Mathematics AA: Standard Level

1. 1.

   Convert 150° to radians, giving your answer as a multiple of π.

   [2 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: Uses the conversion 180° = π radians. 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 150° = 5π/6 radians. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Multiply the degree measure by π/180 and simplify the resulting fraction fully.

   Marking points

   • Uses the conversion 180° = π radians.
   • Obtains 150° = 5π/6 radians.

   Examiner tip: Multiply the degree measure by π/180 and simplify the resulting fraction fully.

2. 2.

   Marking analysis: A learner attempts the following task: “Convert 150° to radians, giving your answer as a multiple of π.” Their response addresses only this point: “Uses the conversion 180° = π radians.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.

   [2 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 conversion 180° = π radians. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   3. Requirement 2: Identifies the missing requirement: Obtains 150° = 5π/6 radians. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   4. 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 conversion 180° = π radians.
   • Identifies the missing requirement: Obtains 150° = 5π/6 radians.

   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. 3.

   Use exact values to evaluate sin(π/3) + cos(π/6), giving your answer in simplest surd form.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Build a supported judgement: identify the claim, use the question's evidence, consider a relevant limitation or alternative, and make the conclusion depend on that evidence. There may be more than one defensible answer.
   2. Work through this mathematical step: States sin(π/3) = √3/2. 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 cos(π/6) = √3/2. 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: Adds the values to obtain √3. 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: Exact values for π/6, π/4 and π/3 should be memorised, since calculators are not permitted for these questions.

   Marking points

   • States sin(π/3) = √3/2.
   • States cos(π/6) = √3/2.
   • Adds the values to obtain √3.

   Examiner tip: Exact values for π/6, π/4 and π/3 should be memorised, since calculators are not permitted for these questions.

4. 4.

   Marking analysis: A learner attempts the following task: “Use exact values to evaluate sin(π/3) + cos(π/6), giving your answer in simplest surd form.” Their response addresses only this point: “States sin(π/3) = √3/2.” 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 sin(π/3) = √3/2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   3. Requirement 2: Identifies the missing requirement: States cos(π/6) = √3/2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   4. Requirement 3: Identifies the missing requirement: Adds the values to obtain √3. 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 sin(π/3) = √3/2.
   • Identifies the missing requirement: States cos(π/6) = √3/2.
   • Identifies the missing requirement: Adds the values to obtain √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.

5. 5.

   Given that sin θ = 3/5 and θ is obtuse, find the exact value of cos θ.

   [4 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 Pythagorean identity sin²θ + cos²θ = 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 cos²θ = 1 − 9/25 = 16/25. 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: Takes the square root to obtain cos θ = ±4/5. 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: Selects the negative value cos θ = −4/5 because θ is obtuse (second quadrant). 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 quadrant given by the angle description (obtuse, acute, reflex) determines the sign; always check this before finalising the square root.

   Marking points

   • Uses the Pythagorean identity sin²θ + cos²θ = 1.
   • Substitutes to obtain cos²θ = 1 − 9/25 = 16/25.
   • Takes the square root to obtain cos θ = ±4/5.
   • Selects the negative value cos θ = −4/5 because θ is obtuse (second quadrant).

   Examiner tip: The quadrant given by the angle description (obtuse, acute, reflex) determines the sign; always check this before finalising the square root.

6. 6.

   Marking analysis: A learner attempts the following task: “Given that sin θ = 3/5 and θ is obtuse, find the exact value of cos θ.” Their response addresses only this point: “Uses the Pythagorean identity sin²θ + cos²θ = 1.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.

   [4 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 Pythagorean identity sin²θ + cos²θ = 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   3. Requirement 2: Identifies the missing requirement: Substitutes to obtain cos²θ = 1 − 9/25 = 16/25. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   4. Requirement 3: Identifies the missing requirement: Takes the square root to obtain cos θ = ±4/5. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   5. Requirement 4: Identifies the missing requirement: Selects the negative value cos θ = −4/5 because θ is obtuse (second quadrant). 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: Uses the Pythagorean identity sin²θ + cos²θ = 1.
   • Identifies the missing requirement: Substitutes to obtain cos²θ = 1 − 9/25 = 16/25.
   • Identifies the missing requirement: Takes the square root to obtain cos θ = ±4/5.
   • Identifies the missing requirement: Selects the negative value cos θ = −4/5 because θ is obtuse (second quadrant).

   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. 7.

