4 A curve \(C\) has polar equation \(r ^ { 2 } = 8 \operatorname { cosec } 2 \theta\) for \(0 < \theta < \frac { 1 } { 2 } \pi\). Find a cartesian equation of \(C\).
Sketch \(C\).
Determine the exact area of the sector bounded by the arc of \(C\) between \(\theta = \frac { 1 } { 6 } \pi\) and \(\theta = \frac { 1 } { 3 } \pi\), the half-line \(\theta = \frac { 1 } { 6 } \pi\) and the half-line \(\theta = \frac { 1 } { 3 } \pi\).
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Question 4:
Answer Marks
Guidance
Working/Answer Marks
Guidance
Using \(x = r\cos\theta\) and \(y = r\sin\theta\) B1
\(r^2 = 8\csc 2\theta \Rightarrow r^2 = \frac{4}{\sin\theta\cos\theta}\) M1
\(\Rightarrow r\cos\theta \cdot r\sin\theta = 4 \Rightarrow xy = 4\) (in simple form) A1 [3]
Sketch: Curve in 1st quadrant with correct concavity, asymptotic to both axes B1B1 [2]
\(\frac{1}{2}\int_{\frac{1}{6}\pi}^{\frac{1}{3}\pi} 8\csc 2\theta\, d\theta = \left[2\ln \tan\theta
\right]_{\frac{1}{6}\pi}^{\frac{1}{3}\pi}\)
\(= 2\left\{\ln \sqrt{3}
- \ln\left
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## Question 4:
| Working/Answer | Marks | Guidance |
|---|---|---|
| Using $x = r\cos\theta$ and $y = r\sin\theta$ | B1 | |
| $r^2 = 8\csc 2\theta \Rightarrow r^2 = \frac{4}{\sin\theta\cos\theta}$ | M1 | |
| $\Rightarrow r\cos\theta \cdot r\sin\theta = 4 \Rightarrow xy = 4$ (in simple form) | A1 [3] | |
| Sketch: Curve in 1st quadrant with correct concavity, asymptotic to **both** axes | B1B1 [2] | |
| $\frac{1}{2}\int_{\frac{1}{6}\pi}^{\frac{1}{3}\pi} 8\csc 2\theta\, d\theta = \left[2\ln|\tan\theta|\right]_{\frac{1}{6}\pi}^{\frac{1}{3}\pi}$ | M1A1 | |
| $= 2\left\{\ln|\sqrt{3}| - \ln\left|\frac{1}{\sqrt{3}}\right|\right\} = 2\ln 3$ or $\ln 9$ | A1 [3] | |
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4 A curve $C$ has polar equation $r ^ { 2 } = 8 \operatorname { cosec } 2 \theta$ for $0 < \theta < \frac { 1 } { 2 } \pi$. Find a cartesian equation of $C$.
Sketch $C$.
Determine the exact area of the sector bounded by the arc of $C$ between $\theta = \frac { 1 } { 6 } \pi$ and $\theta = \frac { 1 } { 3 } \pi$, the half-line $\theta = \frac { 1 } { 6 } \pi$ and the half-line $\theta = \frac { 1 } { 3 } \pi$.\\[0pt]
[It is given that $\int \operatorname { cosec } x \mathrm {~d} x = \ln \left| \tan \frac { 1 } { 2 } x \right| + c$.]
\hfill \mbox{\textit{CAIE FP1 2016 Q4 [8]}}