Mathematics AA: Higher Level

1. 1.

   Given z₁ = 3 + 2i and z₂ = 1 − 4i, calculate z₁ + z₂ and z₁z₂, giving each answer in the form a + bi.

   [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: Adds real and imaginary parts separately to obtain z₁ + z₂ = 4 − 2i. 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: Expands the product (3 + 2i)(1 − 4i) using distribution. 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 i² = −1 to simplify. 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 z₁z₂ = 11 − 10i. 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: Treat i as an algebraic unknown throughout the expansion, then substitute i² = −1 only at the final simplification step.

   Marking points

   • Adds real and imaginary parts separately to obtain z₁ + z₂ = 4 − 2i.
   • Expands the product (3 + 2i)(1 − 4i) using distribution.
   • Uses i² = −1 to simplify.
   • Obtains z₁z₂ = 11 − 10i.

   Examiner tip: Treat i as an algebraic unknown throughout the expansion, then substitute i² = −1 only at the final simplification step.

2. 2.

   Marking analysis: A learner attempts the following task: “Given z₁ = 3 + 2i and z₂ = 1 − 4i, calculate z₁ + z₂ and z₁z₂, giving each answer in the form a + bi.” Their response addresses only this point: “Adds real and imaginary parts separately to obtain z₁ + z₂ = 4 − 2i.” 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: Adds real and imaginary parts separately to obtain z₁ + z₂ = 4 − 2i. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   3. Requirement 2: Identifies the missing requirement: Expands the product (3 + 2i)(1 − 4i) using distribution. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   4. Requirement 3: Identifies the missing requirement: Uses i² = −1 to simplify. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   5. Requirement 4: Identifies the missing requirement: Obtains z₁z₂ = 11 − 10i. 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: Adds real and imaginary parts separately to obtain z₁ + z₂ = 4 − 2i.
   • Identifies the missing requirement: Expands the product (3 + 2i)(1 − 4i) using distribution.
   • Identifies the missing requirement: Uses i² = −1 to simplify.
   • Identifies the missing requirement: Obtains z₁z₂ = 11 − 10i.

   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.

   Express z = −2 + 2√3 i in the polar form r(cos θ + i sin θ), giving r and θ exactly.

   [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: Calculates r = |z| = √((−2)² + (2√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: Obtains r = 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: Identifies that z lies in the second quadrant. 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 the reference angle arctan(2√3/2) = 60°. 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 = 120° (2π/3 radians) and writes z = 4(cos 120° + i sin 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: Always sketch or reason about the correct quadrant before finalising the argument; arctan alone gives only a reference angle.

   Marking points

   • Calculates r = |z| = √((−2)² + (2√3)²).
   • Obtains r = 4.
   • Identifies that z lies in the second quadrant.
   • Calculates the reference angle arctan(2√3/2) = 60°.
   • Obtains θ = 120° (2π/3 radians) and writes z = 4(cos 120° + i sin 120°).

   Examiner tip: Always sketch or reason about the correct quadrant before finalising the argument; arctan alone gives only a reference angle.

4. 4.

   Marking analysis: A learner attempts the following task: “Express z = −2 + 2√3 i in the polar form r(cos θ + i sin θ), giving r and θ exactly.” Their response addresses only this point: “Calculates r = |z| = √((−2)² + (2√3)²).” 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: Calculates r = |z| = √((−2)² + (2√3)²). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   3. Requirement 2: Identifies the missing requirement: Obtains r = 4. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   4. Requirement 3: Identifies the missing requirement: Identifies that z lies in the second quadrant. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   5. Requirement 4: Identifies the missing requirement: Calculates the reference angle arctan(2√3/2) = 60°. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   6. Requirement 5: Identifies the missing requirement: Obtains θ = 120° (2π/3 radians) and writes z = 4(cos 120° + i sin 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: Calculates r = |z| = √((−2)² + (2√3)²).
   • Identifies the missing requirement: Obtains r = 4.
   • Identifies the missing requirement: Identifies that z lies in the second quadrant.
   • Identifies the missing requirement: Calculates the reference angle arctan(2√3/2) = 60°.
   • Identifies the missing requirement: Obtains θ = 120° (2π/3 radians) and writes z = 4(cos 120° + i sin 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.