   Solve the equation 2 sin x − 1 = 0 for 0 ≤ x ≤ 2π, giving all solutions exactly.

   [4 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: Rearranges to sin 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: Identifies the reference angle π/6. 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: Identifies that sine is positive in the first and second quadrants. 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 both solutions x = π/6 and x = 5π/6. 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: Always consider all quadrants consistent with the sign of the trig ratio within the given domain, not just the reference angle.

   Marking points

   • Rearranges to sin x = 1/2.
   • Identifies the reference angle π/6.
   • Identifies that sine is positive in the first and second quadrants.
   • States both solutions x = π/6 and x = 5π/6.

   Examiner tip: Always consider all quadrants consistent with the sign of the trig ratio within the given domain, not just the reference angle.

8. 8.

   Marking analysis: A learner attempts the following task: “Solve the equation 2 sin x − 1 = 0 for 0 ≤ x ≤ 2π, giving all solutions exactly.” Their response addresses only this point: “Rearranges to sin x = 1/2.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.

   [4 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: Rearranges to sin x = 1/2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   3. Requirement 2: Identifies the missing requirement: Identifies the reference angle π/6. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   4. Requirement 3: Identifies the missing requirement: Identifies that sine is positive in the first and second quadrants. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   5. Requirement 4: Identifies the missing requirement: States both solutions x = π/6 and x = 5π/6. 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: Rearranges to sin x = 1/2.
   • Identifies the missing requirement: Identifies the reference angle π/6.
   • Identifies the missing requirement: Identifies that sine is positive in the first and second quadrants.
   • Identifies the missing requirement: States both solutions x = π/6 and x = 5π/6.

   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. 9.

   In triangle ABC, angle A = 40°, side a = 12 cm and side b = 15 cm. Find angle B, giving your answer to one 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: Uses the sine rule sin A/a = sin B/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: Substitutes to obtain sin B = 15 sin(40°)/12. 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 sin B ≈ 0.8035. 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 B ≈ 53.4°. 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: When using the sine rule to find an angle, check whether an obtuse alternative (180° minus the calculated angle) is also geometrically valid.

   Marking points

   • Uses the sine rule sin A/a = sin B/b.
   • Substitutes to obtain sin B = 15 sin(40°)/12.
   • Calculates sin B ≈ 0.8035.
   • Obtains B ≈ 53.4°.

   Examiner tip: When using the sine rule to find an angle, check whether an obtuse alternative (180° minus the calculated angle) is also geometrically valid.

10. 10.

    Marking analysis: A learner attempts the following task: “In triangle ABC, angle A = 40°, side a = 12 cm and side b = 15 cm. Find angle B, giving your answer to one decimal place.” Their response addresses only this point: “Uses the sine rule sin A/a = sin B/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: Uses the sine rule sin A/a = sin B/b. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes to obtain sin B = 15 sin(40°)/12. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Calculates sin B ≈ 0.8035. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains B ≈ 53.4°. 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: Uses the sine rule sin A/a = sin B/b.
    • Identifies the missing requirement: Substitutes to obtain sin B = 15 sin(40°)/12.
    • Identifies the missing requirement: Calculates sin B ≈ 0.8035.
    • Identifies the missing requirement: Obtains B ≈ 53.4°.

    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.

11. 11.

    In triangle PQR, PQ = 7 cm, QR = 9 cm and angle PQR = 65°. Find the length of PR.

    [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: Uses the cosine rule PR² = PQ² + QR² − 2(PQ)(QR)cos(PQR). 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 PR² = 49 + 81 − 2(7)(9)cos(65°). 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 PR² ≈ 76.7. 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 PR ≈ 8.76 cm. 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 cosine rule is used when two sides and the included angle are known, exactly the case here (PQ, QR and the angle between them).

    Marking points

    • Uses the cosine rule PR² = PQ² + QR² − 2(PQ)(QR)cos(PQR).
    • Substitutes to obtain PR² = 49 + 81 − 2(7)(9)cos(65°).
    • Calculates PR² ≈ 76.7.
    • Obtains PR ≈ 8.76 cm.

    Examiner tip: The cosine rule is used when two sides and the included angle are known, exactly the case here (PQ, QR and the angle between them).

12. 12.

