Mathematics: Applications & Interpretation SL
Geometry and trigonometry — Topic 3
- 1.
A ladder of length 10 m leans against a vertical wall, making an angle of 65° with the horizontal ground. Calculate the height the ladder reaches up the wall.
[2 marks] - 2.
Marking analysis: A learner attempts the following task: “A ladder of length 10 m leans against a vertical wall, making an angle of 65° with the horizontal ground. Calculate the height the ladder reaches up the wall.” Their response addresses only this point: “Uses height = 10 × sin(65°).” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] - 3.
A building is 50 m tall. An observer stands 120 m from the base of the building on level ground. Calculate the angle of elevation from the observer to the top of the building.
[2 marks] - 4.
Marking analysis: A learner attempts the following task: “A building is 50 m tall. An observer stands 120 m from the base of the building on level ground. Calculate the angle of elevation from the observer to the top of the building.” Their response addresses only this point: “Uses tan(angle) = 50/120.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] - 5.
In triangle ABC, angle A = 40°, angle B = 65°, and side a = 12 cm. Use the sine rule to find the length of side b.
[3 marks] - 6.
Marking analysis: A learner attempts the following task: “In triangle ABC, angle A = 40°, angle B = 65°, and side a = 12 cm. Use the sine rule to find the length of side b.” Their response addresses only this point: “Writes the sine rule a/sin(A) = b/sin(B).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 7.
In triangle ABC, b = 8 cm, c = 10 cm, and the angle between them A = 55°. Use the cosine rule to find the length of side a.
[3 marks] - 8.
Marking analysis: A learner attempts the following task: “In triangle ABC, b = 8 cm, c = 10 cm, and the angle between them A = 55°. Use the cosine rule to find the length of side a.” Their response addresses only this point: “Writes the cosine rule a² = b² + c² − 2bc·cos(A).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 9.
In triangle ABC, a = 7 cm, b = 9 cm, and c = 12 cm. Use the cosine rule to find the size of angle C, the angle opposite the longest side.
[3 marks] - 10.
Marking analysis: A learner attempts the following task: “In triangle ABC, a = 7 cm, b = 9 cm, and c = 12 cm. Use the cosine rule to find the size of angle C, the angle opposite the longest side.” Their response addresses only this point: “Rearranges the cosine rule to cos(C) = (a² + b² − c²)/(2ab).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 11.
Triangle ABC has sides a = 10 cm and b = 14 cm, with the included angle C = 38°. Calculate the area of the triangle.
[2 marks] - 12.
Marking analysis: A learner attempts the following task: “Triangle ABC has sides a = 10 cm and b = 14 cm, with the included angle C = 38°. Calculate the area of the triangle.” Their response addresses only this point: “Uses the area formula Area = (1/2)ab·sin(C).” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] - 13.
A ship sails 50 km from port A on a bearing of 070° to reach point B, then sails 70 km on a bearing of 140° to reach point C. Calculate the straight-line distance from A to C.
[4 marks] - 14.
Marking analysis: A learner attempts the following task: “A ship sails 50 km from port A on a bearing of 070° to reach point B, then sails 70 km on a bearing of 140° to reach point C. Calculate the straight-line distance from A to C.” Their response addresses only this point: “Recognises that the interior angle at B between BA and BC is 180° − (140° − 70°) = 110°.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 15.
A boat sails from point A to point B, travelling 40 km east and 30 km north. Calculate the bearing of B from A.
[3 marks] - 16.
Marking analysis: A learner attempts the following task: “A boat sails from point A to point B, travelling 40 km east and 30 km north. Calculate the bearing of B from A.” Their response addresses only this point: “Sketches the displacement and identifies the angle from north as arctan(east/north).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 17.
A cone has base radius 6 cm and height 8 cm. (a) Find the slant height of the cone. (b) Find the angle between the slant and the base.
[4 marks] - 18.
Marking analysis: A learner attempts the following task: “A cone has base radius 6 cm and height 8 cm. (a) Find the slant height of the cone. (b) Find the angle between the slant and the base.” Their response addresses only this point: “Uses Pythagoras' theorem: slant² = 6² + 8².” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 19.
A cuboid has length 5 cm, width 7 cm, and height 9 cm. (a) Find the length of the space diagonal of the cuboid. (b) Find the angle this diagonal makes with the base.
[4 marks] - 20.
Marking analysis: A learner attempts the following task: “A cuboid has length 5 cm, width 7 cm, and height 9 cm. (a) Find the length of the space diagonal of the cuboid. (b) Find the angle this diagonal makes with the base.” Their response addresses only this point: “Finds the base diagonal first: √(5² + 7²) ≈ 8.60 cm.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 21.
A circle has radius 15 cm. Find the length of the arc subtended by an angle of 50° at the centre.
[3 marks] - 22.
Marking analysis: A learner attempts the following task: “A circle has radius 15 cm. Find the length of the arc subtended by an angle of 50° at the centre.” Their response addresses only this point: “Converts 50° to radians: 50 × π/180 ≈ 0.873 radians.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 23.
A circular sector has radius 9 cm and central angle 120°. Find the area of the sector.
[3 marks] - 24.
Marking analysis: A learner attempts the following task: “A circular sector has radius 9 cm and central angle 120°. Find the area of the sector.” Their response addresses only this point: “Converts 120° to radians: 120 × π/180 ≈ 2.094 radians.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 25.
Vectors a = (3, 4) and b = (−1, 2) represent two displacements. (a) Find the resultant vector a + b. (b) Find the magnitude of the resultant and the angle it makes with the positive x-axis.
[4 marks] - 26.
Marking analysis: A learner attempts the following task: “Vectors a = (3, 4) and b = (−1, 2) represent two displacements. (a) Find the resultant vector a + b. (b) Find the magnitude of the resultant and the angle it makes with the positive x-axis.” Their response addresses only this point: “Adds the vectors component-wise: a + b = (3 + (−1), 4 + 2) = (2, 6).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 27.
Triangle PQR is similar to triangle XYZ, with a scale factor of 1.5 from XYZ to PQR. If side XY = 8 cm, find the length of the corresponding side PQ.
[2 marks] - 28.
Marking analysis: A learner attempts the following task: “Triangle PQR is similar to triangle XYZ, with a scale factor of 1.5 from XYZ to PQR. If side XY = 8 cm, find the length of the corresponding side PQ.” Their response addresses only this point: “Multiplies the given side by the scale factor: PQ = 8 × 1.5.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] - 29.
From the top of a vertical cliff 45 m high, the angle of depression to a boat at sea is 25°. Calculate the horizontal distance from the base of the cliff to the boat.
[3 marks] - 30.
Marking analysis: A learner attempts the following task: “From the top of a vertical cliff 45 m high, the angle of depression to a boat at sea is 25°. Calculate the horizontal distance from the base of the cliff to the boat.” Their response addresses only this point: “Recognises that the angle of depression equals the angle of elevation from the boat, by alternate angles.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 31.
In triangle ABC, a = 9 cm, b = 11 cm, and angle C = 50°. (a) Use the cosine rule to find side c. (b) Use the sine rule to find angle A. (c) Hence find angle B.
[6 marks] - 32.
Marking analysis: A learner attempts the following task: “In triangle ABC, a = 9 cm, b = 11 cm, and angle C = 50°. (a) Use the cosine rule to find side c. (b) Use the sine rule to find angle A. (c) Hence find angle B.” Their response addresses only this point: “Applies the cosine rule: c² = 9² + 11² − 2(9)(11)cos(50°).” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[6 marks]