5. 5.

   Solve the quadratic equation z² − 4z + 13 = 0, giving both roots in the form a + bi.

   [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 quadratic formula z = [4 ± √(16 − 52)]/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: Simplifies the discriminant to √(−36) = 6i. 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 z = (4 ± 6i)/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: States the roots as z = 2 + 3i and z = 2 − 3i. 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 negative discriminant gives a complex conjugate pair of roots; write √(negative) as i√(positive) before simplifying.

   Marking points

   • Uses the quadratic formula z = [4 ± √(16 − 52)]/2.
   • Simplifies the discriminant to √(−36) = 6i.
   • Obtains z = (4 ± 6i)/2.
   • States the roots as z = 2 + 3i and z = 2 − 3i.

   Examiner tip: A negative discriminant gives a complex conjugate pair of roots; write √(negative) as i√(positive) before simplifying.

6. 6.

   Marking analysis: A learner attempts the following task: “Solve the quadratic equation z² − 4z + 13 = 0, giving both roots in the form a + bi.” Their response addresses only this point: “Uses the quadratic formula z = [4 ± √(16 − 52)]/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: Uses the quadratic formula z = [4 ± √(16 − 52)]/2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   3. Requirement 2: Identifies the missing requirement: Simplifies the discriminant to √(−36) = 6i. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   4. Requirement 3: Identifies the missing requirement: Obtains z = (4 ± 6i)/2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   5. Requirement 4: Identifies the missing requirement: States the roots as z = 2 + 3i and z = 2 − 3i. 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 quadratic formula z = [4 ± √(16 − 52)]/2.
   • Identifies the missing requirement: Simplifies the discriminant to √(−36) = 6i.
   • Identifies the missing requirement: Obtains z = (4 ± 6i)/2.
   • Identifies the missing requirement: States the roots as z = 2 + 3i and z = 2 − 3i.

   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.

   Given z = 2(cos 30° + i sin 30°), use De Moivre's theorem to find z⁶ in the form a + bi.

   [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 De Moivre's theorem zⁿ = rⁿ(cos nθ + i sin nθ). 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 r = 2, θ = 30°, n = 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: Calculates r⁶ = 64 and nθ = 180°. 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 z⁶ = 64(cos 180° + i sin 180°). 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: Simplifies to z⁶ = −64. 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: De Moivre's theorem turns repeated multiplication into a single multiplication of the argument and a single power of the modulus — always convert to polar form first.

   Marking points

   • States De Moivre's theorem zⁿ = rⁿ(cos nθ + i sin nθ).
   • Substitutes r = 2, θ = 30°, n = 6.
   • Calculates r⁶ = 64 and nθ = 180°.
   • States z⁶ = 64(cos 180° + i sin 180°).
   • Simplifies to z⁶ = −64.

   Examiner tip: De Moivre's theorem turns repeated multiplication into a single multiplication of the argument and a single power of the modulus — always convert to polar form first.

8. 8.

   Marking analysis: A learner attempts the following task: “Given z = 2(cos 30° + i sin 30°), use De Moivre's theorem to find z⁶ in the form a + bi.” Their response addresses only this point: “States De Moivre's theorem zⁿ = rⁿ(cos nθ + i sin nθ).” 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 De Moivre's theorem zⁿ = rⁿ(cos nθ + i sin nθ). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   3. Requirement 2: Identifies the missing requirement: Substitutes r = 2, θ = 30°, n = 6. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   4. Requirement 3: Identifies the missing requirement: Calculates r⁶ = 64 and nθ = 180°. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   5. Requirement 4: Identifies the missing requirement: States z⁶ = 64(cos 180° + i sin 180°). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
   6. Requirement 5: Identifies the missing requirement: Simplifies to z⁶ = −64. 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 De Moivre's theorem zⁿ = rⁿ(cos nθ + i sin nθ).
   • Identifies the missing requirement: Substitutes r = 2, θ = 30°, n = 6.
   • Identifies the missing requirement: Calculates r⁶ = 64 and nθ = 180°.
   • Identifies the missing requirement: States z⁶ = 64(cos 180° + i sin 180°).
   • Identifies the missing requirement: Simplifies to z⁶ = −64.