    Marking analysis: A learner attempts the following task: “In triangle PQR, PQ = 7 cm, QR = 9 cm and angle PQR = 65°. Find the length of PR.” Their response addresses only this point: “Uses the cosine rule PR² = PQ² + QR² − 2(PQ)(QR)cos(PQR).” 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: Uses the cosine rule PR² = PQ² + QR² − 2(PQ)(QR)cos(PQR). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes to obtain PR² = 49 + 81 − 2(7)(9)cos(65°). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Calculates PR² ≈ 76.7. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains PR ≈ 8.76 cm. 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: Uses the cosine rule PR² = PQ² + QR² − 2(PQ)(QR)cos(PQR).
    • Identifies the missing requirement: Substitutes to obtain PR² = 49 + 81 − 2(7)(9)cos(65°).
    • Identifies the missing requirement: Calculates PR² ≈ 76.7.
    • Identifies the missing requirement: Obtains PR ≈ 8.76 cm.

    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. 13.

    Triangle XYZ has XY = 8 cm, XZ = 10 cm and angle YXZ = 55°. Find the area of the triangle.

    [3 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: Uses the area formula Area = ½ab sin C with the given sides and included angle. 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 ½(8)(10)sin(55°). 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 an area of approximately 32.8 cm². 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: The angle used in the area formula must be the angle included between the two given sides.

    Marking points

    • Uses the area formula Area = ½ab sin C with the given sides and included angle.
    • Substitutes ½(8)(10)sin(55°).
    • Obtains an area of approximately 32.8 cm².

    Examiner tip: The angle used in the area formula must be the angle included between the two given sides.

14. 14.

    Marking analysis: A learner attempts the following task: “Triangle XYZ has XY = 8 cm, XZ = 10 cm and angle YXZ = 55°. Find the area of the triangle.” Their response addresses only this point: “Uses the area formula Area = ½ab sin C with the given sides and included angle.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [3 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: Uses the area formula Area = ½ab sin C with the given sides and included angle. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes ½(8)(10)sin(55°). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Obtains an area of approximately 32.8 cm². 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: Uses the area formula Area = ½ab sin C with the given sides and included angle.
    • Identifies the missing requirement: Substitutes ½(8)(10)sin(55°).
    • Identifies the missing requirement: Obtains an area of approximately 32.8 cm².

    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. 15.

    The function f(x) = 3 sin(2x) + 1 models a periodic quantity. State the amplitude, period and vertical shift of f.

    [3 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: States the amplitude as 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: States the period as 2π/2 = π. 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 vertical shift as 1 unit upward. 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: For y = a sin(bx) + d, the period is 2π/b, not b itself; this is a common error.

    Marking points

    • States the amplitude as 3.
    • States the period as 2π/2 = π.
    • States the vertical shift as 1 unit upward.

    Examiner tip: For y = a sin(bx) + d, the period is 2π/b, not b itself; this is a common error.

16. 16.

    Marking analysis: A learner attempts the following task: “The function f(x) = 3 sin(2x) + 1 models a periodic quantity. State the amplitude, period and vertical shift of f.” Their response addresses only this point: “States the amplitude as 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: States the amplitude as 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: States the period as 2π/2 = π. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: States the vertical shift as 1 unit upward. 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 the amplitude as 3.
    • Identifies the missing requirement: States the period as 2π/2 = π.
    • Identifies the missing requirement: States the vertical shift as 1 unit upward.

    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. 17.

    Prove the identity (1 − cos²x)/sin x = sin x for sin x ≠ 0.

    [3 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: Uses the Pythagorean identity to rewrite the numerator: 1 − cos²x = sin²x. 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 sin²x/sin 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: Simplifies to sin x, matching 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.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Work from the more complicated side of the identity toward the simpler side, one algebraic step at a time.

    Marking points

    • Uses the Pythagorean identity to rewrite the numerator: 1 − cos²x = sin²x.
    • Substitutes to obtain sin²x/sin x.
    • Simplifies to sin x, matching the right-hand side.

    Examiner tip: Work from the more complicated side of the identity toward the simpler side, one algebraic step at a time.

18. 18.

    Marking analysis: A learner attempts the following task: “Prove the identity (1 − cos²x)/sin x = sin x for sin x ≠ 0.” Their response addresses only this point: “Uses the Pythagorean identity to rewrite the numerator: 1 − cos²x = sin²x.” 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: Uses the Pythagorean identity to rewrite the numerator: 1 − cos²x = sin²x. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes to obtain sin²x/sin x. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Simplifies to sin x, matching the right-hand side. 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: Uses the Pythagorean identity to rewrite the numerator: 1 − cos²x = sin²x.
    • Identifies the missing requirement: Substitutes to obtain sin²x/sin x.
    • Identifies the missing requirement: Simplifies to sin x, matching 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.

