IB · MATH AA HL

Mathematics AA: Higher Level

Geometry and trigonometry — Topic 3

Name: ____________________Date: October 10, 2026

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]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States that the area of the parallelogram equals |a × b|. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Calculates the cross product a × b = (2(4) − 3(1), 3(0) − 1(4), 1(1) − 2(0)) = (5, −4, 1). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Calculates the magnitude |a × b| = √(5² + (−4)² + 1²) = √42. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: States the area as √42 (≈ 6.48) square units. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The cross product's magnitude gives the area of the parallelogram spanned by the two vectors directly; halve it if the question instead asks for the area of the triangle they form.
Marking points
- States that the area of the parallelogram equals |a × b|.
- Calculates the cross product a × b = (2(4) − 3(1), 3(0) − 1(4), 1(1) − 2(0)) = (5, −4, 1).
- Calculates the magnitude |a × b| = √(5² + (−4)² + 1²) = √42.
- States the area as √42 (≈ 6.48) square units.
Examiner tip: The cross product's magnitude gives the area of the parallelogram spanned by the two vectors directly; halve it if the question instead asks for the area of the triangle they form.
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]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that the area of the parallelogram equals |a × b|. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Calculates the cross product a × b = (2(4) − 3(1), 3(0) − 1(4), 1(1) − 2(0)) = (5, −4, 1). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Calculates the magnitude |a × b| = √(5² + (−4)² + 1²) = √42. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: States the area as √42 (≈ 6.48) square units. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that the area of the parallelogram equals |a × b|.
- Identifies the missing requirement: Calculates the cross product a × b = (2(4) − 3(1), 3(0) − 1(4), 1(1) − 2(0)) = (5, −4, 1).
- Identifies the missing requirement: Calculates the magnitude |a × b| = √(5² + (−4)² + 1²) = √42.
- Identifies the missing requirement: States the area as √42 (≈ 6.48) square units.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
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]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Calculates b × c = (3(4) − 0(0), 0(0) − 1(4), 1(0) − 3(0)) = (12, −4, 0). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Calculates the dot product a · (b × c) = 2(12) + 1(−4) + 0(0). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Obtains a · (b × c) = 20. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: States the volume as the absolute value of the scalar triple product, 20 cubic units. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The scalar triple product can be negative depending on the order of the vectors, but volume is always the absolute value of the result — never leave a negative sign in a final volume answer.
Marking points
- Calculates b × c = (3(4) − 0(0), 0(0) − 1(4), 1(0) − 3(0)) = (12, −4, 0).
- Calculates the dot product a · (b × c) = 2(12) + 1(−4) + 0(0).
- Obtains a · (b × c) = 20.
- States the volume as the absolute value of the scalar triple product, 20 cubic units.
Examiner tip: The scalar triple product can be negative depending on the order of the vectors, but volume is always the absolute value of the result — never leave a negative sign in a final volume answer.
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]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Calculates b × c = (3(4) − 0(0), 0(0) − 1(4), 1(0) − 3(0)) = (12, −4, 0). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Calculates the dot product a · (b × c) = 2(12) + 1(−4) + 0(0). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Obtains a · (b × c) = 20. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: States the volume as the absolute value of the scalar triple product, 20 cubic units. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Calculates b × c = (3(4) − 0(0), 0(0) − 1(4), 1(0) − 3(0)) = (12, −4, 0).
- Identifies the missing requirement: Calculates the dot product a · (b × c) = 2(12) + 1(−4) + 0(0).
- Identifies the missing requirement: Obtains a · (b × c) = 20.
- Identifies the missing requirement: States the volume as the absolute value of the scalar triple product, 20 cubic units.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
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]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States that the angle θ between a line and a plane satisfies sin θ = |d · n| / (|d||n|). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Calculates d · n = 1(2) + 2(−1) + 2(2) = 4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Calculates |d| = 3 and |n| = 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Obtains sin θ = 4/9. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: States θ ≈ 26.4°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The angle between a line and a plane uses sine and the line's direction vector with the plane's normal — this is easy to confuse with the angle between two planes, which uses cosine and the two normal vectors directly.