   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.

   Find the three cube roots of z = 8i, giving each in the form r(cos θ + i sin θ) with 0 ≤ θ < 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: Writes z in polar form: 8i = 8(cos 90° + i sin 90°). 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 the nth root formula r^(1/n)(cos((θ + 360°k)/n) + i sin((θ + 360°k)/n)) for k = 0, 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 common modulus 8^(1/3) = 2 for all three roots. 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 angles 30°, 150°, 270° for k = 0, 1, 2. 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 three roots as 2(cos 30° + i sin 30°), 2(cos 150° + i sin 150°), 2(cos 270° + i sin 270°). 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 n roots of a complex number are equally spaced around a circle of radius r^(1/n), exactly 360°/n apart — use this to check no root is missing.

   Marking points

   • Writes z in polar form: 8i = 8(cos 90° + i sin 90°).
   • Uses the nth root formula r^(1/n)(cos((θ + 360°k)/n) + i sin((θ + 360°k)/n)) for k = 0, 1, 2.
   • Calculates the common modulus 8^(1/3) = 2 for all three roots.
   • Obtains angles 30°, 150°, 270° for k = 0, 1, 2.
   • States the three roots as 2(cos 30° + i sin 30°), 2(cos 150° + i sin 150°), 2(cos 270° + i sin 270°).

   Examiner tip: The n roots of a complex number are equally spaced around a circle of radius r^(1/n), exactly 360°/n apart — use this to check no root is missing.

10. 10.

    Marking analysis: A learner attempts the following task: “Find the three cube roots of z = 8i, giving each in the form r(cos θ + i sin θ) with 0 ≤ θ < 360°.” Their response addresses only this point: “Writes z in polar form: 8i = 8(cos 90° + i sin 90°).” 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 z in polar form: 8i = 8(cos 90° + i sin 90°). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Uses the nth root formula r^(1/n)(cos((θ + 360°k)/n) + i sin((θ + 360°k)/n)) for k = 0, 1, 2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Calculates the common modulus 8^(1/3) = 2 for all three roots. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains angles 30°, 150°, 270° for k = 0, 1, 2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: States the three roots as 2(cos 30° + i sin 30°), 2(cos 150° + i sin 150°), 2(cos 270° + i sin 270°). 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 z in polar form: 8i = 8(cos 90° + i sin 90°).
    • Identifies the missing requirement: Uses the nth root formula r^(1/n)(cos((θ + 360°k)/n) + i sin((θ + 360°k)/n)) for k = 0, 1, 2.
    • Identifies the missing requirement: Calculates the common modulus 8^(1/3) = 2 for all three roots.
    • Identifies the missing requirement: Obtains angles 30°, 150°, 270° for k = 0, 1, 2.
    • Identifies the missing requirement: States the three roots as 2(cos 30° + i sin 30°), 2(cos 150° + i sin 150°), 2(cos 270° + i sin 270°).

    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.

    Prove by mathematical induction that 1 + 2 + 3 + … + n = n(n + 1)/2 for all positive integers n.

    [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: States the base case n = 1: LHS = 1, RHS = 1(2)/2 = 1, so the statement holds. 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 inductive hypothesis: assume true for n = k, i.e. 1 + 2 + … + k = k(k + 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: Adds (k + 1) to both sides: 1 + 2 + … + k + (k + 1) = k(k + 1)/2 + (k + 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: Combines the right-hand side over a common denominator to (k + 1)(k + 2)/2. 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: Concludes the statement holds for n = k + 1, and by induction for all positive integers n. 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: Every induction proof needs all three named stages — base case, inductive hypothesis, and inductive step — explicitly stated, not just the algebra.

    Marking points

    • States the base case n = 1: LHS = 1, RHS = 1(2)/2 = 1, so the statement holds.
    • States the inductive hypothesis: assume true for n = k, i.e. 1 + 2 + … + k = k(k + 1)/2.
    • Adds (k + 1) to both sides: 1 + 2 + … + k + (k + 1) = k(k + 1)/2 + (k + 1).
    • Combines the right-hand side over a common denominator to (k + 1)(k + 2)/2.
    • Concludes the statement holds for n = k + 1, and by induction for all positive integers n.

