OCR MEI C4 2008 January — Question 7 18 marks

Exam BoardOCR MEI
ModuleC4 (Core Mathematics 4)
Year2008
SessionJanuary
Marks18
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicVectors 3D & Lines
TypeVector geometry in 3D shapes
DifficultyStandard +0.3 This is a structured multi-part question requiring standard 3D vector techniques: finding vectors from coordinates, calculating magnitudes, verifying perpendicularity via dot products, writing plane equations, finding line intersections, and applying volume formulas. While it has many parts (5 marks worth), each step uses routine methods with clear guidance ('show that', 'hence find'). The pyramid volume calculation is straightforward application of given formulas. Slightly easier than average due to scaffolding and standard techniques.
Spec1.10a Vectors in 2D: i,j notation and column vectors1.10c Magnitude and direction: of vectors4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04f Line-plane intersection: find point
7 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]{9a8332ec-2216-4e1f-9768-ef175c9e159b-3_632_1102_486_520} \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.
\(a + (a + d) = a + 2d \Rightarrow a = d\)
\((a + d) + (a + 2d) = a + 3d \Rightarrow a = 0\)
AnswerMarks
\(a = d = 0\) *M1 M1 E1 [18] 
$a + (a + d) = a + 2d \Rightarrow a = d$

$(a + d) + (a + 2d) = a + 3d \Rightarrow a = 0$

$a = d = 0$ * | M1 M1 E1 [18] |
7 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]{9a8332ec-2216-4e1f-9768-ef175c9e159b-3_632_1102_486_520}
\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 2008 Q7 [18]}}
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Q1 7 Q2 8 Q3 5 Q4 7 Q5 6 Q6 3 Q7 18 Q8 18