19. 19.

    A vertical cliff is observed from two points A and B on level ground, with B further from the cliff than A. AB = 50 m, the angle of elevation of the top of the cliff from A is 42° and from B is 27°. Find the height of the cliff.

    [6 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: Sets up a triangle with the cliff top, A and B, identifying the angle at the cliff top using angle properties. 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 the angle at the cliff top as 42° − 27° = 15° (exterior angle result). 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 sine rule in triangle (top, A, B) to find the distance from A to the cliff top: distance/sin(27°) = 50/sin(15°). 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: Calculates this distance ≈ 87.7 m. 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: Uses right-angled trigonometry in the vertical triangle: height = distance × sin(42°). 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: Obtains a cliff height of approximately 58.7 m. 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: This two-triangle problem requires the sine rule first to find a slant distance, then simple right-angled trigonometry to extract the vertical height.

    Marking points

    • Sets up a triangle with the cliff top, A and B, identifying the angle at the cliff top using angle properties.
    • Finds the angle at the cliff top as 42° − 27° = 15° (exterior angle result).
    • Uses the sine rule in triangle (top, A, B) to find the distance from A to the cliff top: distance/sin(27°) = 50/sin(15°).
    • Calculates this distance ≈ 87.7 m.
    • Uses right-angled trigonometry in the vertical triangle: height = distance × sin(42°).
    • Obtains a cliff height of approximately 58.7 m.

    Examiner tip: This two-triangle problem requires the sine rule first to find a slant distance, then simple right-angled trigonometry to extract the vertical height.

20. 20.

    Marking analysis: A learner attempts the following task: “A vertical cliff is observed from two points A and B on level ground, with B further from the cliff than A. AB = 50 m, the angle of elevation of the top of the cliff from A is 42° and from B is 27°. Find the height of the cliff.” Their response addresses only this point: “Sets up a triangle with the cliff top, A and B, identifying the angle at the cliff top using angle properties.” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [6 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: Sets up a triangle with the cliff top, A and B, identifying the angle at the cliff top using angle properties. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Finds the angle at the cliff top as 42° − 27° = 15° (exterior angle result). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Uses the sine rule in triangle (top, A, B) to find the distance from A to the cliff top: distance/sin(27°) = 50/sin(15°). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Calculates this distance ≈ 87.7 m. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: Uses right-angled trigonometry in the vertical triangle: height = distance × sin(42°). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    7. Requirement 6: Identifies the missing requirement: Obtains a cliff height of approximately 58.7 m. 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: Sets up a triangle with the cliff top, A and B, identifying the angle at the cliff top using angle properties.
    • Identifies the missing requirement: Finds the angle at the cliff top as 42° − 27° = 15° (exterior angle result).
    • Identifies the missing requirement: Uses the sine rule in triangle (top, A, B) to find the distance from A to the cliff top: distance/sin(27°) = 50/sin(15°).
    • Identifies the missing requirement: Calculates this distance ≈ 87.7 m.
    • Identifies the missing requirement: Uses right-angled trigonometry in the vertical triangle: height = distance × sin(42°).
    • Identifies the missing requirement: Obtains a cliff height of approximately 58.7 m.

    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. 21.

    Points A(1, 2, 0) and B(4, −2, 3) are given. Find the vector AB and hence determine the distance |AB|.

    [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: Uses AB = OB − OA. 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 AB = (3, −4, 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: Uses |AB| = √(3² + (−4)² + 3²). 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 |AB| = √34 ≈ 5.83. 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: Subtract the initial point from the terminal point component-wise; the order matters for direction, not for length.

    Marking points

    • Uses AB = OB − OA.
    • Obtains AB = (3, −4, 3).
    • Uses |AB| = √(3² + (−4)² + 3²).
    • Obtains |AB| = √34 ≈ 5.83.

    Examiner tip: Subtract the initial point from the terminal point component-wise; the order matters for direction, not for length.

22. 22.

