General Mathematics · Past QCAA questions All simple familiar questions
2023 · Paper 1 · Simple familiar

Question 20

[4 marks] Scientific calculator
Topic guidesPremiumTopic guide 11 · Planar graphs and Euler’s formula
Game plansPremiumGame plan 6 · Graphs and networks
The question

Graphs 1 and 2 are shown.

1 2
a)

Show that Euler’s formula applies to graph 1.

[2 marks]
b)

What feature in the drawing should be changed to represent graph 2 as planar?

[1 mark]
c)

Redraw graph 2 to represent it as a planar graph.

[1 mark]
Watch the situation first

The outside counts as a face, and a crossing is only a drawing problem

First watch graph 1 get counted, outside face included. Then watch one edge of graph 2 swing around the outside so that nothing crosses.

Graph 1: count vertices, edges and faces Graph 2: two edges cross in this drawing Redrawn with no crossings, so graph 2 is planar 1 2 3 4 1 2 3 the outside v = 4 e = 5 f = 3 v + f − e = 4 + 3 − 5 = 2 The crossing point is not a vertex. Change the drawing, keep the graph. 1 2 3 4 the outside v = 5, e = 7, f = 4 5 + 4 − 7 = 2

Euler’s formula only works if you count the outside region as a face. Graph 2 was planar all along. Its first drawing just hid that.

Now for the mathematics

Count graph 1 honestly, then fix graph 2 without changing it

For graph 1, choose your counts and watch Euler’s formula respond. For graph 2, the vertices have letters so you can name them. Choose what the feature is, then try a change and read the check panel.

Part a) Graph 1
1 2 3
Vertices, $v$
Edges, $e$
Faces, $f$

Parts b) and c) Graph 2
P Q R S T
b) Which feature of the drawing should change?

c) Try a change to the drawing
Vertices Edges Edge crossings Degrees Same graph as graph 2 Faces

QCAA marking guide · 4 marks

Count, substitute, then redraw with nothing added or lost

Try each step yourself before you reveal it. Part a) has a mark for the counts and one for Euler’s formula. Parts b) and c) are one mark each.

Step 1 · part a)1 mark

Count the dots, the lines and the regions. Graph 1 encloses two triangles, and the region outside the graph is a face as well.

1 2 3
$$v=4,\qquad f=3,\qquad e=5$$

Write all three values down, labelled. The mark is for identifying all three correctly.

Marker: correctly identifies the number of vertices, faces and edges for graph 1

Step 2 · part a)1 mark

Euler’s formula from the formula book is $v+f-e=2$. Substitute your counts and show that you get 2.

$$v+f-e=4+3-5=2$$

“Show that” means the working is the answer, so write the substitution in full. Follow-through applies to this mark, so substituting your own counts correctly still earns it if a count in Step 1 slipped.

Marker: applies Euler’s formula to graph 1

Step 3 · part b)1 mark

Two edges of graph 2 cross inside the pentagon. A planar drawing has no edges crossing, so that is the feature to change.

Redraw it so that no edges cross.

The marking guide’s sample is just “Show no crossing edges.” The 2023 subject report highlighted answers that said the drawing should be changed so that no two edges cross. Answers like “remove an edge” change the graph itself, so stick to the crossing.

Marker: correctly identifies the feature to be changed

Step 4 · part c)1 mark

Keep the pentagon and one diagonal. Take the other diagonal around the outside of the graph so that it meets nothing on the way.

5 vertices, 7 edges, none crossing
Degrees 2, 3, 3, 3, 3

The marking guide’s sample moves the vertices around and uses one curved edge, which is just as good. Any drawing earns the mark if it has the same five vertices, the same seven edges, no crossings, and one vertex of degree 2 with four of degree 3. Do not add a dot where the edges used to cross, and do not drop an edge.

Marker: correctly draws graph 2 as a simple connected graph with seven edges that do not cross and five vertices (one degree 2 vertex, four degree 3 vertices)

Putting it all together

Each phrase of the question gave you something

“Graphs 1 and 2 are shown.”
Two drawings. Graph 1 has no crossings. Graph 2’s drawing has one.
“Show that Euler’s formula applies to graph 1” [2 marks]
One mark for stating $v$, $f$ and $e$, one for substituting into $v+f-e=2$. Count the outside face.
“What feature in the drawing should be changed”
It asks about how graph 2 is drawn. The feature is the pair of crossing edges.
“to represent graph 2 as planar”
Planar means it can be drawn with no edges crossing.
“Redraw graph 2”
Same five vertices, same seven edges, so every degree stays the same. Only the routes of the edges change.
What makes this simple familiar

Euler’s formula is in the formula book, the graphs are tiny, and the redraw needs just one edge moved. The 2023 subject report listed applying Euler’s formula, and describing what feature in a drawing should be changed to make a graph planar, among the things students did well.

If I had to guess where you might lose marks, it would be the faces in part a). Forget the outside and you get $4+2-5=1$, and the formula you were asked to confirm seems to fail. In part c), redraw rather than edit. Deleting one of the crossing edges, or putting a dot where they cross, gives a different graph, and the marker checks for exactly seven edges and five vertices.

Only mark this done when you could do it without help. Reading the solution does not count.

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Question wording and marking-guide steps are from the 2023 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.

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