The minimum number of colours required to colour the following graph, such that no two adjacent vertices are assigned the same colour, is

Let $$A$$ be and n$$ \times $$n matrix of the folowing form.

What is the value of the determinant of $$A$$?

How many solutions does the following system of linear equations have?

- x + 5y = - 1
x - y = 2
x + 3y = 3

The recurrence equation
$$\,\,\,\,\,\,\,T\left( 1 \right) = 1$$
$$\,\,\,\,\,\,T\left( n \right) = 2T\left( {n - 1} \right) + n,\,n \ge 2$$
evaluates to