Marking points
- States that the angle θ between a line and a plane satisfies sin θ = |d · n| / (|d||n|).
- Calculates d · n = 1(2) + 2(−1) + 2(2) = 4.
- Calculates |d| = 3 and |n| = 3.
- Obtains sin θ = 4/9.
- States θ ≈ 26.4°.
Examiner tip: The angle between a line and a plane uses sine and the line's direction vector with the plane's normal — this is easy to confuse with the angle between two planes, which uses cosine and the two normal vectors directly.
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]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that the angle θ between a line and a plane satisfies sin θ = |d · n| / (|d||n|). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Calculates d · n = 1(2) + 2(−1) + 2(2) = 4. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Calculates |d| = 3 and |n| = 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Obtains sin θ = 4/9. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: States θ ≈ 26.4°. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that the angle θ between a line and a plane satisfies sin θ = |d · n| / (|d||n|).
- Identifies the missing requirement: Calculates d · n = 1(2) + 2(−1) + 2(2) = 4.
- Identifies the missing requirement: Calculates |d| = 3 and |n| = 3.
- Identifies the missing requirement: Obtains sin θ = 4/9.
- Identifies the missing requirement: States θ ≈ 26.4°.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
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
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Writes the parametric coordinates of the line: x = 1 + t, y = t, z = 2 − t. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Substitutes these into the plane equation: (1 + t) + 2(t) − (2 − t) = 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Simplifies to obtain 4t − 1 = 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Solves to find t = 1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Substitutes t = 1 back into the line to obtain the point of intersection (2, 1, 1). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Always substitute the parametric coordinates into the Cartesian plane equation, solve for the single parameter t, then substitute back — a common error is stopping at the value of t instead of finding the actual point.
Marking points
- Writes the parametric coordinates of the line: x = 1 + t, y = t, z = 2 − t.
- Substitutes these into the plane equation: (1 + t) + 2(t) − (2 − t) = 3.
- Simplifies to obtain 4t − 1 = 3.
- Solves to find t = 1.
- Substitutes t = 1 back into the line to obtain the point of intersection (2, 1, 1).
Examiner tip: Always substitute the parametric coordinates into the Cartesian plane equation, solve for the single parameter t, then substitute back — a common error is stopping at the value of t instead of finding the actual point.
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
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Writes the parametric coordinates of the line: x = 1 + t, y = t, z = 2 − t. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Substitutes these into the plane equation: (1 + t) + 2(t) − (2 − t) = 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Simplifies to obtain 4t − 1 = 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Solves to find t = 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: Substitutes t = 1 back into the line to obtain the point of intersection (2, 1, 1). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Writes the parametric coordinates of the line: x = 1 + t, y = t, z = 2 − t.
- Identifies the missing requirement: Substitutes these into the plane equation: (1 + t) + 2(t) − (2 − t) = 3.
- Identifies the missing requirement: Simplifies to obtain 4t − 1 = 3.
- Identifies the missing requirement: Solves to find t = 1.
- Identifies the missing requirement: Substitutes t = 1 back into the line to obtain the point of intersection (2, 1, 1).
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
9.
Solve sin(x + 30°) = 0.5 for 0° ≤ x < 360°.
[5 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States the general solutions of sin θ = 0.5 as θ = 30° or θ = 150° (plus multiples of 360°). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Sets x + 30° = 30° and x + 30° = 150° as the two cases within one full rotation. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Solves the first case to obtain x = 0°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Solves the second case to obtain x = 120°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Checks that no further solutions fall within 0° ≤ x < 360° and states the final answers x = 0°, 120°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Shift the interval for the compound angle (here x + 30°) to match the substitution before listing general solutions, then convert back to x at the very end.
Marking points
- States the general solutions of sin θ = 0.5 as θ = 30° or θ = 150° (plus multiples of 360°).