    Examiner tip: Every induction proof needs all three named stages — base case, inductive hypothesis, and inductive step — explicitly stated, not just the algebra.

12. 12.

    Marking analysis: A learner attempts the following task: “Prove by mathematical induction that 1 + 2 + 3 + … + n = n(n + 1)/2 for all positive integers n.” Their response addresses only this point: “States the base case n = 1: LHS = 1, RHS = 1(2)/2 = 1, so the statement holds.” 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 base case n = 1: LHS = 1, RHS = 1(2)/2 = 1, so the statement holds. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: States the inductive hypothesis: assume true for n = k, i.e. 1 + 2 + … + k = k(k + 1)/2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Adds (k + 1) to both sides: 1 + 2 + … + k + (k + 1) = k(k + 1)/2 + (k + 1). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Combines the right-hand side over a common denominator to (k + 1)(k + 2)/2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: Concludes the statement holds for n = k + 1, and by induction for all positive integers n. 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 base case n = 1: LHS = 1, RHS = 1(2)/2 = 1, so the statement holds.
    • Identifies the missing requirement: States the inductive hypothesis: assume true for n = k, i.e. 1 + 2 + … + k = k(k + 1)/2.
    • Identifies the missing requirement: Adds (k + 1) to both sides: 1 + 2 + … + k + (k + 1) = k(k + 1)/2 + (k + 1).
    • Identifies the missing requirement: Combines the right-hand side over a common denominator to (k + 1)(k + 2)/2.
    • Identifies the missing requirement: Concludes the statement holds for n = k + 1, and by induction for all positive integers n.

    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.

    The polynomial p(x) = x³ − 5x² + 9x − 5 has 2 + i as one root. Find the other two roots.

    [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 that since p(x) has real coefficients, complex roots occur in conjugate pairs, so 2 − i is also a root. 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: Forms the quadratic factor from these two roots: (x − (2 + i))(x − (2 − i)) = x² − 4x + 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: Divides p(x) by x² − 4x + 5 to obtain the linear quotient x − 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: Sets the linear factor equal to zero to find the third root. 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 all three roots: 2 + i, 2 − i, and x = 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: For a real-coefficient polynomial, forming the conjugate pair's quadratic factor first and dividing it out is far faster than searching for roots directly.

    Marking points

    • States that since p(x) has real coefficients, complex roots occur in conjugate pairs, so 2 − i is also a root.
    • Forms the quadratic factor from these two roots: (x − (2 + i))(x − (2 − i)) = x² − 4x + 5.
    • Divides p(x) by x² − 4x + 5 to obtain the linear quotient x − 1.
    • Sets the linear factor equal to zero to find the third root.
    • States all three roots: 2 + i, 2 − i, and x = 1.

    Examiner tip: For a real-coefficient polynomial, forming the conjugate pair's quadratic factor first and dividing it out is far faster than searching for roots directly.

14. 14.

    Marking analysis: A learner attempts the following task: “The polynomial p(x) = x³ − 5x² + 9x − 5 has 2 + i as one root. Find the other two roots.” Their response addresses only this point: “States that since p(x) has real coefficients, complex roots occur in conjugate pairs, so 2 − i is also a root.” 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 that since p(x) has real coefficients, complex roots occur in conjugate pairs, so 2 − i is also a root. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Forms the quadratic factor from these two roots: (x − (2 + i))(x − (2 − i)) = x² − 4x + 5. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Divides p(x) by x² − 4x + 5 to obtain the linear quotient x − 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Sets the linear factor equal to zero to find the third root. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: States all three roots: 2 + i, 2 − i, and x = 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 that since p(x) has real coefficients, complex roots occur in conjugate pairs, so 2 − i is also a root.
    • Identifies the missing requirement: Forms the quadratic factor from these two roots: (x − (2 + i))(x − (2 − i)) = x² − 4x + 5.
    • Identifies the missing requirement: Divides p(x) by x² − 4x + 5 to obtain the linear quotient x − 1.
    • Identifies the missing requirement: Sets the linear factor equal to zero to find the third root.
    • Identifies the missing requirement: States all three roots: 2 + i, 2 − i, and x = 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.

