3 The curve $C$ has parametric equations $x = 2 t ^ { 3 } - 6 t , y = 6 t ^ { 2 }$.\\
(i) Find the length of the arc of $C$ for which $0 \leqslant t \leqslant 1$.\\
(ii) Find the area of the surface generated when the arc of $C$ for which $0 \leqslant t \leqslant 1$ is rotated through $2 \pi$ radians about the $x$-axis.\\
(iii) Show that the equation of the normal to $C$ at the point with parameter $t$ is

$$y = \frac { 1 } { 2 } \left( \frac { 1 } { t } - t \right) x + 2 t ^ { 2 } + t ^ { 4 } + 3$$

(iv) Find the cartesian equation of the envelope of the normals to $C$.\\
(v) The point $\mathrm { P } ( 64 , a )$ is the centre of curvature corresponding to a point on $C$. Find $a$.

\hfill \mbox{\textit{OCR MEI FP3  Q3}}