7. Two helicopters \(P\) and \(Q\) are moving in the same horizontal plane. They are modelled as particles moving in straight lines with constant speeds. At noon \(P\) is at the point with position vector \(( 20 \mathbf { i } + 35 \mathbf { j } ) \mathrm { km }\) with respect to a fixed origin \(O\). At time \(t\) hours after noon the position vector of \(P\) is \(\mathbf { p } \mathrm { km }\). When \(t = \frac { 1 } { 2 }\) the position vector of \(P\) is \(( 50 \mathbf { i } - 25 \mathbf { j } ) \mathrm { km }\). Find

1. the velocity of \(P\) in the form \(( a \mathbf { i } + b \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\),
2. an expression for \(\mathbf { p }\) in terms of \(t\). At noon \(Q\) is at \(O\) and at time \(t\) hours after noon the position vector of \(Q\) is \(\mathbf { q } \mathrm { km }\). The velocity of \(Q\) has magnitude \(120 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction of \(4 \mathbf { i } - 3 \mathbf { j }\). Find
3. an expression for \(\mathbf { q }\) in terms of \(t\),
4. the distance, to the nearest km , between \(P\) and \(Q\) when \(t = 2\). \section*{8.} \section*{Figure 4}
   \includegraphics[max width=\textwidth, alt={}]{14703bfa-abd8-4a8d-bc18-20d66eea409e-6_695_1153_322_562}
   Two particles \(A\) and \(B\), of mass \(m \mathrm {~kg}\) and 3 kg respectively, are connected by a light inextensible string. The particle \(A\) is held resting on a smooth fixed plane inclined at \(30 ^ { \circ }\) to the horizontal. The string passes over a smooth pulley \(P\) fixed at the top of the plane. The portion \(A P\) of the string lies along a line of greatest slope of the plane and \(B\) hangs freely from the pulley, as shown in Fig. 4. The system is released from rest with \(B\) at a height of 0.25 m above horizontal ground. Immediately after release, \(B\) descends with an acceleration of \(\frac { 2 } { 5 } g\). Given that \(A\) does not reach \(P\), calculate
5. the tension in the string while \(B\) is descending,
6. the value of \(m\). The particle \(B\) strikes the ground and does not rebound. Find
7. the magnitude of the impulse exerted by \(B\) on the ground,
8. the time between the instant when \(B\) strikes the ground and the instant when \(A\) reaches its highest point.