15. 15.

    Solve the system of linear equations 2x + y − z = 3, x − y + 2z = 4, 3x + 2y − z = 7 using row reduction, or state if it has no unique solution.

    [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: Writes the augmented matrix for the system. 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 row operations to eliminate x from the second and third rows. 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: Continues row reduction to eliminate y from the resulting third row. 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: Back-substitutes to find z, then y, then x consistently. 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 unique solution as an ordered triple (x, y, z), or identifies a row of the form 0 = k (k ≠ 0) as evidence of no solution. 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: A row that reduces to 0 = k with k ≠ 0 means the system is inconsistent; a row that reduces to 0 = 0 means infinitely many solutions — always check for these before assuming a unique answer.

    Marking points

    • Writes the augmented matrix for the system.
    • Uses row operations to eliminate x from the second and third rows.
    • Continues row reduction to eliminate y from the resulting third row.
    • Back-substitutes to find z, then y, then x consistently.
    • States the unique solution as an ordered triple (x, y, z), or identifies a row of the form 0 = k (k ≠ 0) as evidence of no solution.

    Examiner tip: A row that reduces to 0 = k with k ≠ 0 means the system is inconsistent; a row that reduces to 0 = 0 means infinitely many solutions — always check for these before assuming a unique answer.

16. 16.

    Marking analysis: A learner attempts the following task: “Solve the system of linear equations 2x + y − z = 3, x − y + 2z = 4, 3x + 2y − z = 7 using row reduction, or state if it has no unique solution.” Their response addresses only this point: “Writes the augmented matrix for the system.” 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: Writes the augmented matrix for the system. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Uses row operations to eliminate x from the second and third rows. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Continues row reduction to eliminate y from the resulting third row. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Back-substitutes to find z, then y, then x consistently. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: States the unique solution as an ordered triple (x, y, z), or identifies a row of the form 0 = k (k ≠ 0) as evidence of no solution. 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 augmented matrix for the system.
    • Identifies the missing requirement: Uses row operations to eliminate x from the second and third rows.
    • Identifies the missing requirement: Continues row reduction to eliminate y from the resulting third row.
    • Identifies the missing requirement: Back-substitutes to find z, then y, then x consistently.
    • Identifies the missing requirement: States the unique solution as an ordered triple (x, y, z), or identifies a row of the form 0 = k (k ≠ 0) as evidence of no solution.

    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.

    Find the number of distinct arrangements of the letters in the word STATS.

    [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 that STATS has 5 letters, with S repeated twice and T repeated twice. 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 formula for arrangements with repetition: 5!/(2!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 5! = 120 and 2!2! = 4. 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 number of distinct arrangements as 30. 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: Divide the total factorial by the factorial of each letter's repeat count — forgetting to divide is the most common error in this type of question.

    Marking points

    • Identifies that STATS has 5 letters, with S repeated twice and T repeated twice.
    • States the formula for arrangements with repetition: 5!/(2!2!).
    • Calculates 5! = 120 and 2!2! = 4.
    • States the number of distinct arrangements as 30.

    Examiner tip: Divide the total factorial by the factorial of each letter's repeat count — forgetting to divide is the most common error in this type of question.

18. 18.

    Marking analysis: A learner attempts the following task: “Find the number of distinct arrangements of the letters in the word STATS.” Their response addresses only this point: “Identifies that STATS has 5 letters, with S repeated twice and T repeated twice.” 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 that STATS has 5 letters, with S repeated twice and T repeated twice. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: States the formula for arrangements with repetition: 5!/(2!2!). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Calculates 5! = 120 and 2!2! = 4. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: States the number of distinct arrangements as 30. 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 that STATS has 5 letters, with S repeated twice and T repeated twice.
    • Identifies the missing requirement: States the formula for arrangements with repetition: 5!/(2!2!).
    • Identifies the missing requirement: Calculates 5! = 120 and 2!2! = 4.
    • Identifies the missing requirement: States the number of distinct arrangements as 30.

