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

At time \(t = 0\), a particle \(P\) is projected with velocity ( \(4 \mathbf { i } + 9 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) from a fixed point \(O\) on horizontal ground. The particle moves freely under gravity. When \(P\) is at the point \(H\) on its path, \(P\) is at its greatest height above the ground.

1. Find the time taken by \(P\) to reach \(H\). At the point \(A\) on its path, the position vector of \(P\) relative to \(O\) is \(( k \mathbf { i } + k \mathbf { j } ) \mathrm { m }\), where \(k\) is a positive constant.
2. Find the value of \(k\).
   (4)
3. Find, in terms of \(k\), the position vector of the other point on the path of \(P\) which is at the same vertical height above the ground as the point \(A\).
   (3) At time \(T\) seconds the particle is at the point \(B\) and is moving perpendicular to \(( 4 \mathbf { i } + 9 \mathbf { j } )\)
4. Find the value of \(T\).