    Marking analysis: A learner attempts the following task: “Points A(1, 2, 0) and B(4, −2, 3) are given. Find the vector AB and hence determine the distance |AB|.” Their response addresses only this point: “Uses AB = OB − OA.” 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: Uses AB = OB − OA. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Obtains AB = (3, −4, 3). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Uses |AB| = √(3² + (−4)² + 3²). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains |AB| = √34 ≈ 5.83. 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: Uses AB = OB − OA.
    • Identifies the missing requirement: Obtains AB = (3, −4, 3).
    • Identifies the missing requirement: Uses |AB| = √(3² + (−4)² + 3²).
    • Identifies the missing requirement: Obtains |AB| = √34 ≈ 5.83.

    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. 23.

    Vectors a = 2i − j + 2k and b = i + 2j − 2k are given. Calculate a · b and hence find the angle between a and b.

    [6 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: Uses a · b = (2)(1) + (−1)(2) + (2)(−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: Obtains a · b = −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 |a| = 3 and |b| = 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: Uses cos θ = (a · b)/(|a||b|). 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 cos θ = −4/9. 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: Obtains θ ≈ 116°. 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: A negative dot product means the angle between the vectors is obtuse; check this matches your final angle.

    Marking points

    • Uses a · b = (2)(1) + (−1)(2) + (2)(−2).
    • Obtains a · b = −4.
    • Calculates |a| = 3 and |b| = 3.
    • Uses cos θ = (a · b)/(|a||b|).
    • Obtains cos θ = −4/9.
    • Obtains θ ≈ 116°.

    Examiner tip: A negative dot product means the angle between the vectors is obtuse; check this matches your final angle.

24. 24.

    Marking analysis: A learner attempts the following task: “Vectors a = 2i − j + 2k and b = i + 2j − 2k are given. Calculate a · b and hence find the angle between a and b.” Their response addresses only this point: “Uses a · b = (2)(1) + (−1)(2) + (2)(−2).” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [6 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: Uses a · b = (2)(1) + (−1)(2) + (2)(−2). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Obtains a · b = −4. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Calculates |a| = 3 and |b| = 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Uses cos θ = (a · b)/(|a||b|). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: Obtains cos θ = −4/9. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    7. Requirement 6: Identifies the missing requirement: Obtains θ ≈ 116°. 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 a · b = (2)(1) + (−1)(2) + (2)(−2).
    • Identifies the missing requirement: Obtains a · b = −4.
    • Identifies the missing requirement: Calculates |a| = 3 and |b| = 3.
    • Identifies the missing requirement: Uses cos θ = (a · b)/(|a||b|).
    • Identifies the missing requirement: Obtains cos θ = −4/9.
    • Identifies the missing requirement: Obtains θ ≈ 116°.

    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. 25.

    A line passes through the point (1, 0, −2) and is parallel to the vector (2, 1, 3). Write a vector equation for the line and state the coordinates of the point on the line when the parameter equals 2.

    [3 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: Writes r = (1, 0, −2) + t(2, 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: Substitutes t = 2 into each component. 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 point (5, 2, 4). 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: A vector equation needs both a point on the line and a direction vector; neither alone is sufficient.

    Marking points

    • Writes r = (1, 0, −2) + t(2, 1, 3).
    • Substitutes t = 2 into each component.
    • States the point (5, 2, 4).

    Examiner tip: A vector equation needs both a point on the line and a direction vector; neither alone is sufficient.

26. 26.

    Marking analysis: A learner attempts the following task: “A line passes through the point (1, 0, −2) and is parallel to the vector (2, 1, 3). Write a vector equation for the line and state the coordinates of the point on the line when the parameter equals 2.” Their response addresses only this point: “Writes r = (1, 0, −2) + t(2, 1, 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: Writes r = (1, 0, −2) + t(2, 1, 3). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes t = 2 into each component. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: States the point (5, 2, 4). 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: Writes r = (1, 0, −2) + t(2, 1, 3).
    • Identifies the missing requirement: Substitutes t = 2 into each component.
    • Identifies the missing requirement: States the point (5, 2, 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.

27. 27.

    Find a unit vector in the same direction as v = (3, −4, 12).

    [3 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 |v| = √(3² + (−4)² + 12²). 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 |v| = 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 unit vector as (3/13, −4/13, 12/13). 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: A unit vector is the original vector divided by its own magnitude, keeping the same direction.

    Marking points

    • Calculates |v| = √(3² + (−4)² + 12²).
    • Obtains |v| = 13.
    • States the unit vector as (3/13, −4/13, 12/13).

    Examiner tip: A unit vector is the original vector divided by its own magnitude, keeping the same direction.

