School / IB / MATH AA HL / Geometry and trigonometry Exam-style + marking analysis
Geometry and trigonometry 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.
16 activities ≈ 52 minutes
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. 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 No 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). 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 No 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. 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 No 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. 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°. 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°. 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. 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 No calculator + 15 Prove the identity sec θ − cos θ = sin θ tan θ. 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 + Self-assessed 0 / 0
Set total 78
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