6 A curve is defined parametrically by
$$x = t ^ { 3 } + 5 , \quad y = 6 t ^ { 2 } - 1$$
The arc length between the points where \(t = 0\) and \(t = 3\) on the curve is \(s\).
- Show that \(s = \int _ { 0 } ^ { 3 } 3 t \sqrt { t ^ { 2 } + A } \mathrm {~d} t\), stating the value of the constant \(A\).
- Hence show that \(s = 61\).
\(7 \quad\) The polynomial \(\mathrm { p } ( n )\) is given by \(\mathrm { p } ( n ) = ( n - 1 ) ^ { 3 } + n ^ { 3 } + ( n + 1 ) ^ { 3 }\). - Show that \(\mathrm { p } ( k + 1 ) - \mathrm { p } ( k )\), where \(k\) is a positive integer, is a multiple of 9 .
- Prove by induction that \(\mathrm { p } ( n )\) is a multiple of 9 for all integers \(n \geqslant 1\).
- Using the result from part (a)(ii), show that \(n \left( n ^ { 2 } + 2 \right)\) is a multiple of 3 for any positive integer \(n\).