- Sets x + 30° = 30° and x + 30° = 150° as the two cases within one full rotation.
- Solves the first case to obtain x = 0°.
- Solves the second case to obtain x = 120°.
- Checks that no further solutions fall within 0° ≤ x < 360° and states the final answers x = 0°, 120°.
Examiner tip: Shift the interval for the compound angle (here x + 30°) to match the substitution before listing general solutions, then convert back to x at the very end.
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
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States the general solutions of sin θ = 0.5 as θ = 30° or θ = 150° (plus multiples of 360°). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Sets x + 30° = 30° and x + 30° = 150° as the two cases within one full rotation. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Solves the first case to obtain x = 0°. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Solves the second case to obtain x = 120°. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: Checks that no further solutions fall within 0° ≤ x < 360° and states the final answers x = 0°, 120°. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States the general solutions of sin θ = 0.5 as θ = 30° or θ = 150° (plus multiples of 360°).
- Identifies the missing requirement: Sets x + 30° = 30° and x + 30° = 150° as the two cases within one full rotation.
- Identifies the missing requirement: Solves the first case to obtain x = 0°.
- Identifies the missing requirement: Solves the second case to obtain x = 120°.
- Identifies the missing requirement: Checks that no further solutions fall within 0° ≤ x < 360° and states the final answers x = 0°, 120°.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
11.
Solve cos(2x) = cos(x) for 0° ≤ x < 360°.
[6 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Uses the double angle identity cos(2x) = 2cos²x − 1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Substitutes to obtain 2cos²x − 1 = cos x. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Rearranges into a quadratic in cos x: 2cos²x − cos x − 1 = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Factorises to (2cos x + 1)(cos x − 1) = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Solves cos x = 1 to obtain x = 0°, and cos x = −1/2 to obtain x = 120° or x = 240°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Work through this mathematical step: States the complete solution set x = 0°, 120°, 240°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
8. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Whenever an equation mixes cos(2x) with cos(x), rewrite cos(2x) using a double angle identity to turn it into a single quadratic in cos x rather than trying to solve it directly.
Marking points
- Uses the double angle identity cos(2x) = 2cos²x − 1.
- Substitutes to obtain 2cos²x − 1 = cos x.
- Rearranges into a quadratic in cos x: 2cos²x − cos x − 1 = 0.
- Factorises to (2cos x + 1)(cos x − 1) = 0.
- Solves cos x = 1 to obtain x = 0°, and cos x = −1/2 to obtain x = 120° or x = 240°.
- States the complete solution set x = 0°, 120°, 240°.
Examiner tip: Whenever an equation mixes cos(2x) with cos(x), rewrite cos(2x) using a double angle identity to turn it into a single quadratic in cos x rather than trying to solve it directly.
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
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Uses the double angle identity cos(2x) = 2cos²x − 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Substitutes to obtain 2cos²x − 1 = cos x. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Rearranges into a quadratic in cos x: 2cos²x − cos x − 1 = 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Factorises to (2cos x + 1)(cos x − 1) = 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: Solves cos x = 1 to obtain x = 0°, and cos x = −1/2 to obtain x = 120° or x = 240°. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Requirement 6: Identifies the missing requirement: States the complete solution set x = 0°, 120°, 240°. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
8. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Uses the double angle identity cos(2x) = 2cos²x − 1.
- Identifies the missing requirement: Substitutes to obtain 2cos²x − 1 = cos x.
- Identifies the missing requirement: Rearranges into a quadratic in cos x: 2cos²x − cos x − 1 = 0.
- Identifies the missing requirement: Factorises to (2cos x + 1)(cos x − 1) = 0.
- Identifies the missing requirement: Solves cos x = 1 to obtain x = 0°, and cos x = −1/2 to obtain x = 120° or x = 240°.
- Identifies the missing requirement: States the complete solution set x = 0°, 120°, 240°.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
13.
Find the distance from the point (1, 2, 3) to the plane 2x − y + 2z = 5.
