10 free IB Mathematics: Analysis & Approaches Geometry & Trigonometry questions written in the style of IB Mathematics: Analysis & Approaches 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, Sine & cosine rules, Radians, arcs & sectors, Trig identities & exact values, Trig equations & graphs, Compound angles & reciprocal trig, Vectors, lines & planes. Work through the questions below, then generate a full Geometry & Trigonometry paper with a fresh set every time.
Generate a full Geometry & Trigonometry paper →In triangle ABC, BC = 8.2 cm, AC = 10 cm and BÂC = 40°.
Find the two possible values of AB̂C.
Find the difference between the areas of the two possible triangles.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 51.6° or 128° sin AB̂C10 = sin 40°8.2 (M1)(A1) 51.6° A1 180° − 51.6° = 128° A1 | 4 |
| (b) | 32.7 cm2 AĈB = 88.4° or 11.6° (A1) 12(8.2)(10)sin C for each (M1) 32.7 cm2 A1 | 3 |
A sector of a circle with radius 18 cm and angle 4 radians is folded so that its two straight edges meet, forming the curved surface of a cone.
Find the base radius of the cone.
Find the volume of the cone.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 11.5 cm arc length = 18 × 4 = 72.0 (A1) arc length = circumference of base M1 r = 11.5 cm A1 | 3 |
| (b) | 1909 cm3 h = √182 − r2 = 13.9 (M1)(A1) 1909 cm3 A1 | 3 |
Consider the equation 3sin(2x) − 1 = k, for 0 ≤ x ≤ π.
Find the set of values of k for which the equation has exactly two solutions.
| Part | Answer and marking points | Marks |
|---|---|---|
| −4 < k < −1, −1 < k < 2 sketch of y = 3sin(2x) − 1 over one full period M1 range from −4 to 2 A1 k = −1 gives three solutions (x = 0, π2, π) R1 k = 2 or k = −4 gives one solution R1 −4 < k < −1 or −1 < k < 2 A1 | 5 |
A ship leaves port P and sails 18 km on a bearing of 070° to point A. It then sails 8 km on a bearing of 165° to point B.
Find the size of angle PÂB.
Find the distance PB.
Find the bearing of P from B.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 85° use of parallel north lines (M1) 85° A1 | 2 |
| (b) | 19.0 km PB2 = 182 + 82 − 2(18)(8)cos 85° (M1)(A1) 19.0 km A1 | 3 |
| (c) | 275° sine rule or coordinates to find an angle at B or P (M1) correct angle (A1) 274.7° A1 | 3 |
A sector has radius 5 cm and perimeter 40 cm.
Find the angle of the sector in radians.
Find the area of the segment formed by the chord joining the ends of the arc.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 6.00 2(5) + 5θ = 40 (M1) 6.00 A1 | 2 |
| (b) | 78.5 cm2 sector area (M1) minus triangle area (M1) 78.5 A1 | 3 |
Let sin θ = 725, where π2 < θ < π.
Find the exact value of cos θ.
Find the exact value of sin 2θ.
Find the exact value of tan 2θ.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | −2425 use of sin2θ + cos2θ = 1 (M1) cos θ = −2425 (negative in the second quadrant) A1 | 2 |
| (b) | −336625 2(725)(−2425) (M1) −336625 A1 | 2 |
| (c) | −336527 cos 2θ = 527625 (M1)(A1) tan 2θ = −336527 A1 | 3 |
The height h metres of a seat on a Ferris wheel above the ground, t minutes after the ride starts, is h(t) = 35 − 18cos(π6t), t ≥ 0.
Write down the maximum height of the seat and the time for one full revolution.
Find the height of the seat after 3 minutes.
Find the length of time during one revolution for which the seat is more than 39 m above the ground.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 53 m; 12 minutes 53 m A1 12 minutes A1 | 2 |
| (b) | 35.0 m h(3) = 35 − 18cos(π2) (M1) 35.0 m A1 | 2 |
| (c) | 5.14 minutes 35 − 18cos(π6t) = 39 (M1) t = 3.43, t = 8.57 (A1)(A1) 5.14 minutes A1 | 4 |
Show that cos4x − sin4x = cos 2x.
| Part | Answer and marking points | Marks |
|---|---|---|
| shown difference of squares: (cos2x − sin2x)(cos2x + sin2x) M1 cos2x + sin2x = 1 A1 cos2x − sin2x = cos 2x AG A1 | 3 |
In triangle ABC, BC = 7 cm, AC = 11 cm and angle ACB = 40°.
Find AB.
Find the area of the triangle.
Find angle ABC.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 7.21 cm cosine rule (M1) correct substitution (A1) 7.21 cm A1 | 3 |
| (b) | 24.7 cm2 ½ab sin C (M1) 24.7 cm2 A1 | 2 |
| (c) | 78.6° sine rule (M1) 78.6° A1 | 2 |
The plane Π has equation −x − 2y + 2z = −13. The line L has equation r = 313−4 + λ−2−30.
Find the coordinates of the point where L meets Π.
Find the shortest distance from the point (0, 0, 0) to the plane Π.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | (−3, 4, −4) substituting the parametric form of L into the equation of Π M1 −37 + 8λ = −13 A1 λ = 3 A1 (−3, 4, −4) A1 | 4 |
| (b) | 13√9 distance = |d| ÷ |n| for the plane r·n = d (M1) |n| = √9 (A1) 13√9 A1 | 3 |
In the SL & HL syllabus, Geometry & Trigonometry covers 3D geometry, Sine & cosine rules, Radians, arcs & sectors, Trig identities & exact values, Trig equations & graphs, Compound angles & reciprocal trig, Vectors, lines & planes.
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