Find the cosine of angle \(A O B\).
The position vector of \(C\) is given by \(\overrightarrow { O C } = \left( \begin{array} { c } k - 2 k 2 k - 3 \end{array} \right)\).
Given that \(A B\) and \(O C\) have the same length, find the possible values of \(k\).
5 A piece of wire of length 24 cm is bent to form the perimeter of a sector of a circle of radius \(r \mathrm {~cm}\).
Show that the area of the sector, \(A \mathrm {~cm} ^ { 2 }\), is given by \(A = 12 r - r ^ { 2 }\).
Express \(A\) in the form \(a - ( r - b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
Given that \(r\) can vary, state the greatest value of \(A\) and find the corresponding angle of the sector. [2]
6 The line with gradient - 2 passing through the point \(P ( 3 t , 2 t )\) intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
Find the area of triangle \(A O B\) in terms of \(t\).
The line through \(P\) perpendicular to \(A B\) intersects the \(x\)-axis at \(C\).
Show that the mid-point of \(P C\) lies on the line \(y = x\).