Write down the matrix, \(\mathbf { A }\), that represents a shear with \(x\)-axis invariant in which the image of the point \(( 1,1 )\) is \(( 4,1 )\).
The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { c c } \sqrt { 3 } & 0 0 & \sqrt { 3 } \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { B }\).
The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { l l } 2 & 6 0 & 2 \end{array} \right)\).
(a) Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { C }\).
(b) Write down the determinant of \(\mathbf { C }\) and explain briefly how this value relates to the transformation represented by \(\mathbf { C }\).
8 The quadratic equation \(2 x ^ { 2 } - x + 3 = 0\) has roots \(\alpha\) and \(\beta\), and the quadratic equation \(x ^ { 2 } - p x + q = 0\) has roots \(\alpha + \frac { 1 } { \alpha }\) and \(\beta + \frac { 1 } { \beta }\).
Show that \(p = \frac { 5 } { 6 }\).
Find the value of \(q\).
9 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r r } a & - a & 1 3 & a & 1 4 & 2 & 1 \end{array} \right)\).
Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
Hence find the values of \(a\) for which \(\mathbf { M } ^ { - 1 }\) does not exist.
Determine whether the simultaneous equations
$$\begin{aligned}
& 6 x - 6 y + z = 3 k
& 3 x + 6 y + z = 0
& 4 x + 2 y + z = k
\end{aligned}$$
where \(k\) is a non-zero constant, have a unique solution, no solution or an infinite number of solutions, justifying your answer.
Show that \(\frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 1 ) ( r + 2 ) }\).
Hence find an expression, in terms of \(n\), for
$$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$
Show that \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }\).