1. \hspace{0pt} [In this question, the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a horizontal plane.]

A particle \(Q\) of mass 1.5 kg is moving on a smooth horizontal plane under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds ( \(t \geqslant 0\) ), the position vector of \(Q\), relative to a fixed point \(O\), is \(\mathbf { r }\) metres and the velocity of \(Q\) is \(\mathbf { v } \mathrm { ms } ^ { - 1 }\)
It is given that $$\mathbf { v } = \left( 3 t ^ { 2 } + 2 t \right) \mathbf { i } + \left( t ^ { 3 } + k t \right) \mathbf { j }$$ where \(k\) is a constant.
Given that when \(t = 2\) particle \(Q\) is moving in the direction of the vector \(\mathbf { i } + \mathbf { j }\)

1. show that \(k = 4\)
2. find the magnitude of \(\mathbf { F }\) when \(t = 2\) Given that \(\mathbf { r } = 3 \mathbf { i } + 4 \mathbf { j }\) when \(t = 0\)
3. find \(\mathbf { r }\) when \(t = 2\)