1. At time \(t\) seconds \(( t \geqslant 0 )\) a particle \(P\) has position vector \(\mathbf { r }\) metres, with respect to a fixed origin \(O\), where

$$\mathbf { r } = \left( \frac { 1 } { 8 } t ^ { 4 } - 2 \lambda t ^ { 2 } + 5 \right) \mathbf { i } + \left( 5 t ^ { 2 } - \lambda t \right) \mathbf { j }$$ and \(\lambda\) is a constant. When \(t = 4 , P\) is moving parallel to the vector \(\mathbf { j }\).

1. Show that \(\lambda = 2\)
2. Find the speed of \(P\) when \(t = 4\)
3. Find the acceleration of \(P\) when \(t = 4\) When \(t = 0 , P\) is at the point \(A\). When \(t = 4 , P\) is at the point \(B\).
4. Find the distance \(A B\).