    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 committee of 4 people is chosen from a group of 6 men and 5 women. Find the number of ways of forming a committee containing exactly 2 men and 2 women.

    [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: Recognises that order does not matter, so combinations are used for each group. 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 number of ways to choose 2 men from 6: ⁶C₂ = 15. 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 number of ways to choose 2 women from 5: ⁵C₂ = 10. 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: Multiplies the two results using the counting principle: 15 × 10 = 150. 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 a selection is made independently from two separate groups, multiply the combinations for each group — do not add them.

    Marking points

    • Recognises that order does not matter, so combinations are used for each group.
    • Calculates the number of ways to choose 2 men from 6: ⁶C₂ = 15.
    • Calculates the number of ways to choose 2 women from 5: ⁵C₂ = 10.
    • Multiplies the two results using the counting principle: 15 × 10 = 150.

    Examiner tip: When a selection is made independently from two separate groups, multiply the combinations for each group — do not add them.

20. 20.

    Marking analysis: A learner attempts the following task: “A committee of 4 people is chosen from a group of 6 men and 5 women. Find the number of ways of forming a committee containing exactly 2 men and 2 women.” Their response addresses only this point: “Recognises that order does not matter, so combinations are used for each group.” 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: Recognises that order does not matter, so combinations are used for each group. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Calculates the number of ways to choose 2 men from 6: ⁶C₂ = 15. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Calculates the number of ways to choose 2 women from 5: ⁵C₂ = 10. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Multiplies the two results using the counting principle: 15 × 10 = 150. 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: Recognises that order does not matter, so combinations are used for each group.
    • Identifies the missing requirement: Calculates the number of ways to choose 2 men from 6: ⁶C₂ = 15.
    • Identifies the missing requirement: Calculates the number of ways to choose 2 women from 5: ⁵C₂ = 10.
    • Identifies the missing requirement: Multiplies the two results using the counting principle: 15 × 10 = 150.

    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.

    Find the coefficient of x³ in the binomial expansion of (2x − 3)⁵.

    [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 the general term of the binomial expansion: ⁿCᵣ aⁿ⁻ʳ bʳ. 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 that the term in x³ requires r = 2, since a = 2x contributes exponent 5 − r = 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: Substitutes ⁵C₂(2x)³(−3)² = 10 × 8x³ × 9. 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 coefficient = 720. 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: Identify which value of r gives the required power first, then substitute — trying every term in order wastes time when only one specific power is needed.

    Marking points

    • States the general term of the binomial expansion: ⁿCᵣ aⁿ⁻ʳ bʳ.
    • Identifies that the term in x³ requires r = 2, since a = 2x contributes exponent 5 − r = 3.
    • Substitutes ⁵C₂(2x)³(−3)² = 10 × 8x³ × 9.
    • Obtains coefficient = 720.

    Examiner tip: Identify which value of r gives the required power first, then substitute — trying every term in order wastes time when only one specific power is needed.

22. 22.

    Marking analysis: A learner attempts the following task: “Find the coefficient of x³ in the binomial expansion of (2x − 3)⁵.” Their response addresses only this point: “States the general term of the binomial expansion: ⁿCᵣ aⁿ⁻ʳ bʳ.” 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 the general term of the binomial expansion: ⁿCᵣ aⁿ⁻ʳ bʳ. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Identifies that the term in x³ requires r = 2, since a = 2x contributes exponent 5 − r = 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Substitutes ⁵C₂(2x)³(−3)² = 10 × 8x³ × 9. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains coefficient = 720. 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 the general term of the binomial expansion: ⁿCᵣ aⁿ⁻ʳ bʳ.
    • Identifies the missing requirement: Identifies that the term in x³ requires r = 2, since a = 2x contributes exponent 5 − r = 3.
    • Identifies the missing requirement: Substitutes ⁵C₂(2x)³(−3)² = 10 × 8x³ × 9.
    • Identifies the missing requirement: Obtains coefficient = 720.

    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.

