6 Given that \(\operatorname { cosec } ^ { 2 } \theta - \cot \theta = 3\), show that \(\cot ^ { 2 } \theta - \cot \theta - 2 = 0\).
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Question 6:
\(\cosec^2\theta = 1 + \cot^2\theta\)
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Guidance
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Guidance
\(1 + \cot^2\theta - \cot\theta = 3\)* E1
Clear use of \(1 + \cot^2\theta = \cosec^2\theta\)
\(\cot^2\theta - \cot\theta - 2 = 0\) \((\cot\theta - 2)(\cot\theta + 1) = 0\) M1
Factorising or formula
A1 Roots 2, \(-1\)
\(\cot\theta = 2\), \(\tan\theta = \frac{1}{2}\), \(\theta = 26.57°\) M1
\(\cot = 1/\tan\) used
\(\cot\theta = -1\), \(\tan\theta = -1\), \(\theta = 135°\) A1
\(\theta = 26.57°\)
A1 \(\theta = 135°\) (penalise extra solutions in the range (\(-1\)))
[6]
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## Question 6:
$\cosec^2\theta = 1 + \cot^2\theta$
| Answer | Mark | Guidance |
|--------|------|----------|
| $1 + \cot^2\theta - \cot\theta = 3$* | E1 | Clear use of $1 + \cot^2\theta = \cosec^2\theta$ |
| $\cot^2\theta - \cot\theta - 2 = 0$ | | |
| $(\cot\theta - 2)(\cot\theta + 1) = 0$ | M1 | Factorising or formula |
| | A1 | Roots 2, $-1$ |
| $\cot\theta = 2$, $\tan\theta = \frac{1}{2}$, $\theta = 26.57°$ | M1 | $\cot = 1/\tan$ used |
| $\cot\theta = -1$, $\tan\theta = -1$, $\theta = 135°$ | A1 | $\theta = 26.57°$ |
| | A1 | $\theta = 135°$ (penalise extra solutions in the range ($-1$)) |
| | **[6]** | |
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6 Given that $\operatorname { cosec } ^ { 2 } \theta - \cot \theta = 3$, show that $\cot ^ { 2 } \theta - \cot \theta - 2 = 0$.\\
Hence solve the equation $\operatorname { cosec } ^ { 2 } \theta - \cot \theta = 3$ for $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$.
\hfill \mbox{\textit{OCR MEI C4 Q6 [6]}}