28. 28.

    Marking analysis: A learner attempts the following task: “Find a unit vector in the same direction as v = (3, −4, 12).” Their response addresses only this point: “Calculates |v| = √(3² + (−4)² + 12²).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [3 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 |v| = √(3² + (−4)² + 12²). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Obtains |v| = 13. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: States the unit vector as (3/13, −4/13, 12/13). 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 |v| = √(3² + (−4)² + 12²).
    • Identifies the missing requirement: Obtains |v| = 13.
    • Identifies the missing requirement: States the unit vector as (3/13, −4/13, 12/13).

    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. 29.

    Points A(2, 1, −1), B(4, 0, 2) and C(3, 3, 0) form a triangle. Find the vectors AB and AC, and hence calculate the area of triangle ABC using the cross product.

    [6 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: Obtains AB = (2, −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: Obtains AC = (1, 2, 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 cross product AB × AC component-wise. 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 AB × AC = (−7, 1, 5). 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: Calculates |AB × AC| = √(49 + 1 + 25) = √75. 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 area as ½√75 ≈ 4.33 square units. 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: The area of a triangle formed by two vectors from a common vertex is exactly half the magnitude of their cross product.

    Marking points

    • Obtains AB = (2, −1, 3).
    • Obtains AC = (1, 2, 1).
    • Calculates the cross product AB × AC component-wise.
    • Obtains AB × AC = (−7, 1, 5).
    • Calculates |AB × AC| = √(49 + 1 + 25) = √75.
    • States the area as ½√75 ≈ 4.33 square units.

    Examiner tip: The area of a triangle formed by two vectors from a common vertex is exactly half the magnitude of their cross product.

30. 30.

    Marking analysis: A learner attempts the following task: “Points A(2, 1, −1), B(4, 0, 2) and C(3, 3, 0) form a triangle. Find the vectors AB and AC, and hence calculate the area of triangle ABC using the cross product.” Their response addresses only this point: “Obtains AB = (2, −1, 3).” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [6 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: Obtains AB = (2, −1, 3). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Obtains AC = (1, 2, 1). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Calculates the cross product AB × AC component-wise. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains AB × AC = (−7, 1, 5). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: Calculates |AB × AC| = √(49 + 1 + 25) = √75. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    7. Requirement 6: Identifies the missing requirement: States the area as ½√75 ≈ 4.33 square units. 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: Obtains AB = (2, −1, 3).
    • Identifies the missing requirement: Obtains AC = (1, 2, 1).
    • Identifies the missing requirement: Calculates the cross product AB × AC component-wise.
    • Identifies the missing requirement: Obtains AB × AC = (−7, 1, 5).
    • Identifies the missing requirement: Calculates |AB × AC| = √(49 + 1 + 25) = √75.
    • Identifies the missing requirement: States the area as ½√75 ≈ 4.33 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.

31. 31.

    Two lines have equations r = (1, 2, 0) + s(1, −1, 2) and r = (3, 0, 4) + t(2, 1, −1). Show that the lines intersect and find the point of intersection.

    [6 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: Equates corresponding components: 1 + s = 3 + 2t, 2 − s = t, 2s = 4 − 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: Solves the first two equations simultaneously, e.g. substituting t = 2 − s into the first equation: 1 + s = 3 + 2(2 − s). 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 s = 2 and t = 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: Verifies these values satisfy the third equation: 2(2) = 4 − 0, which holds, confirming the lines intersect. 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 s = 2 into the first line's equation. 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 point of intersection as (3, 0, 4). 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: Two lines in 3D generally do not intersect; the third equation must be checked as a consistency test, not assumed.

    Marking points

    • Equates corresponding components: 1 + s = 3 + 2t, 2 − s = t, 2s = 4 − t.
    • Solves the first two equations simultaneously, e.g. substituting t = 2 − s into the first equation: 1 + s = 3 + 2(2 − s).
    • Obtains s = 2 and t = 0.
    • Verifies these values satisfy the third equation: 2(2) = 4 − 0, which holds, confirming the lines intersect.
    • Substitutes s = 2 into the first line's equation.
    • States the point of intersection as (3, 0, 4).

    Examiner tip: Two lines in 3D generally do not intersect; the third equation must be checked as a consistency test, not assumed.

32. 32.