    An infinite geometric series has first term 8 and common ratio 0.5. Calculate the sum to infinity, stating the condition required for this sum to exist.

    [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: States that the condition |r| < 1 is required for the sum to infinity to exist. 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 S∞ = a/(1 − r). 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: Substitutes 8/(1 − 0.5) to obtain S∞ = 16. 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: If |r| ≥ 1, successive terms do not shrink towards zero, so the partial sums never settle on a finite value — the sum to infinity simply does not exist in that case.

    Marking points

    • States that the condition |r| < 1 is required for the sum to infinity to exist.
    • Uses S∞ = a/(1 − r).
    • Substitutes 8/(1 − 0.5) to obtain S∞ = 16.

    Examiner tip: If |r| ≥ 1, successive terms do not shrink towards zero, so the partial sums never settle on a finite value — the sum to infinity simply does not exist in that case.

24. 24.

    Marking analysis: A learner attempts the following task: “An infinite geometric series has first term 8 and common ratio 0.5. Calculate the sum to infinity, stating the condition required for this sum to exist.” Their response addresses only this point: “States that the condition |r| < 1 is required for the sum to infinity to exist.” 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: States that the condition |r| < 1 is required for the sum to infinity to exist. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Uses S∞ = a/(1 − r). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Substitutes 8/(1 − 0.5) to obtain S∞ = 16. 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 the condition |r| < 1 is required for the sum to infinity to exist.
    • Identifies the missing requirement: Uses S∞ = a/(1 − r).
    • Identifies the missing requirement: Substitutes 8/(1 − 0.5) to obtain S∞ = 16.

    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.

    The quadratic equation 2x² − 7x + 3 = 0 has roots α and β. Without solving for the roots directly, calculate the value of α + β and αβ.

    [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: States that for ax² + bx + c = 0, the sum of roots α + β = −b/a. 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 α + β = −(−7)/2 = 3.5. 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 the product of roots αβ = c/a = 3/2 = 1.5. 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: These relationships between a quadratic's coefficients and the sum/product of its roots (Vieta's formulas) let you answer many questions without ever solving the quadratic formula.

    Marking points

    • States that for ax² + bx + c = 0, the sum of roots α + β = −b/a.
    • Substitutes to obtain α + β = −(−7)/2 = 3.5.
    • States that the product of roots αβ = c/a = 3/2 = 1.5.

    Examiner tip: These relationships between a quadratic's coefficients and the sum/product of its roots (Vieta's formulas) let you answer many questions without ever solving the quadratic formula.

26. 26.

    Marking analysis: A learner attempts the following task: “The quadratic equation 2x² − 7x + 3 = 0 has roots α and β. Without solving for the roots directly, calculate the value of α + β and αβ.” Their response addresses only this point: “States that for ax² + bx + c = 0, the sum of roots α + β = −b/a.” 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: States that for ax² + bx + c = 0, the sum of roots α + β = −b/a. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes to obtain α + β = −(−7)/2 = 3.5. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: States that the product of roots αβ = c/a = 3/2 = 1.5. 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 for ax² + bx + c = 0, the sum of roots α + β = −b/a.
    • Identifies the missing requirement: Substitutes to obtain α + β = −(−7)/2 = 3.5.
    • Identifies the missing requirement: States that the product of roots αβ = c/a = 3/2 = 1.5.

    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.

    Prove by contradiction that √2 is an irrational number.

    [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: Assumes, for contradiction, that √2 is rational, so √2 = p/q where p and q are integers with no common factors (in lowest terms) and q ≠ 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: Squares both sides: 2 = p²/q², so p² = 2q², meaning p² is even, so p itself must be even (write p = 2k). 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: Substitutes p = 2k: (2k)² = 2q² → 4k² = 2q² → q² = 2k², meaning q² is even, so q itself must also be even. 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 that if both p and q are even, they share a common factor of 2, contradicting the assumption that p/q was in lowest terms. 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: Concludes that the original assumption must be false, so √2 is irrational. 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: Proof by contradiction always follows the same pattern: assume the opposite of what you want to prove, derive a logical impossibility, then conclude the original statement must be true.

