These Ten-Year-Series (TYS) worked solutions with video explanations for 2022 A Level H2 Mathematics Paper 1 Question 11 are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
2022 A Level H2 Math Paper 1 Question 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 $\left( 0,0,0 \right)$ 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 $\left( 1136,\,\,92,\,\,p \right)$ and $Q$ has coordinates $\left( 200,\,\,20,\,\,-15 \right)$.
(a)
The length of the pipeline $PQ$ is $939$ m. Given that the level of $P$ is below that of $Q$, find the value of $p$.
[3]
(a) The length of the pipeline $PQ$ is $939$ m. Given that the level of $P$ is below that of $Q$, find the value of $p$.
[3]
A thin layer of rock lies below the ground. This layer is modelled as a plane. Three points in this plane are $\left( 400,\,600,\,-20 \right)$, $\left( 500,\,200,\,-70 \right)$ and $\left( 600,\,-340,\,-50 \right)$.
(b)
Find the cartesian equation of this plane.
[4]
(b) Find the cartesian equation of this plane.
[4]
(c)
Hence find the coordinates of the point where the pipeline meets the rock.
[4]
(c) Hence find the coordinates of the point where the pipeline meets the rock.
[4]
(d)
Find the angle that the pipeline between the points $P$ and $Q$ makes with the horizontal.
[2]
(d) Find the angle that the pipeline between the points $P$ and $Q$ makes with the horizontal.
[2]
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