IB Questionbank

User interface language: English | Español

Date	May Example question	Marks available	2	Reference code	EXM.2.SL.TZ0.4
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	State and Give a reason	Question number	4	Adapted from	N/A

Question

A king rules a small mountain kingdom which is in the form of a square of length 4 kilometres. The square is described by the co-ordinate system $0 \leqslant x \leqslant 4{\text{,}}\,\,0 \leqslant y \leqslant 4$ .

The king has four adult children, each of which has a luxury palace located at the points $\left( {1{\text{,}}\,\,1} \right){\text{,}}\,\left( {3{\text{,}}\,\,1} \right){\text{,}}\,\left( {1{\text{,}}\,\,3} \right){\text{,}}\,\left( {3{\text{,}}\,\,3} \right)$ . Each child owns all the land that is nearer their palace than any other palace.

The king has a fifth (youngest) child who is now just growing up. He installs her in a new palace situated at point (2, 2). The rule about ownership of land is then reapplied.

Sketch a Voronoi diagram to represent this information.

[2]

Sketch a new Voronoi diagram to represent this new situation.

[2]

State what the shape of the land, owned by the youngest child, is.

[1]

Find the area of the youngest child’s land.

[4]

Find how much land an older child has lost.

[1]

State, with a reason, if all five children now own an equal amount of land.

[2]

Markscheme

[2 marks]

By symmetry a square A1

[1 mark]

Distance from (2, 2) to (1, 3) is $\sqrt {{1^2} + {1^2}} = \sqrt 2$ M1A1

So length of youngest child’s square is $2\frac{{\sqrt 2 }}{2} = \sqrt 2$ and thus area is 2. M1A1

[4 marks]

By symmetry each older child must lose $\frac{2}{4} = \frac{1}{2}$ A1

[1 mark]

No, youngest child has less as each older child has $2 \times 2 - \frac{1}{2} = 3\tfrac{1}{2}$ A1R1

[2 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3—Geometry and trigonometry » SL 3.6—Voronoi diagrams

Show 27 related questions

22M.3.AHL.TZ1.2c.i:
Find the equation of the perpendicular bisector of $[IF]$ .
22M.3.AHL.TZ1.2b:
Sketch a Voronoi diagram showing the regions within the metropolitan area that are closest to each town.
22M.3.AHL.TZ1.2c.iii:
Sketch this new Voronoi diagram showing the regions within the metropolitan area which are closest to each town.
EXM.2.SL.TZ0.4a:
Sketch a Voronoi diagram to represent this information.
22M.3.AHL.TZ1.2e.ii:
Make one possible criticism of the council’s choice of location.
22M.3.AHL.TZ1.2c.ii:
Given that the coordinates of one vertex of the new Voronoi diagram are $(20, 37.5)$ , find the coordinates of the other two vertices within the metropolitan area.
22M.3.AHL.TZ1.2e.i:
Find the location of the toxic waste dump, given that this location is not on the edge of the metropolitan area.
21M.1.SL.TZ1.5a:
Explain why Tim will receive the strongest signal from tower $T4$ .
21N.1.SL.TZ0.7b.i:
Write down the equation of the perpendicular bisector of $[PC]$ .
EXN.2.SL.TZ0.5e:
A new office is to be built within the triangle formed by $A$ , $B$ and $C$ , at a point as far as possible from the other three offices.

Find the distance of this office from each of the other offices.
22M.2.SL.TZ2.3a:
Find the midpoint of $[BD]$ .
22M.3.AHL.TZ1.2a:
The model assumes that each town is positioned at a single point. Describe possible circumstances in which this modelling assumption is reasonable.
EXM.2.SL.TZ0.4b:
Sketch a new Voronoi diagram to represent this new situation.
EXM.2.SL.TZ0.4c:
State what the shape of the land, owned by the youngest child, is.
SPM.1.SL.TZ0.7c:
In the context of the question, explain the significance of the Voronoi cell containing site E.
SPM.1.SL.TZ0.7a:
Calculate the gradient of the line segment AE.
EXM.2.SL.TZ0.4e:
Find how much land an older child has lost.
EXM.1.SL.TZ0.4:
The diagram below is part of a Voronoi diagram.

Diagram not to scale

A and B are sites with B having the co-ordinates of (4, 6). L is an edge; the equation of this perpendicular bisector of the line segment from A to B is $y = - 2x + 9$

Find the co-ordinates of the point A.
EXN.2.SL.TZ0.5d:
State a further assumption that must be made in order to use area covered as a measure of whether or not the manager of office $C$ is responsible for more students than the managers of offices $A$ and $B$ .
21M.1.SL.TZ1.5b:
Write down the coordinates of tower $T4$ .
21M.1.SL.TZ1.5c:
Tower $T1$ has coordinates $(- 13, 3)$ .

Find the gradient of the edge of the Voronoi diagram between towers $T1$ and $T2$ .
SPM.1.SL.TZ0.7b:
Find the equation of the line which would complete the Voronoi cell containing site E.

Give your answer in the form $ax + by + d = 0$ where $a$ , $b$ , $d \in \mathbb{Z}$ .
EXM.2.SL.TZ0.4d:
Find the area of the youngest child’s land.
EXN.2.SL.TZ0.5b:
Find the area of campus managed by office $C$ .
EXN.2.SL.TZ0.5c:
Hence or otherwise find the areas managed by offices $A$ and $B$ .
21N.1.SL.TZ0.7a:
Show that the new station should be built at $P$ .
21N.1.SL.TZ0.7b.ii:
Hence draw the missing boundaries of the region around $P$ on the following diagram.

Hide related questions

Topic 3—Geometry and trigonometry

Question

Markscheme

Examiners report

Syllabus sections

View options