    Marking points

    • Assumes, for contradiction, that √2 is rational, so √2 = p/q where p and q are integers with no common factors (in lowest terms) and q ≠ 0.
    • Squares both sides: 2 = p²/q², so p² = 2q², meaning p² is even, so p itself must be even (write p = 2k).
    • Substitutes p = 2k: (2k)² = 2q² → 4k² = 2q² → q² = 2k², meaning q² is even, so q itself must also be even.
    • States that if both p and q are even, they share a common factor of 2, contradicting the assumption that p/q was in lowest terms.
    • Concludes that the original assumption must be false, so √2 is irrational.

    Examiner tip: Proof by contradiction always follows the same pattern: assume the opposite of what you want to prove, derive a logical impossibility, then conclude the original statement must be true.

28. 28.

    Marking analysis: A learner attempts the following task: “Prove by contradiction that √2 is an irrational number.” Their response addresses only this point: “Assumes, for contradiction, that √2 is rational, so √2 = p/q where p and q are integers with no common factors (in lowest terms) and q ≠ 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] · 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: Assumes, for contradiction, that √2 is rational, so √2 = p/q where p and q are integers with no common factors (in lowest terms) and q ≠ 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Squares both sides: 2 = p²/q², so p² = 2q², meaning p² is even, so p itself must be even (write p = 2k). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Substitutes p = 2k: (2k)² = 2q² → 4k² = 2q² → q² = 2k², meaning q² is even, so q itself must also be even. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: States that if both p and q are even, they share a common factor of 2, contradicting the assumption that p/q was in lowest terms. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: Concludes that the original assumption must be false, so √2 is irrational. 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: Assumes, for contradiction, that √2 is rational, so √2 = p/q where p and q are integers with no common factors (in lowest terms) and q ≠ 0.
    • Identifies the missing requirement: Squares both sides: 2 = p²/q², so p² = 2q², meaning p² is even, so p itself must be even (write p = 2k).
    • Identifies the missing requirement: Substitutes p = 2k: (2k)² = 2q² → 4k² = 2q² → q² = 2k², meaning q² is even, so q itself must also be even.
    • Identifies the missing requirement: States that if both p and q are even, they share a common factor of 2, contradicting the assumption that p/q was in lowest terms.
    • Identifies the missing requirement: Concludes that the original assumption must be false, so √2 is irrational.

    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.

    Find the number of ways in which 5 people can be seated around a circular table, where rotations of the same arrangement are considered identical.

    [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: States that for circular arrangements, one person's position can be fixed to remove rotational duplicates, leaving (n − 1)! distinct arrangements. 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 n = 5, giving (5 − 1)! = 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: Obtains 24 distinct seating arrangements. 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: Circular arrangements always have fewer distinct outcomes than linear arrangements of the same objects, since rotating the whole circle produces an arrangement that looks identical, not a new one.

    Marking points

    • States that for circular arrangements, one person's position can be fixed to remove rotational duplicates, leaving (n − 1)! distinct arrangements.
    • Substitutes n = 5, giving (5 − 1)! = 4!.
    • Obtains 24 distinct seating arrangements.

    Examiner tip: Circular arrangements always have fewer distinct outcomes than linear arrangements of the same objects, since rotating the whole circle produces an arrangement that looks identical, not a new one.

30. 30.

    Marking analysis: A learner attempts the following task: “Find the number of ways in which 5 people can be seated around a circular table, where rotations of the same arrangement are considered identical.” Their response addresses only this point: “States that for circular arrangements, one person's position can be fixed to remove rotational duplicates, leaving (n − 1)! distinct arrangements.” 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: States that for circular arrangements, one person's position can be fixed to remove rotational duplicates, leaving (n − 1)! distinct arrangements. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes n = 5, giving (5 − 1)! = 4!. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Obtains 24 distinct seating arrangements. 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 for circular arrangements, one person's position can be fixed to remove rotational duplicates, leaving (n − 1)! distinct arrangements.
    • Identifies the missing requirement: Substitutes n = 5, giving (5 − 1)! = 4!.
    • Identifies the missing requirement: Obtains 24 distinct seating arrangements.

    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.