2021 HCI J2 BT Q7 [Modified]

For a school camp, Tom and Jerry stay in a tent which can be modelled by using sheets of cardboard. Cardboard \(OABC\) is rectangular in shape with length \(OC=6\)m and width \(OA=5\)m. Cardboard \(ABQP\) is a trapezium with parallel sides \(AB\) and \(PQ\) of lengths \(6\)m and \(8\)m respectively. The vectors \(\mathbf{i}\), \(\mathbf{j}\), \(\mathbf{k}\) are perpendicular vectors as shown below, each with a magnitude of \(1\) metre. The model is positioned such that \(OCQP\) is a horizontal base on the \(x\)-\(y\) plane and \(OC\) is parallel to the vector \(\mathbf{j}\).

1. Given that \(A\) is \(3\)m directly above the horizontal base, find the acute angle that \(OABC\) makes with the horizontal base.[2]
2. Given that \(Q\) has position vector \(6\mathbf{i}+7\mathbf{j}\), find the cartesian equation of the plane \(ABQP\).[3]
3. A lamp is placed on the ground inside the tent at the point \(K\) with coordinates \(\left( k,k,0 \right)\).
   Find the exact value of \(k\) such that both \(OABC\) and \(ABQP\) are equidistant to the lamp.[3]
4. Tom and Jerry have a squabble and want to split the base \(OCQP\) equally. Jerry suggests a fair way to do so is to use a straight rope to divide the base into two, by attaching the rope from \(P\) to a point \(R\) lying on the line segment \(CQ\) such that the points \(O\) and \(Q\) have the same perpendicular distance to the rope \(PR\). With the help of a clearly labelled diagram, suggest whether Jerry’s suggestion is fair. Justify your answer.[2]
5. Tom decides to go with Jerry’s suggestion and they proceed to attach a rope from \(P\) to \(R\) on \(CQ\) where \(R\) divides \(CQ\) in the ratio \(t:1-t\), where \(0 < t < 1\). By considering a suitable point on \(OQ\), find the value of \(t\).[3]