Mr Neo wants to build a shed in his future garden as shown in the diagrams below. The planes \(ABCD\

Question

Mr Neo wants to build a shed in his future garden as shown in the diagrams below.

The planes $ABCD$ and $EFGH$ are parallel to the horizontal plane, represented by the $xy$-plane. The triangles $APD$ and $BQC$ are congruent isosceles triangles and the lengths of the pillars $AE$, $BF$, $CG$ and $DH$ are of the same height.

The equation of the plane $ABQP$ is given by $-2x-y+3z=2$ and the point $P$ has position vector $4\mathbf{i}+2\mathbf{j}+4\mathbf{k}$.

Explain why plane $CDPQ$ is perpendicular to $2\mathbf{i}+\mathbf{j}+3\mathbf{k}$. Hence show that the equation of the plane $CDPQ$ is $2x+y+3z=22$.[2]
Find the angle the roof $ABQP$ makes with the horizontal plane.[2]
Find the vector equation of the line $PQ$ and hence find the coordinates of the point $Q$ given that $PQ$ has length $3\sqrt{5}$ units.[5]
When the shed was completed, Mr Neo discovered a hole in the roof $ABQP$. When light shines perpendicularly onto the plane $ABQP$, the light passes through the hole and hits the ground at the point with coordinates $(3,6,0)$.
Find the coordinates of the hole in the roof $ABQP$.[3]

Accepted Answer

(i) $2x+y+3z=22$ (ii) $36.7^\circ$ (iii) $\mathbf r=\begin{pmatrix}4\2\4\end{pmatrix}+\lambda\begin{pmatrix}-1\2\0\end{pmatrix}$, $\lambda\in\mathbb R$, $Q=(1,8,4)$ (iv) $(1,5,3)$

2021 ACJC P1 Q11

2021 ACJC P1 Q11

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