\)M1
\(\Rightarrow \tan^2\theta = 3\) \(\Rightarrow \tan\theta = \sqrt{3}\) or \(-\sqrt{3}\) M1
\(\pm\sqrt{3}\)
\(\Rightarrow \theta = \pi/3, 2\pi/3\) A1 A1
Allow unsupported answers
[4]
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## Question 8:
$\sec^2\theta = 4$
| Answer | Mark | Guidance |
|--------|------|----------|
| $\Rightarrow \frac{1}{\cos^2\theta} = 4$ | M1 | $\sec\theta = 1/\cos\theta$ used |
| $\Rightarrow \cos^2\theta = \frac{1}{4}$ | | |
| $\Rightarrow \cos\theta = \frac{1}{2}$ or $-\frac{1}{2}$ | M1 | $\pm\frac{1}{2}$ |
| $\Rightarrow \theta = \pi/3, 2\pi/3$ | A1 A1 | Allow unsupported answers |
| **OR** $\sec^2\theta = 1 + \tan^2\theta$ | M1 | |
| $\Rightarrow \tan^2\theta = 3$ | | |
| $\Rightarrow \tan\theta = \sqrt{3}$ or $-\sqrt{3}$ | M1 | $\pm\sqrt{3}$ |
| $\Rightarrow \theta = \pi/3, 2\pi/3$ | A1 A1 | Allow unsupported answers |
| | **[4]** | |
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8 Solve the equation
$$\sec ^ { 2 } \theta = 4 , \quad 0 \leqslant \theta \leqslant \pi ,$$
giving your answers in terms of $\pi$.
\hfill \mbox{\textit{OCR MEI C4 Q8 [4]}}