Polar curve with substitution integral

7 The curve \(C\) has polar equation \(r ^ { 2 } = ( \pi - \theta ) \tan ^ { - 1 } ( \pi - \theta )\), for \(0 \leqslant \theta \leqslant \pi\).

1. Sketch \(C\) and state the polar coordinates of the point of \(C\) furthest from the pole.
2. Using the substitution \(u = \pi - \theta\), or otherwise, find the area of the region enclosed by \(C\) and the initial line.
3. Show that, at the point of \(C\) furthest from the initial line, $$2 ( \pi - \theta ) \tan ^ { - 1 } ( \pi - \theta ) \cot \theta - \frac { \pi - \theta } { 1 + ( \pi - \theta ) ^ { 2 } } - \tan ^ { - 1 } ( \pi - \theta ) = 0$$ and verify that this equation has a root for \(\theta\) between 1.2 and 1.3.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

5 The curve \(C\) has polar equation \(r = \ln ( 1 + \pi - \theta )\), for \(0 \leqslant \theta \leqslant \pi\).

1. Sketch \(C\) and state the polar coordinates of the point of \(C\) furthest from the pole.
2. Using the substitution \(u = 1 + \pi - \theta\), or otherwise, show that the area of the region enclosed by \(C\) and the initial line is $$\frac { 1 } { 2 } ( 1 + \pi ) \ln ( 1 + \pi ) ( \ln ( 1 + \pi ) - 2 ) + \pi$$
3. Show that, at the point of \(C\) furthest from the initial line, $$( 1 + \pi - \theta ) \ln ( 1 + \pi - \theta ) - \tan \theta = 0$$ and verify that this equation has a root between 1.2 and 1.3.

8 The diagram shows a sketch of a curve.
\includegraphics[max width=\textwidth, alt={}, center]{f05737eb-adb1-4228-aebf-6b5c7f26a434-5_464_574_402_726} The polar equation of the curve is $$r = \sin 2 \theta \sqrt { \left( 2 + \frac { 1 } { 2 } \cos \theta \right) } , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ The point \(P\) is the point of the curve at which \(\theta = \frac { \pi } { 3 }\). The perpendicular from \(P\) to the initial line meets the initial line at the point \(N\).

1. 1. Find the exact value of \(r\) when \(\theta = \frac { \pi } { 3 }\).
   2. Show that the polar equation of the line \(P N\) is \(r = \frac { 3 \sqrt { 3 } } { 8 } \sec \theta\).
   3. Find the area of triangle \(O N P\) in the form \(\frac { k \sqrt { 3 } } { 128 }\), where \(k\) is an integer.
2. 1. Using the substitution \(u = \sin \theta\), or otherwise, find \(\int \sin ^ { n } \theta \cos \theta \mathrm {~d} \theta\), where \(n \geqslant 2\).
   2. Find the area of the shaded region bounded by the line \(O P\) and the arc \(O P\) of the curve. Give your answer in the form \(a \pi + b \sqrt { 3 } + c\), where \(a , b\) and \(c\) are constants.
      (8 marks)

3 The diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line.
\includegraphics[max width=\textwidth, alt={}, center]{c4bce668-61f1-4be0-97ee-c635df7e1fc6-2_380_735_1827_648} The polar equation of \(C\) is $$r = 2 \sqrt { 1 + \tan \theta } , \quad - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 }$$ Show that the area of the shaded region, bounded by the curve \(C\) and the initial line, is \(\frac { \pi } { 2 } - \ln 2\).
(4 marks)

8 The diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line. The curve \(C\) intersects the initial line at the point \(P\).
\includegraphics[max width=\textwidth, alt={}, center]{0eb3e96e-528c-4a99-b164-31cc865f0d68-20_432_949_402_525} The polar equation of \(C\) is \(r = \left( 1 - \tan ^ { 2 } \theta \right) \sec \theta , - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 }\).

1. Show that the area of the region bounded by the curve \(C\) is \(\frac { 8 } { 15 }\).
2. The curve whose polar equation is $$r = \frac { 1 } { 2 } \sec ^ { 3 } \theta , \quad - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 }$$ intersects \(C\) at the points \(A\) and \(B\).
   1. Find the polar coordinates of \(A\) and \(B\).
   2. Given that angle \(O A P =\) angle \(O B P = \alpha\), show that \(\tan \alpha = k \sqrt { 3 }\), where \(k\) is an integer.
   3. Using your value of \(k\) from part (b)(ii), state whether the point \(A\) lies inside or lies outside the circle whose diameter is \(O P\). Give a reason for your answer.
      [0pt] [1 mark]
      \includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-21_2484_1707_221_153}
      \includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-23_2484_1707_221_153}