Mathematics AA: Higher Level
Geometry and trigonometry — Topic 3
- 1.
Vectors a = (1, 2, 3) and b = (0, 1, 4) form two sides of a parallelogram. Find the area of the parallelogram.
[4 marks] - 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.
[4 marks] - 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).
[4 marks] - 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.
[4 marks] - 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.
[5 marks] - 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.
[5 marks] - 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.
[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.
[5 marks] · no calculator - 9.
Solve sin(x + 30°) = 0.5 for 0° ≤ x < 360°.
[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.
[5 marks] · no calculator - 11.
Solve cos(2x) = cos(x) for 0° ≤ x < 360°.
[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.
[6 marks] · no calculator - 13.
Find the distance from the point (1, 2, 3) to the plane 2x − y + 2z = 5.
[5 marks] - 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.
[5 marks] - 15.
Prove the identity sec θ − cos θ = sin θ tan θ.
[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.
[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).
[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.
[5 marks] · no calculator - 19.
Find the angle between the planes x + 2y − z = 3 and 2x − y + z = 5, using their normal vectors.
[5 marks] - 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.
[5 marks] - 21.
Find a vector equation of the line passing through the points A(1, 2, −1) and B(3, −1, 2).
[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.
[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).
[5 marks] - 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.
[5 marks] - 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.
[5 marks] - 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.
[5 marks] - 27.
Solve sec x = 2 for 0° ≤ x < 360°.
[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.
[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.
[4 marks] - 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.
[4 marks]