[5 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States the point-to-plane distance formula d = |ax₀ + by₀ + cz₀ − k| / √(a² + b² + c²) for a plane ax + by + cz = k. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Substitutes the point and plane coefficients: |2(1) − 1(2) + 2(3) − 5| / √(2² + (−1)² + 2²). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Simplifies the numerator to |2 − 2 + 6 − 5| = 1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Simplifies the denominator to √9 = 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: States the distance as 1/3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The plane's normal vector coefficients (a, b, c) come directly from the Cartesian equation — write the plane in the form ax + by + cz = k first if it is given in a different arrangement.
Marking points
- States the point-to-plane distance formula d = |ax₀ + by₀ + cz₀ − k| / √(a² + b² + c²) for a plane ax + by + cz = k.
- Substitutes the point and plane coefficients: |2(1) − 1(2) + 2(3) − 5| / √(2² + (−1)² + 2²).
- Simplifies the numerator to |2 − 2 + 6 − 5| = 1.
- Simplifies the denominator to √9 = 3.
- States the distance as 1/3.
Examiner tip: The plane's normal vector coefficients (a, b, c) come directly from the Cartesian equation — write the plane in the form ax + by + cz = k first if it is given in a different arrangement.
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]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States the point-to-plane distance formula d = |ax₀ + by₀ + cz₀ − k| / √(a² + b² + c²) for a plane ax + by + cz = k. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Substitutes the point and plane coefficients: |2(1) − 1(2) + 2(3) − 5| / √(2² + (−1)² + 2²). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Simplifies the numerator to |2 − 2 + 6 − 5| = 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Simplifies the denominator to √9 = 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: States the distance as 1/3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States the point-to-plane distance formula d = |ax₀ + by₀ + cz₀ − k| / √(a² + b² + c²) for a plane ax + by + cz = k.
- Identifies the missing requirement: Substitutes the point and plane coefficients: |2(1) − 1(2) + 2(3) − 5| / √(2² + (−1)² + 2²).
- Identifies the missing requirement: Simplifies the numerator to |2 − 2 + 6 − 5| = 1.
- Identifies the missing requirement: Simplifies the denominator to √9 = 3.
- Identifies the missing requirement: States the distance as 1/3.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
15.
Prove the identity sec θ − cos θ = sin θ tan θ.
[5 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Work through this mathematical step: Rewrites sec θ as 1/cos θ on the left-hand side. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Combines the two terms over a common denominator: (1 − cos²θ)/cos θ. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Uses the Pythagorean identity 1 − cos²θ = sin²θ. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Rewrites sin²θ/cos θ as sin θ · (sin θ/cos θ). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Identifies sin θ/cos θ as tan θ, obtaining sin θ tan θ, which equals the right-hand side. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Proving a trig identity almost always starts by converting sec, csc, and cot into expressions using only sin and cos, then combining fractions over a common denominator.
Marking points
- Rewrites sec θ as 1/cos θ on the left-hand side.
- Combines the two terms over a common denominator: (1 − cos²θ)/cos θ.
- Uses the Pythagorean identity 1 − cos²θ = sin²θ.
- Rewrites sin²θ/cos θ as sin θ · (sin θ/cos θ).
- Identifies sin θ/cos θ as tan θ, obtaining sin θ tan θ, which equals the right-hand side.
Examiner tip: Proving a trig identity almost always starts by converting sec, csc, and cot into expressions using only sin and cos, then combining fractions over a common denominator.
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
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Rewrites sec θ as 1/cos θ on the left-hand side. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Combines the two terms over a common denominator: (1 − cos²θ)/cos θ. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Uses the Pythagorean identity 1 − cos²θ = sin²θ. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Rewrites sin²θ/cos θ as sin θ · (sin θ/cos θ). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: Identifies sin θ/cos θ as tan θ, obtaining sin θ tan θ, which equals the right-hand side. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Rewrites sec θ as 1/cos θ on the left-hand side.
- Identifies the missing requirement: Combines the two terms over a common denominator: (1 − cos²θ)/cos θ.
