9758/2022/P1/Q11

11) A gas company has plans to install a pipeline from a gas field to a storage facility. One part of the route for the pipeline has to pass under a river. This part of the pipeline is in a straight line between two points, $P$ and $Q$.

Points are defined relative to an origin $(0,0,0)$ at the gas field. The $x-, y$ – and $z$-axes are in the directions east, north and vertically upwards respectively, with units in metres. $P$ has coordinates $(1136,92, p)$ and $Q$ has coordinates $(200,20,-15)$.
(a) The length of the pipeline $P Q$ is $939 \mathrm{~m}$. Given that the level of $P$ is below that of $Q$, find the value of $p$.

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A thin layer of rock lies below the ground. This layer is modelled as a plane. Three points in this plane are $(400,600,-20),(500,200,-70)$ and $(600,-340,-50)$.
(b) Find the cartesian equation of this plane.

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(c) Hence find the coordinates of the point where the pipeline meets the rock.

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(d) Find the angle that the pipeline between the points $P$ and $Q$ makes with the horizontal.

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