9 Roberto is solving this mathematics problem: The curve \(C _ { 1 }\) has polar equation $$r ^ { 2 } = 9 \sin 2 \theta$$ for all possible values of \(\theta\)
Find the area enclosed by \(C _ { 1 }\) Roberto's solution is as follows: $$\begin{aligned} A & = \frac { 1 } { 2 } \int _ { - \pi } ^ { \pi } 9 \sin 2 \theta \mathrm {~d} \theta
& = \left[ - \frac { 9 } { 4 } \cos 2 \theta \right] _ { - \pi } ^ { \pi }
& = 0 \end{aligned}$$ 9

1. \(\quad\) Sketch the curve \(C _ { 1 }\) 9
2. Explain what Roberto has done wrong.
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3. \(\quad\) Find the area enclosed by \(C _ { 1 }\)
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4. \(\quad P\) and \(Q\) are distinct points on \(C _ { 1 }\) for which \(r\) is a maximum. \(P\) is above the initial line. Find the polar coordinates of \(P\) and \(Q\)
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5. The matrix \(\mathbf { M } = \left[ \begin{array} { l l } 1 & 2
   0 & 1 \end{array} \right]\) represents the transformation T T maps \(C _ { 1 }\) onto a curve \(C _ { 2 }\)
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6. 1. T maps \(P\) onto the point \(P ^ { \prime }\)
      Find the polar coordinates of \(P ^ { \prime }\)
      [0pt] [4 marks]
      9
7. (ii) Find the area enclosed by \(C _ { 2 }\) Fully justify your answer.