OCR MEI C4 — Question 1 18 marks

Exam BoardOCR MEI
ModuleC4 (Core Mathematics 4)
Marks18
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicVectors 3D & Lines
TypeVector geometry in 3D shapes
DifficultyStandard +0.3 This is a structured multi-part question with clear scaffolding through standard vector operations (finding vectors from coordinates, magnitude, perpendicularity check, plane equation, line intersection). While it involves 3D geometry and multiple steps, each part uses routine techniques with explicit guidance. The volume calculation at the end is straightforward application of a given formula. Slightly easier than average due to the scaffolding and standard methods.
Spec1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04c Scalar product: calculate and use for angles
1 A glass ornament OABCDEFG is a truncated pyramid on a rectangular base (see Fig. 7). All dimensions are in centimetres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7a52b6ce-a0cc-421d-8eae-3b6cf098e381-1_625_1109_416_522} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Write down the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\).
  2. Find the length of the edge CD.
  3. Show that the vector \(4 \mathbf { i } + \mathbf { k }\) is perpendicular to the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\). Hence find the cartesian equation of the plane BCDE.
  4. Write down vector equations for the lines OG and AF . Show that they meet at the point P with coordinates (5, 10, 40). You may assume that the lines CD and BE also meet at the point P .
    The volume of a pyramid is \(\frac { 1 } { 3 } \times\) area of base × height.
  5. Find the volumes of the pyramids POABC and PDEFG . Hence find the volume of the ornament.
1 A glass ornament OABCDEFG is a truncated pyramid on a rectangular base (see Fig. 7). All dimensions are in centimetres.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{7a52b6ce-a0cc-421d-8eae-3b6cf098e381-1_625_1109_416_522}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{center}
\end{figure}

(i) Write down the vectors $\overrightarrow { \mathrm { CD } }$ and $\overrightarrow { \mathrm { CB } }$.\\
(ii) Find the length of the edge CD.\\
(iii) Show that the vector $4 \mathbf { i } + \mathbf { k }$ is perpendicular to the vectors $\overrightarrow { \mathrm { CD } }$ and $\overrightarrow { \mathrm { CB } }$. Hence find the cartesian equation of the plane BCDE.\\
(iv) Write down vector equations for the lines OG and AF .

Show that they meet at the point P with coordinates (5, 10, 40).

You may assume that the lines CD and BE also meet at the point P .\\
The volume of a pyramid is $\frac { 1 } { 3 } \times$ area of base × height.\\
(v) Find the volumes of the pyramids POABC and PDEFG .

Hence find the volume of the ornament.

\hfill \mbox{\textit{OCR MEI C4  Q1 [18]}}
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Q1 18 Q2 17 Q3 18 Q4 18