OCR FP1 AS (Further Pure 1 AS) 2021 June

1 In this question you must show detailed reasoning.
The cubic equation \(2 x ^ { 3 } + 3 x ^ { 2 } - 5 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). By making an appropriate substitution, or otherwise, find a cubic equation with integer coefficients whose roots are \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).

2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r c } 2 & 1 & 2 \\ 1 & - 1 & 1 \\ 2 & 2 & a \end{array} \right)\).

1. Show that \(\operatorname { det } \mathbf { A } = 6 - 3 a\).
2. State the value of \(a\) for which \(\mathbf { A }\) is singular.
3. Given that \(\mathbf { A }\) is non-singular find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).

3 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } t & 6 \\ t & - 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 2 t & 4 \\ t & - 2 \end{array} \right)\) where \(t\) is a constant.

1. Show that \(| \mathrm { A } | = | \mathrm { B } |\).
2. Verify that \(| \mathrm { AB } | = | \mathrm { A } | | \mathrm { B } |\).
3. Given that \(| \mathbf { A B } | = - 1\) explain what this means about the constant \(t\). The \(2 \times 2\) matrix \(A\) represents a transformation \(T\) which has the following properties.
   The transformation \(S\) is represented by the matrix \(B\) where \(B = \left( \begin{array} { l l } 3 & 1 \\ 2 & 2 \end{array} \right)\).
   (b) Find the equation of the line of invariant points of S .
   (c) Show that any line of the form \(y = x + c\) is an invariant line of S .