- Identifies the missing requirement: Uses the Pythagorean identity 1 − cos²θ = sin²θ.
- Identifies the missing requirement: Rewrites sin²θ/cos θ as sin θ · (sin θ/cos θ).
- Identifies the missing requirement: Identifies sin θ/cos θ as tan θ, obtaining sin θ tan θ, which equals the right-hand side.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
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
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Finds sin A = 4/5, using the Pythagorean triple (3, 4, 5). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Finds cos B = 12/13, using the Pythagorean triple (5, 12, 13). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: States the compound angle formula sin(A + B) = sin A cos B + cos A sin B. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Substitutes (4/5)(12/13) + (3/5)(5/13). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Obtains sin(A + B) = 63/65. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Recognising 3-4-5 and 5-12-13 as Pythagorean triples immediately gives the missing side ratio without needing a calculator or square roots — a pattern worth memorising.
Marking points
- Finds sin A = 4/5, using the Pythagorean triple (3, 4, 5).
- Finds cos B = 12/13, using the Pythagorean triple (5, 12, 13).
- States the compound angle formula sin(A + B) = sin A cos B + cos A sin B.
- Substitutes (4/5)(12/13) + (3/5)(5/13).
- Obtains sin(A + B) = 63/65.
Examiner tip: Recognising 3-4-5 and 5-12-13 as Pythagorean triples immediately gives the missing side ratio without needing a calculator or square roots — a pattern worth memorising.
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
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Finds sin A = 4/5, using the Pythagorean triple (3, 4, 5). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Finds cos B = 12/13, using the Pythagorean triple (5, 12, 13). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: States the compound angle formula sin(A + B) = sin A cos B + cos A sin B. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Substitutes (4/5)(12/13) + (3/5)(5/13). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: Obtains sin(A + B) = 63/65. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Finds sin A = 4/5, using the Pythagorean triple (3, 4, 5).
- Identifies the missing requirement: Finds cos B = 12/13, using the Pythagorean triple (5, 12, 13).
- Identifies the missing requirement: States the compound angle formula sin(A + B) = sin A cos B + cos A sin B.
- Identifies the missing requirement: Substitutes (4/5)(12/13) + (3/5)(5/13).
- Identifies the missing requirement: Obtains sin(A + B) = 63/65.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
19.
Find the angle between the planes x + 2y − z = 3 and 2x − y + z = 5, using their normal vectors.
[5 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Identifies the normal vectors n₁ = (1, 2, −1) and n₂ = (2, −1, 1) from the plane equations. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Calculates n₁ · n₂ = 1(2) + 2(−1) + (−1)(1) = −1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Calculates |n₁| = √6 and |n₂| = √6. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Uses cos θ = |n₁ · n₂| / (|n₁||n₂|) = 1/6. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Obtains θ ≈ 80.4°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The angle between two planes uses the cosine formula applied directly to their normal vectors — this is the counterpart to the sine formula used for the angle between a line and a plane.
Marking points
- Identifies the normal vectors n₁ = (1, 2, −1) and n₂ = (2, −1, 1) from the plane equations.
- Calculates n₁ · n₂ = 1(2) + 2(−1) + (−1)(1) = −1.
- Calculates |n₁| = √6 and |n₂| = √6.
- Uses cos θ = |n₁ · n₂| / (|n₁||n₂|) = 1/6.
- Obtains θ ≈ 80.4°.
Examiner tip: The angle between two planes uses the cosine formula applied directly to their normal vectors — this is the counterpart to the sine formula used for the angle between a line and a plane.
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]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Identifies the normal vectors n₁ = (1, 2, −1) and n₂ = (2, −1, 1) from the plane equations. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Calculates n₁ · n₂ = 1(2) + 2(−1) + (−1)(1) = −1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Calculates |n₁| = √6 and |n₂| = √6. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Uses cos θ = |n₁ · n₂| / (|n₁||n₂|) = 1/6. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: Obtains θ ≈ 80.4°. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Identifies the normal vectors n₁ = (1, 2, −1) and n₂ = (2, −1, 1) from the plane equations.
