Show parametric identity

1. A curve has parametric equations
   where \(a\) is a positive constant.

$$\begin{aligned} & x = a ( \theta - \sin \theta ) \\ & y = a ( 1 - \cos \theta ) \end{aligned}$$

1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = k a ^ { 2 } \sin ^ { 2 } \frac { \theta } { 2 }$$ where \(k\) is a constant to be determined. The part of the curve from \(\theta = 0\) to \(\theta = 2 \pi\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Determine the area of the surface generated, giving your answer in terms of \(\pi\) and \(a\).
   [0pt] [Solutions relying on calculator technology are not acceptable.]

6. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(C\) with parametric equations $$x = 2 \cos \theta - \cos 2 \theta , y = 2 \sin \theta - \sin 2 \theta , \quad 0 \leqslant \theta \leqslant \pi$$

1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = 8 ( 1 - \cos \theta )$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Find the area of the surface generated, giving your answer in the form \(k \pi\), where \(k\) is a rational number.

A curve \(C\) has parametric equations $$x = 1 - 3 t ^ { 2 } , \quad y = t \left( 1 - 3 t ^ { 2 } \right) , \quad \text { for } 0 \leqslant t \leqslant \frac { 1 } { \sqrt { 3 } }$$ Show that \(\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + 9 t ^ { 2 } \right) ^ { 2 }\). Hence find

1. the arc length of \(C\),
2. the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Use the fact that \(t = \frac { y } { x }\) to find a cartesian equation of \(C\). Hence show that the polar equation of \(C\) is \(r = \sec \theta \left( 1 - 3 \tan ^ { 2 } \theta \right)\), and state the domain of \(\theta\). Find the area of the region enclosed between \(C\) and the initial line. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }