10 free IB Mathematics: Applications & Interpretation Geometry & Trigonometry questions written in the style of IB Mathematics: Applications & Interpretation papers, each with its answer and mark scheme. 4 of them are A* challenge questions, built to catch the mistakes strong students still make.
This topic covers 3D geometry, Trigonometry & bearings, Voronoi diagrams, Vectors & kinematics, Graph theory, Matrix transformations. Work through the questions below, then generate a full Geometry & Trigonometry paper with a fresh set every time.
Generate a full Geometry & Trigonometry paper →Three schools are located at P(0, 2), Q(8, 4) and R(2, 2). A new library is to be built at the point that is the same distance from all three schools.
Find the coordinates of the library and its distance from each school.
| Part | Answer and marking points | Marks |
|---|---|---|
| (1, 15); 13.0 perpendicular bisector of one pair of schools M1 a second perpendicular bisector M1 correct equations A1A1 (1, 15) A1 distance 13.0 A1 | 6 |
The weights of edges in a complete graph are shown.
| A | B | C | D | E | |
|---|---|---|---|---|---|
| A | – | 16 | 4 | 15 | 6 |
| B | 16 | – | 7 | 18 | 9 |
| C | 4 | 7 | – | 15 | 5 |
| D | 15 | 18 | 15 | – | 16 |
| E | 6 | 9 | 5 | 16 | – |
Use Prim’s algorithm, starting at A, to find a minimum spanning tree, listing the edges in the order they are added, and state its weight.
| Part | Answer and marking points | Marks |
|---|---|---|
| 31 Prim’s algorithm applied M1 order: AC (4), CE (5), CB (7), AD (15) A3 tree drawn or listed A1 weight 31 A1 | 6 |
A courier must visit five towns, starting and finishing at A. The table shows the travel times in minutes between the towns.
| A | B | C | D | E | |
|---|---|---|---|---|---|
| A | – | 114 | 100 | 41 | 120 |
| B | 114 | – | 85 | 104 | 10 |
| C | 100 | 85 | – | 63 | 82 |
| D | 41 | 104 | 63 | – | 108 |
| E | 120 | 10 | 82 | 108 | – |
Explain why the table represents a complete graph.
Use the nearest neighbour algorithm, starting at A, to find an upper bound for the travelling salesman problem.
By deleting vertex A, use the deleted vertex algorithm to find a lower bound.
Write down an interval for the length T of the optimal route.
Find the number of different Hamiltonian cycles starting and finishing at A, and explain why checking all of them is not practical for larger networks.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | every pair joined every pair of towns has a direct route R1 | 1 |
| (b) | 310 nearest neighbour applied correctly M1 route ADCEBA A2 upper bound 310 A1 | 4 |
| (c) | 296 minimum spanning tree of the remaining graph M1 weight 155 A1 adding the two shortest edges from A M1 41 + 100 A1 lower bound 296 A1 | 5 |
| (d) | 296 ≤ T ≤ 310 296 ≤ T ≤ 310 A1 | 1 |
| (e) | 12 (n − 1)!2 (M1) 12 A1 factorial growth R1 | 3 |
A room is a cuboid 6 m long, 4 m wide and 2.5 m high. A string is stretched from a bottom corner to the diagonally opposite top corner.
Find the length of the string.
Find the angle between the string and the floor.
Find the angle between the string and the 6 m by 2.5 m wall containing its starting point.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 7.63 m √l2 + w2 + h2 (M1) 7.63 m A1 | 2 |
| (b) | 19.1° sin θ = 2.57.63 (M1) 19.1° A1 | 2 |
| (c) | 31.6° projection onto the wall (M1) 31.6° A1 | 2 |
Three post offices are at A(0, 0), B(4, 0) and C(0, 4).
Find the equations of the perpendicular bisectors of AB and AC.
Find the coordinates of the vertex of the Voronoi diagram, and explain its significance.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | x = 2; y = 2 x = 2 A1 y = 2 A1 | 2 |
| (b) | (2, 2) (2, 2) A1 equidistant from all three post offices R1 | 2 |
A storage tank is a cylinder of radius 6 m and height 12 m with a hemisphere on top.
Find the volume of the tank.
Paint costs $4.50 per m2. Find the cost of painting the curved surface and the hemisphere.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 1810 m3 cylinder (M1) hemisphere (M1) 1810 A1 | 3 |
| (b) | $3053.63 2πrh (M1) 2πr2 (M1) $3053.63 A1 | 3 |
Let R be the matrix representing an anticlockwise rotation of 180° about the origin, and E the matrix representing an enlargement, scale factor 2, centre the origin.
Write down R and E.
Find the single matrix representing the rotation followed by the enlargement.
A triangle of area 8 units2 is transformed by this matrix. Find the area of its image.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | R = −1 00 −1, E = 2 00 2 R = −1 00 −1 A1 E = 2 00 2 A1 | 2 |
| (b) | −2 00 −2 ER (order matters) (M1) −2 00 −2 A1 | 2 |
| (c) | 32 |det| = 4 (M1) 32 A1 | 2 |
A boat leaves point P with position vector −4−2 km and travels with constant velocity 63 km h−1. A lighthouse is at L(7, 7.5).
Find the speed of the boat.
Find the time at which the boat is due south of the lighthouse, and the distance between them at that time.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 6.71 km h−1 |63| (M1) 6.71 km h−1 A1 | 2 |
| (b) | t = 1.83 h; 4.00 km −4 + 6t = 7 (M1) t = 1.83 A1 position (7, 3.50) (A1) 4.00 km A1 | 4 |
A ship sails 12 km on a bearing of 040° and then 8 km on a bearing of 130°.
Find the distance of the ship from its starting point.
Find the bearing of the ship from its starting point.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 14.4 km angle between legs (A1) cosine rule (M1) 14.4 km A1 | 3 |
| (b) | 73.7° sine rule or coordinates (M1) 73.7° A1 | 2 |
From a point A on level ground, the angle of elevation of the top T of a vertical tower is 20°. From a point B, 50 m closer to the tower, the angle of elevation is 38°.
Find the size of angle ATB.
Find the length BT.
Find the height of the tower.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 18° 38° − 20° = 18° A1 | 1 |
| (b) | 55.3 m BTsin 20° = 50sin 18° (M1)(A1) 55.3 m A1 | 3 |
| (c) | 34.1 m BT sin 38° (M1) 34.1 m A1 | 2 |
In the SL & HL syllabus, Geometry & Trigonometry covers 3D geometry, Trigonometry & bearings, Voronoi diagrams, Vectors & kinematics, Graph theory, Matrix transformations.
This page has 10 Geometry & Trigonometry questions with answers. For more, the free generator on mentorriyaz.in builds a new Geometry & Trigonometry paper with a full mark scheme each time you click.
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Meet Riyaz →Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in