- Identifies the missing requirement: Calculates n₁ · n₂ = 1(2) + 2(−1) + (−1)(1) = −1.
- Identifies the missing requirement: Calculates |n₁| = √6 and |n₂| = √6.
- Identifies the missing requirement: Uses cos θ = |n₁ · n₂| / (|n₁||n₂|) = 1/6.
- Identifies the missing requirement: Obtains θ ≈ 80.4°.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
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
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Calculates the direction vector d = B − A = (2, −3, 3). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Uses point A = (1, 2, −1) as a fixed point on the line. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Writes the vector equation r = (1, 2, −1) + t(2, −3, 3), t ∈ ℝ. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Any point on the line can be used as the fixed point, and either A→B or B→A can be used as the direction vector — different valid vector equations can describe exactly the same line.
Marking points
- Calculates the direction vector d = B − A = (2, −3, 3).
- Uses point A = (1, 2, −1) as a fixed point on the line.
- Writes the vector equation r = (1, 2, −1) + t(2, −3, 3), t ∈ ℝ.
Examiner tip: Any point on the line can be used as the fixed point, and either A→B or B→A can be used as the direction vector — different valid vector equations can describe exactly the same line.
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
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Calculates the direction vector d = B − A = (2, −3, 3). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Uses point A = (1, 2, −1) as a fixed point on the line. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Writes the vector equation r = (1, 2, −1) + t(2, −3, 3), t ∈ ℝ. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Calculates the direction vector d = B − A = (2, −3, 3).
- Identifies the missing requirement: Uses point A = (1, 2, −1) as a fixed point on the line.
- Identifies the missing requirement: Writes the vector equation r = (1, 2, −1) + t(2, −3, 3), t ∈ ℝ.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
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]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Identifies a point on the line A = (0, 1, 0) and its direction vector d = (1, 1, 0). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Calculates the vector AP = P − A = (1, −1, 2). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Calculates the cross product AP × d = (−2, 2, 2). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Uses distance = |AP × d| / |d|. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Obtains distance = √6 (≈ 2.45 units). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: This cross-product distance formula works because |AP × d| gives the area of the parallelogram spanned by AP and d, and dividing by the base length |d| converts that area into the perpendicular height — exactly the shortest distance to the line.
Marking points
- Identifies a point on the line A = (0, 1, 0) and its direction vector d = (1, 1, 0).
- Calculates the vector AP = P − A = (1, −1, 2).
- Calculates the cross product AP × d = (−2, 2, 2).
- Uses distance = |AP × d| / |d|.
- Obtains distance = √6 (≈ 2.45 units).
Examiner tip: This cross-product distance formula works because |AP × d| gives the area of the parallelogram spanned by AP and d, and dividing by the base length |d| converts that area into the perpendicular height — exactly the shortest distance to the line.
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]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Identifies a point on the line A = (0, 1, 0) and its direction vector d = (1, 1, 0). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Calculates the vector AP = P − A = (1, −1, 2). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Calculates the cross product AP × d = (−2, 2, 2). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Uses distance = |AP × d| / |d|. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: Obtains distance = √6 (≈ 2.45 units). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Identifies a point on the line A = (0, 1, 0) and its direction vector d = (1, 1, 0).
- Identifies the missing requirement: Calculates the vector AP = P − A = (1, −1, 2).
- Identifies the missing requirement: Calculates the cross product AP × d = (−2, 2, 2).
- Identifies the missing requirement: Uses distance = |AP × d| / |d|.
- Identifies the missing requirement: Obtains distance = √6 (≈ 2.45 units).
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
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]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Work through this mathematical step: States R = √(a² + b²), with a = 3, b = 4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Obtains R = 5. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: States that tan α = b/a = 4/3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Obtains α ≈ 53.1°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Writes 3 sin x + 4 cos x = 5 sin(x + 53.1°). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: This R sin(x + α) form is especially useful for finding the maximum and minimum values of an expression like a sin x + b cos x directly, since R is exactly that maximum value (and −R the minimum).
