IB Maths AI SL Topic 3: Geometry and trigonometry questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1 Paper 2

Difficulty

Easy Medium Hard

Question 1

Paper 2Medium6 marks

A designer is planning a pavilion with a square-based roof shaped as a right pyramid. The apex $A$ is vertically above the centre $O$ of the square base. Each side of the base is $8.4\text{ m}$ and the current vertical height $AO$ is $4.60\text{ m}$ The design requires the angle between each sloping edge and the plane of the base to be at least $38^\circ$ Let $P$ be a corner of the base.

(a)

Show that $OP=5.94\text{ m}$

[2]

(b)

Calculate the angle between $AP$ and the plane of the base, giving your answer to the nearest $0.1^\circ$

[1]

(c)

Determine the minimum vertical height required to meet the design requirement and hence state whether the current roof meets it.

[3]

Question 2

Paper 2Medium6 marks

Two coastal stations, $A$ and $B$ lie on a straight east–west coastline. Station $B$ is $5.7\text{ km}$ due east of station $A$ A boat at $C$ north of the coastline, is observed on a bearing of $062^\circ$ from $A$ and a bearing of $323^\circ$ from $B$

(a)

Draw a labelled sketch of the situation and determine the interior angles of triangle $ABC$ at $A$ and $B$

[2]

(b)

Determine the distance $AC$ giving your answer to 3 significant figures.

[2]

(c)

Hence, using your answer to part (b), determine the shortest distance from the boat to the coastline. Give your answer to 3 significant figures.

[2]

Question 3

Paper 2Medium6 marks

Three weather stations record annual rainfall. Their coordinates are measured in kilometres.

Station	Coordinates	Annual rainfall (mm)
A	$(1,1)$	540
B	$(7,3)$	680
C	$(3,7)$	610

The Voronoi diagram assigns each location to its nearest station.

(a)

Find the equation of the boundary between the cells for stations A and B, in the form $y=mx+c$

[2]

(b)

The boundary between the cells for stations A and C has equation $x+3y=14$ The boundaries between the cells for A and B, and A and C, meet at a Voronoi vertex. Using your answer to part (a), find the coordinates of this vertex.

[3]

(c)

Estimate the annual rainfall at $Q=(2,5)$ using nearest-neighbour interpolation.

[1]

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