Question 1 - A-Level Maths

OCR MEI D1 2012 January — Question 1 8 marks

Exam Board	OCR MEI
Module	D1 (Decision Mathematics 1)
Year	2012
Session	January
Marks	8
Paper	Download PDF ↗
Topic	Graph Theory Fundamentals
Type	Physical space modeling
Difficulty	Moderate -0.5 This is a straightforward application of graph theory to a concrete geometric situation. Part (i) requires drawing a simple graph for a cube (6 vertices forming an octahedron graph), and part (ii) involves pattern recognition from the given example to extend to three cubes. The concept is clearly explained with a worked example, requiring only careful counting and basic understanding of face-adjacency rather than abstract reasoning or proof.
Spec	7.02a Graphs: vertices (nodes) and arcs (edges)7.02b Graph terminology: tree, simple, connected, simply connected 7.02r Graph modelling: model and solve problems using graphs

1 A graph is obtained from a solid by producing a vertex for each exterior face. Vertices in the graph are connected if their corresponding faces in the original solid share an edge. The diagram shows a solid followed by its graph. The solid is made up of two cubes stacked one on top of the other. This solid has 10 exterior faces, which correspond to the 10 vertices in the graph. (Note that in this question it is the exterior faces of the cubes that are being counted.) \includegraphics[max width=\textwidth, alt={}, center]{3239d012-5699-4789-ba64-f1295f4b4642-2_455_309_571_653} \includegraphics[max width=\textwidth, alt={}, center]{3239d012-5699-4789-ba64-f1295f4b4642-2_444_286_573_1135}

Draw the graph for a cube.
Obtain the number of vertices and the number of edges for the graph of three cubes stacked on top of each other. \includegraphics[max width=\textwidth, alt={}, center]{3239d012-5699-4789-ba64-f1295f4b4642-2_643_305_1302_881}

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1 A graph is obtained from a solid by producing a vertex for each exterior face. Vertices in the graph are connected if their corresponding faces in the original solid share an edge. The diagram shows a solid followed by its graph. The solid is made up of two cubes stacked one on top of the other. This solid has 10 exterior faces, which correspond to the 10 vertices in the graph. (Note that in this question it is the exterior faces of the cubes that are being counted.)\\
\includegraphics[max width=\textwidth, alt={}, center]{3239d012-5699-4789-ba64-f1295f4b4642-2_455_309_571_653}\\
\includegraphics[max width=\textwidth, alt={}, center]{3239d012-5699-4789-ba64-f1295f4b4642-2_444_286_573_1135}\\
(i) Draw the graph for a cube.\\
(ii) Obtain the number of vertices and the number of edges for the graph of three cubes stacked on top of each other.\\
\includegraphics[max width=\textwidth, alt={}, center]{3239d012-5699-4789-ba64-f1295f4b4642-2_643_305_1302_881}

\hfill \mbox{\textit{OCR MEI D1 2012 Q1 [8]}}

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Q1 8 Q2 8 Q3 8 Q4 16 Q5 16 Q6 16