4 The points \(A , B , C\) and \(P\) have coordinates ( \(a , 0,0\) ), ( \(0 , b , 0\) ), ( \(0,0 , c\) ) and ( \(a , b , c\) ) respectively, where \(a , b\) and \(c\) are positive constants.
The plane \(\Pi\) contains \(A , B\) and \(C\).

1. (a) Use the scalar triple product to determine
   • the volume of tetrahedron \(O A B C\),
   • the volume of tetrahedron PABC.
     (b) Hence show that the distance from \(P\) to \(\Pi\) is twice the distance from \(O\) to \(\Pi\).
   • (a) Determine a vector which is normal to \(\Pi\).
     (b) Hence determine, in terms of \(a , b\) and \(c\) only, the distance from \(P\) to \(\Pi\). consisting of the powers of a particular, non-singular \(2 \times 2\) real matrix \(\mathbf { M } = \left( \begin{array} { l l } a & b
     c & d \end{array} \right)\), under the operation of
     matrix multiplication. matrix multiplication.
   • Explain why such a group is only possible if \(\operatorname { det } ( \mathbf { M } ) = 1\) or - 1 .
   • Write down values of \(a , b , c\) and \(d\) that would give a suitable matrix \(\mathbf { M }\) for which \(\mathbf { M } ^ { 6 } = \mathbf { I }\) and
   Student Q observes that their class has already found a group of order 6 in a previous task; a group
   Student Q observes that their class has already found a group of order 6 in a previous task; a group consisting of the powers of a particular, non-singular \(2 \times 2\) real matrix \(\mathbf { M } = \left( \begin{array} { l l } a & b
   c & d \end{array} \right)\), under the operation of
   matrix multiplication. \(\operatorname { det } ( \mathbf { M } ) = 1\).
   Explain why such a group is only possible if \(\operatorname { det } ( \mathbf { M } ) = 1\) or - 1 . Student Q believes that it is possible to construct a rational function f in the form \(\mathrm { f } ( x ) = \frac { a x + b } { c x + d }\) so that the group of functions is isomorphic to the matrix group which is generated by the matrix \(\mathbf { M }\) of part (iii).
2. (a) Write down and simplify the function f that, according to Student Q , corresponds to \(\mathbf { M }\).
   (b) By calculating \(\mathbf { M } ^ { 3 }\), show that Student Q's suggestion does not work.
   (c) Find a different function \(f\) that will satisfy the requirements of the task.