OCR MEI C4 — Question 8

Exam BoardOCR MEI
ModuleC4 (Core Mathematics 4)
PaperDownload PDF ↗
TopicVectors: Lines & Planes
Type3D geometry applications
DifficultyStandard +0.3 This is a structured multi-part question on 3D coordinate geometry with clear scaffolding. Parts (i)-(ii) involve routine verification and calculation of plane equations using standard methods. Part (iii) requires the angle between planes formula (standard technique). Part (iv) involves writing a line equation perpendicular to a plane and finding intersection—all standard C4/FM procedures with no novel problem-solving required. The context adds complexity but the mathematical steps are textbook exercises.
Spec4.04b Plane equations: cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04d Angles: between planes and between line and plane
8 The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{070e9904-12b9-4458-b8f2-60c89b31b828-093_1013_1399_488_372} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Relative to axes \(\mathrm { O } x\) (due east), \(\mathrm { O } y\) (due north) and \(\mathrm { O } z\) (vertically upwards), the coordinates of the points are as follows.
A: (0, 0, -15)
B: (100, 0, -30)
C: (0, 100, -25)
D: (0, 0, -40)
E: (100, 0, -50)
F: (0, 100, -35)
  1. Verify that the cartesian equation of the plane ABC is \(3 x + 2 y + 20 z + 300 = 0\).
  2. Find the vectors \(\overrightarrow { \mathrm { DE } }\) and \(\overrightarrow { \mathrm { DF } }\). Show that the vector \(2 \mathbf { i } - \mathbf { j } + 20 \mathbf { k }\) is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF .
  3. By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. It is decided to drill down to the seam from a point \(\mathrm { R } ( 15,34,0 )\) in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S .
  4. Write down a vector equation of the line RS. Calculate the coordinates of S.
8 The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{070e9904-12b9-4458-b8f2-60c89b31b828-093_1013_1399_488_372}
\captionsetup{labelformat=empty}
\caption{Fig. 8}
\end{center}
\end{figure}

Relative to axes $\mathrm { O } x$ (due east), $\mathrm { O } y$ (due north) and $\mathrm { O } z$ (vertically upwards), the coordinates of the points are as follows.\\
A: (0, 0, -15)\\
B: (100, 0, -30)\\
C: (0, 100, -25)\\
D: (0, 0, -40)\\
E: (100, 0, -50)\\
F: (0, 100, -35)\\
(i) Verify that the cartesian equation of the plane ABC is $3 x + 2 y + 20 z + 300 = 0$.\\
(ii) Find the vectors $\overrightarrow { \mathrm { DE } }$ and $\overrightarrow { \mathrm { DF } }$. Show that the vector $2 \mathbf { i } - \mathbf { j } + 20 \mathbf { k }$ is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF .\\
(iii) By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF.

It is decided to drill down to the seam from a point $\mathrm { R } ( 15,34,0 )$ in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S .\\
(iv) Write down a vector equation of the line RS.

Calculate the coordinates of S.

\hfill \mbox{\textit{OCR MEI C4  Q8}}
This paper (7 questions)
View full paper
Q1 Q2 Q3 Q4 Q5 Q6 4 Q8