1 In this question you must show detailed reasoning.
A surface \(S\) is defined by \(z = f ( x , y )\) where \(f ( x , y ) = x ^ { 3 } + x ^ { 2 } y - 2 y ^ { 2 }\).

1. On the coordinate axes in the Printed Answer Booklet, sketch the section \(z = f ( 2 , y )\) giving the coordinates of any turning points and any points of intersection with the axes.
2. Find the stationary points on \(S\).
   \(2 G\) is a group of order 8.
3. Explain why there is no subgroup of \(G\) of order 6 . You are now given that \(G\) is a cyclic group with the following features:
   • \(e\) is the identity element of \(G\),
   • \(g\) is a generator of \(G\),
   • \(H\) is the subgroup of \(G\) of order 4.
   • Write down the possible generators of \(H\).
     \(M\) is the group ( \(\{ 0,1,2,3,4,5,6,7 \} , + _ { 8 }\) ) where \(+ _ { 8 }\) denotes the binary operation of addition modulo 8. You are given that \(M\) is isomorphic to \(G\).
   • Specify all possible isomorphisms between \(M\) and \(G\).