    Marking analysis: A learner attempts the following task: “Two lines have equations r = (1, 2, 0) + s(1, −1, 2) and r = (3, 0, 4) + t(2, 1, −1). Show that the lines intersect and find the point of intersection.” Their response addresses only this point: “Equates corresponding components: 1 + s = 3 + 2t, 2 − s = t, 2s = 4 − t.” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [6 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: Equates corresponding components: 1 + s = 3 + 2t, 2 − s = t, 2s = 4 − t. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Solves the first two equations simultaneously, e.g. substituting t = 2 − s into the first equation: 1 + s = 3 + 2(2 − s). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Obtains s = 2 and t = 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Verifies these values satisfy the third equation: 2(2) = 4 − 0, which holds, confirming the lines intersect. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: Substitutes s = 2 into the first line's equation. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    7. Requirement 6: Identifies the missing requirement: States the point of intersection as (3, 0, 4). 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: Equates corresponding components: 1 + s = 3 + 2t, 2 − s = t, 2s = 4 − t.
    • Identifies the missing requirement: Solves the first two equations simultaneously, e.g. substituting t = 2 − s into the first equation: 1 + s = 3 + 2(2 − s).
    • Identifies the missing requirement: Obtains s = 2 and t = 0.
    • Identifies the missing requirement: Verifies these values satisfy the third equation: 2(2) = 4 − 0, which holds, confirming the lines intersect.
    • Identifies the missing requirement: Substitutes s = 2 into the first line's equation.
    • Identifies the missing requirement: States the point of intersection as (3, 0, 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.

33. 33.

    Find the angle that the vector d = (1, 1, 0) makes with the positive x-axis.

    [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: Identifies the x-axis direction vector as i = (1, 0, 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: Uses cos θ = (d · i)/(|d||i|). 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 · i = 1, |d| = √2, |i| = 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: Obtains θ = 45°. 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 angle a vector makes with an axis is found using the scalar product with the axis's own unit vector.

    Marking points

    • Identifies the x-axis direction vector as i = (1, 0, 0).
    • Uses cos θ = (d · i)/(|d||i|).
    • Calculates d · i = 1, |d| = √2, |i| = 1.
    • Obtains θ = 45°.

    Examiner tip: The angle a vector makes with an axis is found using the scalar product with the axis's own unit vector.

34. 34.

    Marking analysis: A learner attempts the following task: “Find the angle that the vector d = (1, 1, 0) makes with the positive x-axis.” Their response addresses only this point: “Identifies the x-axis direction vector as i = (1, 0, 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: Identifies the x-axis direction vector as i = (1, 0, 0). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Uses cos θ = (d · i)/(|d||i|). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Calculates d · i = 1, |d| = √2, |i| = 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains θ = 45°. 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: Identifies the x-axis direction vector as i = (1, 0, 0).
    • Identifies the missing requirement: Uses cos θ = (d · i)/(|d||i|).
    • Identifies the missing requirement: Calculates d · i = 1, |d| = √2, |i| = 1.
    • Identifies the missing requirement: Obtains θ = 45°.

    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.

35. 35.

    The vector equation of a line is r = (2, −1, 3) + t(1, 2, −2). Determine whether the point (5, 5, −3) lies on this line.

    [4 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: Sets up three equations: 2 + t = 5, −1 + 2t = 5, 3 − 2t = −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: Solves the first equation to obtain 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: Checks t = 3 in the second equation: −1 + 6 = 5, which is true. 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: Checks t = 3 in the third equation: 3 − 6 = −3, which is also true, so the point lies on the line. 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: A single value of the parameter must satisfy all three component equations simultaneously for the point to lie on the line.

    Marking points

    • Sets up three equations: 2 + t = 5, −1 + 2t = 5, 3 − 2t = −3.
    • Solves the first equation to obtain t = 3.
    • Checks t = 3 in the second equation: −1 + 6 = 5, which is true.
    • Checks t = 3 in the third equation: 3 − 6 = −3, which is also true, so the point lies on the line.

    Examiner tip: A single value of the parameter must satisfy all three component equations simultaneously for the point to lie on the line.

36. 36.

