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 $\mathbf{j}$.
![mf 27 2021 HCI J2 BT Q7 [Modified]](https://timganmath.b-cdn.net/wp-content/uploads/2021-HCI-J2-BT-Q7-Modified-diagram-.png)
(i)
Given that $A$ is $3$m directly above the horizontal base, find the acute angle that $OABC$ makes with the horizontal base.
[2]
(ii)
Given that $Q$ has position vector $6\mathbf{i}+7\mathbf{j}$, find the cartesian equation of the plane $ABQP$.
[3]
(iii)
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]
(iv)
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]
(v)
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]
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