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

A particle \(P\) of mass 4 kg is at rest at the point \(A\) on a smooth horizontal plane.
At time \(t = 0\), two forces, \(\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( \lambda \mathbf { i } + \mu \mathbf { j } ) \mathrm { N }\), where \(\lambda\) and \(\mu\) are constants, are applied to \(P\) Given that \(P\) moves in the direction of the vector ( \(3 \mathbf { i } + \mathbf { j }\) )

1. show that $$\lambda - 3 \mu + 7 = 0$$ At time \(t = 4\) seconds, \(P\) passes through the point \(B\).
   Given that \(\lambda = 2\)
2. find the length of \(A B\).