6 Solve the equation \(\tan \left( \theta + 45 ^ { \circ } \right) = 1 - 2 \tan \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }\).
Section B (36 marks)
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Question 6:
Answer Marks
Guidance
Answer/Working Marks
Guidance
\(\tan(\theta+45) = \frac{\tan\theta + \tan 45}{1 - \tan\theta\tan 45} = \frac{\tan\theta + 1}{1 - \tan\theta}\) M1
oe using sin/cos
\(\frac{\tan\theta+1}{1-\tan\theta} = 1 - 2\tan\theta\) A1
\(1 + \tan\theta = (1-2\tan\theta)(1-\tan\theta) = 1 - 3\tan\theta + 2\tan^2\theta\) M1, A1
multiplying up and expanding; any correct one line equation
\(0 = 2\tan^2\theta - 4\tan\theta = 2\tan\theta(\tan\theta - 2)\) M1
solving quadratic for \(\tan\theta\) oe
\(\tan\theta = 0\) or \(2\) \(\theta = 0\) or \(63.43°\) A1, A1 [7]
www; \(-1\) extra solutions in the range
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# Question 6:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\tan(\theta+45) = \frac{\tan\theta + \tan 45}{1 - \tan\theta\tan 45} = \frac{\tan\theta + 1}{1 - \tan\theta}$ | M1 | oe using sin/cos |
| $\frac{\tan\theta+1}{1-\tan\theta} = 1 - 2\tan\theta$ | A1 | |
| $1 + \tan\theta = (1-2\tan\theta)(1-\tan\theta) = 1 - 3\tan\theta + 2\tan^2\theta$ | M1, A1 | multiplying up and expanding; any correct one line equation |
| $0 = 2\tan^2\theta - 4\tan\theta = 2\tan\theta(\tan\theta - 2)$ | M1 | solving quadratic for $\tan\theta$ oe |
| $\tan\theta = 0$ or $2$ | | |
| $\theta = 0$ or $63.43°$ | A1, A1 [7] | www; $-1$ extra solutions in the range |
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6 Solve the equation $\tan \left( \theta + 45 ^ { \circ } \right) = 1 - 2 \tan \theta$, for $0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }$.
Section B (36 marks)\\
\hfill \mbox{\textit{OCR MEI C4 2010 Q6 [7]}}