Marking points
- States R = √(a² + b²), with a = 3, b = 4.
- Obtains R = 5.
- States that tan α = b/a = 4/3.
- Obtains α ≈ 53.1°.
- Writes 3 sin x + 4 cos x = 5 sin(x + 53.1°).
Examiner tip: This R sin(x + α) form is especially useful for finding the maximum and minimum values of an expression like a sin x + b cos x directly, since R is exactly that maximum value (and −R the minimum).
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]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States R = √(a² + b²), with a = 3, b = 4. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Obtains R = 5. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: States that tan α = b/a = 4/3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Obtains α ≈ 53.1°. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: Writes 3 sin x + 4 cos x = 5 sin(x + 53.1°). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States R = √(a² + b²), with a = 3, b = 4.
- Identifies the missing requirement: Obtains R = 5.
- Identifies the missing requirement: States that tan α = b/a = 4/3.
- Identifies the missing requirement: Obtains α ≈ 53.1°.
- Identifies the missing requirement: Writes 3 sin x + 4 cos x = 5 sin(x + 53.1°).
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
27.
Solve sec x = 2 for 0° ≤ x < 360°.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States that sec x = 1/cos x, so cos x = 1/2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: States the general solutions where cos x = 1/2 within one rotation: x = 60° and x = 360° − 60° = 300°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: States the final answers x = 60°, 300°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Always convert reciprocal trig functions (sec, csc, cot) back to sin, cos or tan before solving — this avoids errors and lets you apply the familiar unit-circle solution pattern directly.
Marking points
- States that sec x = 1/cos x, so cos x = 1/2.
- States the general solutions where cos x = 1/2 within one rotation: x = 60° and x = 360° − 60° = 300°.
- States the final answers x = 60°, 300°.
Examiner tip: Always convert reciprocal trig functions (sec, csc, cot) back to sin, cos or tan before solving — this avoids errors and lets you apply the familiar unit-circle solution pattern directly.
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
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that sec x = 1/cos x, so cos x = 1/2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States the general solutions where cos x = 1/2 within one rotation: x = 60° and x = 360° − 60° = 300°. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: States the final answers x = 60°, 300°. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that sec x = 1/cos x, so cos x = 1/2.
- Identifies the missing requirement: States the general solutions where cos x = 1/2 within one rotation: x = 60° and x = 360° − 60° = 300°.
- Identifies the missing requirement: States the final answers x = 60°, 300°.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
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]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Calculates a · b = 3(1) + 4(0) + 0(1) = 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Calculates |a| = 5 and |b| = √2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Uses cos θ = (a · b)/(|a||b|) = 3/(5√2). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Obtains θ ≈ 64.9°. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The dot product formula for the angle between two vectors works in any number of dimensions — the same cos θ = (a·b)/(|a||b|) relationship applies whether the vectors are in 2D, 3D, or higher.
Marking points
- Calculates a · b = 3(1) + 4(0) + 0(1) = 3.
- Calculates |a| = 5 and |b| = √2.
- Uses cos θ = (a · b)/(|a||b|) = 3/(5√2).
- Obtains θ ≈ 64.9°.
Examiner tip: The dot product formula for the angle between two vectors works in any number of dimensions — the same cos θ = (a·b)/(|a||b|) relationship applies whether the vectors are in 2D, 3D, or higher.
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]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Calculates a · b = 3(1) + 4(0) + 0(1) = 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Calculates |a| = 5 and |b| = √2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Uses cos θ = (a · b)/(|a||b|) = 3/(5√2). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Obtains θ ≈ 64.9°. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Calculates a · b = 3(1) + 4(0) + 0(1) = 3.
- Identifies the missing requirement: Calculates |a| = 5 and |b| = √2.
- Identifies the missing requirement: Uses cos θ = (a · b)/(|a||b|) = 3/(5√2).
- Identifies the missing requirement: Obtains θ ≈ 64.9°.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.