1 Solve the equation \(4 \sin \theta + \tan \theta = 0\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
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Question 1:
Answer Marks
Guidance
Answer Marks
Guidance
\(4\sin\theta + \tan\theta = 0 \Rightarrow 4\sin\theta + \frac{\sin\theta}{\cos\theta} [= 0]\) M1
For use of \(\tan\theta = \frac{\sin\theta}{\cos\theta}\). BOD if \(\theta\) missing.
\(\Rightarrow \sin\theta(4\cos\theta + 1)[=0] \Rightarrow \sin\theta = 0\) or \(\cos\theta = -\frac{1}{4}\) M1
WWW Factorise, not divide by \(\sin\theta\) or \(\tan\theta\). May see \(\tan\theta(4\cos\theta+1)[=0]\) or \(\sin\theta(4+\sec\theta)[=0]\).
\(\theta = 104.5°\) A1
AWRT 1.82 rads A0. Ignore answers outside \((0°, 180°)\). If M1 M0, SC B1 for \(\theta = 104.5°\) max 2/3.
Total: 3
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## Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4\sin\theta + \tan\theta = 0 \Rightarrow 4\sin\theta + \frac{\sin\theta}{\cos\theta} [= 0]$ | M1 | For use of $\tan\theta = \frac{\sin\theta}{\cos\theta}$. BOD if $\theta$ missing. |
| $\Rightarrow \sin\theta(4\cos\theta + 1)[=0] \Rightarrow \sin\theta = 0$ or $\cos\theta = -\frac{1}{4}$ | M1 | WWW Factorise, not divide by $\sin\theta$ or $\tan\theta$. May see $\tan\theta(4\cos\theta+1)[=0]$ or $\sin\theta(4+\sec\theta)[=0]$. |
| $\theta = 104.5°$ | A1 | AWRT 1.82 rads A0. Ignore answers outside $(0°, 180°)$. If M1 M0, **SC B1** for $\theta = 104.5°$ max 2/3. |
| **Total: 3** | | |
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1 Solve the equation $4 \sin \theta + \tan \theta = 0$ for $0 ^ { \circ } < \theta < 180 ^ { \circ }$.\\
\hfill \mbox{\textit{CAIE P1 2023 Q1 [3]}}