School / IB / MATH AA HL / Geometry and trigonometry Question practice
Geometry and trigonometry About this practice HL-only depth: the vector cross and scalar triple products, the geometry of lines and planes in three dimensions, and compound and double angle identities.
Mathematics AA: Higher Level Geometry and trigonometry
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1 Vectors a = (1, 2, 3) and b = (0, 1, 4) form two sides of a parallelogram. Find the area of the parallelogram. Paper 2 style Medium 4 marks Calculator + 2 Marking analysis: A learner attempts the following task: “Vectors a = (1, 2, 3) and b = (0, 1, 4) form two sides of a parallelogram. Find the area of the parallelogram.” Their response addresses only this point: “States that the area of the parallelogram equals |a × b|.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks Calculator + 3 Vectors a = (2, 1, 0), b = (1, 3, 0) and c = (0, 0, 4) define the edges of a parallelepiped from a common vertex. Find its volume using the scalar triple product a · (b × c). Paper 2 style Medium 4 marks Calculator + 4 Marking analysis: A learner attempts the following task: “Vectors a = (2, 1, 0), b = (1, 3, 0) and c = (0, 0, 4) define the edges of a parallelepiped from a common vertex. Find its volume using the scalar triple product a · (b × c).” Their response addresses only this point: “Calculates b × c = (3(4) − 0(0), 0(0) − 1(4), 1(0) − 3(0)) = (12, −4, 0).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks Calculator + 5 A line has direction vector d = (1, 2, 2). A plane has normal vector n = (2, −1, 2). Find the angle between the line and the plane. Paper 2 style Medium 5 marks Calculator + 6 Marking analysis: A learner attempts the following task: “A line has direction vector d = (1, 2, 2). A plane has normal vector n = (2, −1, 2). Find the angle between the line and the plane.” Their response addresses only this point: “States that the angle θ between a line and a plane satisfies sin θ = |d · n| / (|d||n|).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + 7 A line has vector equation r = (1, 0, 2) + t(1, 1, −1). Find the point where this line intersects the plane x + 2y − z = 3. Paper 1 style Medium 5 marks No calculator + 8 Marking analysis: A learner attempts the following task: “A line has vector equation r = (1, 0, 2) + t(1, 1, −1). Find the point where this line intersects the plane x + 2y − z = 3.” Their response addresses only this point: “Writes the parametric coordinates of the line: x = 1 + t, y = t, z = 2 − t.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 9 Solve sin(x + 30°) = 0.5 for 0° ≤ x < 360°. Paper 1 style Medium 5 marks No calculator + 10 Marking analysis: A learner attempts the following task: “Solve sin(x + 30°) = 0.5 for 0° ≤ x < 360°.” Their response addresses only this point: “States the general solutions of sin θ = 0.5 as θ = 30° or θ = 150° (plus multiples of 360°).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 11 Solve cos(2x) = cos(x) for 0° ≤ x < 360°. Paper 1 style Hard 6 marks No calculator + 12 Marking analysis: A learner attempts the following task: “Solve cos(2x) = cos(x) for 0° ≤ x < 360°.” Their response addresses only this point: “Uses the double angle identity cos(2x) = 2cos²x − 1.” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Hard 6 marks No calculator + 13 Find the distance from the point (1, 2, 3) to the plane 2x − y + 2z = 5. Paper 2 style Medium 5 marks Calculator + 14 Marking analysis: A learner attempts the following task: “Find the distance from the point (1, 2, 3) to the plane 2x − y + 2z = 5.” Their response addresses only this point: “States the point-to-plane distance formula d = |ax₀ + by₀ + cz₀ − k| / √(a² + b² + c²) for a plane ax + by + cz = k.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + 15 Prove the identity sec θ − cos θ = sin θ tan θ. Paper 1 style Medium 5 marks No calculator + 16 Marking analysis: A learner attempts the following task: “Prove the identity sec θ − cos θ = sin θ tan θ.” Their response addresses only this point: “Rewrites sec θ as 1/cos θ on the left-hand side.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 17 Given that cos A = 3/5 with A acute, and sin B = 5/13 with B acute, find the exact value of sin(A + B). Paper 1 style Medium 5 marks No calculator + 18 Marking analysis: A learner attempts the following task: “Given that cos A = 3/5 with A acute, and sin B = 5/13 with B acute, find the exact value of sin(A + B).” Their response addresses only this point: “Finds sin A = 4/5, using the Pythagorean triple (3, 4, 5).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 19 Find the angle between the planes x + 2y − z = 3 and 2x − y + z = 5, using their normal vectors. Paper 2 style Medium 5 marks Calculator + 20 Marking analysis: A learner attempts the following task: “Find the angle between the planes x + 2y − z = 3 and 2x − y + z = 5, using their normal vectors.” Their response addresses only this point: “Identifies the normal vectors n₁ = (1, 2, −1) and n₂ = (2, −1, 1) from the plane equations.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + 21 Find a vector equation of the line passing through the points A(1, 2, −1) and B(3, −1, 2). Paper 1 style Easy 3 marks No calculator + 22 Marking analysis: A learner attempts the following task: “Find a vector equation of the line passing through the points A(1, 2, −1) and B(3, −1, 2).” Their response addresses only this point: “Calculates the direction vector d = B − A = (2, −3, 3).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Easy 3 marks No calculator + 23 Find the shortest distance from the point P(1, 0, 2) to the line with vector equation r = (0, 1, 0) + t(1, 1, 0). Paper 2 style Medium 5 marks Calculator + 24 Marking analysis: A learner attempts the following task: “Find the shortest distance from the point P(1, 0, 2) to the line with vector equation r = (0, 1, 0) + t(1, 1, 0).” Their response addresses only this point: “Identifies a point on the line A = (0, 1, 0) and its direction vector d = (1, 1, 0).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + 25 Express 3 sin x + 4 cos x in the form R sin(x + α), where R > 0 and 0° < α < 90°, giving R exactly and α to 1 decimal place. Paper 2 style Medium 5 marks Calculator + 26 Marking analysis: A learner attempts the following task: “Express 3 sin x + 4 cos x in the form R sin(x + α), where R > 0 and 0° < α < 90°, giving R exactly and α to 1 decimal place.” Their response addresses only this point: “States R = √(a² + b²), with a = 3, b = 4.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + 27 Solve sec x = 2 for 0° ≤ x < 360°. Paper 1 style Easy 3 marks No calculator + 28 Marking analysis: A learner attempts the following task: “Solve sec x = 2 for 0° ≤ x < 360°.” Their response addresses only this point: “States that sec x = 1/cos x, so cos x = 1/2.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Easy 3 marks No calculator + 29 Find the angle between the vectors a = (3, 4, 0) and b = (1, 0, 1), giving your answer to 1 decimal place. Paper 2 style Medium 4 marks Calculator + 30 Marking analysis: A learner attempts the following task: “Find the angle between the vectors a = (3, 4, 0) and b = (1, 0, 1), giving your answer to 1 decimal place.” Their response addresses only this point: “Calculates a · b = 3(1) + 4(0) + 0(1) = 3.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks Calculator + Self-assessed Not marked yet
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