    Marking analysis: A learner attempts the following task: “The vector equation of a line is r = (2, −1, 3) + t(1, 2, −2). Determine whether the point (5, 5, −3) lies on this line.” Their response addresses only this point: “Sets up three equations: 2 + t = 5, −1 + 2t = 5, 3 − 2t = −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] · 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: Sets up three equations: 2 + t = 5, −1 + 2t = 5, 3 − 2t = −3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Solves the first equation to obtain t = 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Checks t = 3 in the second equation: −1 + 6 = 5, which is true. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Checks t = 3 in the third equation: 3 − 6 = −3, which is also true, so the point lies on the line. 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: Sets up three equations: 2 + t = 5, −1 + 2t = 5, 3 − 2t = −3.
    • Identifies the missing requirement: Solves the first equation to obtain t = 3.
    • Identifies the missing requirement: Checks t = 3 in the second equation: −1 + 6 = 5, which is true.
    • Identifies the missing requirement: Checks t = 3 in the third equation: 3 − 6 = −3, which is also true, so the point lies on the line.

    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.

37. 37.

    Vectors p and q are such that |p| = 5, |q| = 3 and the angle between them is 60°. Calculate |p + q|.

    [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: Uses |p + q|² = |p|² + |q|² + 2|p||q|cos θ. 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 given values: 25 + 9 + 2(5)(3)cos 60°. 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 2(5)(3)(0.5) = 15. 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 |p + q|² = 49. 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 |p + q| = 7. 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 is the vector form of the cosine rule; it applies directly whenever the angle between two vectors is known.

    Marking points

    • Uses |p + q|² = |p|² + |q|² + 2|p||q|cos θ.
    • Substitutes the given values: 25 + 9 + 2(5)(3)cos 60°.
    • Calculates 2(5)(3)(0.5) = 15.
    • Obtains |p + q|² = 49.
    • States |p + q| = 7.

    Examiner tip: This is the vector form of the cosine rule; it applies directly whenever the angle between two vectors is known.

38. 38.

    Marking analysis: A learner attempts the following task: “Vectors p and q are such that |p| = 5, |q| = 3 and the angle between them is 60°. Calculate |p + q|.” Their response addresses only this point: “Uses |p + q|² = |p|² + |q|² + 2|p||q|cos θ.” 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: Uses |p + q|² = |p|² + |q|² + 2|p||q|cos θ. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes the given values: 25 + 9 + 2(5)(3)cos 60°. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Calculates 2(5)(3)(0.5) = 15. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains |p + q|² = 49. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: States |p + q| = 7. 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: Uses |p + q|² = |p|² + |q|² + 2|p||q|cos θ.
    • Identifies the missing requirement: Substitutes the given values: 25 + 9 + 2(5)(3)cos 60°.
    • Identifies the missing requirement: Calculates 2(5)(3)(0.5) = 15.
    • Identifies the missing requirement: Obtains |p + q|² = 49.
    • Identifies the missing requirement: States |p + q| = 7.

    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.

39. 39.

    A plane contains the point (1, 0, 2) and has normal vector n = (2, −1, 3). Write the Cartesian equation of the plane.

    [3 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: Uses the form n · (r − r₀) = 0 with the given point and normal. 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 2(x − 1) − 1(y − 0) + 3(z − 2) = 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: Simplifies to the Cartesian equation 2x − y + 3z = 8. 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: The coefficients of x, y and z in the Cartesian plane equation are exactly the components of the normal vector.

    Marking points

    • Uses the form n · (r − r₀) = 0 with the given point and normal.
    • Substitutes to obtain 2(x − 1) − 1(y − 0) + 3(z − 2) = 0.
    • Simplifies to the Cartesian equation 2x − y + 3z = 8.

    Examiner tip: The coefficients of x, y and z in the Cartesian plane equation are exactly the components of the normal vector.

40. 40.

    Marking analysis: A learner attempts the following task: “A plane contains the point (1, 0, 2) and has normal vector n = (2, −1, 3). Write the Cartesian equation of the plane.” Their response addresses only this point: “Uses the form n · (r − r₀) = 0 with the given point and normal.” 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: Uses the form n · (r − r₀) = 0 with the given point and normal. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes to obtain 2(x − 1) − 1(y − 0) + 3(z − 2) = 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Simplifies to the Cartesian equation 2x − y + 3z = 8. 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: Uses the form n · (r − r₀) = 0 with the given point and normal.
    • Identifies the missing requirement: Substitutes to obtain 2(x − 1) − 1(y − 0) + 3(z − 2) = 0.
    • Identifies the missing requirement: Simplifies to the Cartesian equation 2x − y + 3z = 8.

    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.

Original, unofficial practice questions from SubjectScout — not official past-paper questions.