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IB · MATH AA SL

Mathematics AA: Standard Level

Number and algebra — Topic 1

Name: ____________________Date: October 10, 2026
  1. 1.

    An arithmetic sequence has first term 5 and common difference 3. Find the 20th term and the sum of the first 20 terms.

    [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 uₙ = u₁ + (n − 1)d. 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 u₂₀ = 5 + 19(3) = 62. 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 Sₙ = n/2(u₁ + uₙ). 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 S₂₀ = 10(5 + 62) = 670. 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: Once the last required term is found, the sum formula using first and last term is usually faster than the alternative form.

    Marking points

    • Uses uₙ = u₁ + (n − 1)d.
    • Obtains u₂₀ = 5 + 19(3) = 62.
    • Uses Sₙ = n/2(u₁ + uₙ).
    • Obtains S₂₀ = 10(5 + 62) = 670.

    Examiner tip: Once the last required term is found, the sum formula using first and last term is usually faster than the alternative form.

  2. 2.

    Marking analysis: A learner attempts the following task: “An arithmetic sequence has first term 5 and common difference 3. Find the 20th term and the sum of the first 20 terms.” Their response addresses only this point: “Uses uₙ = u₁ + (n − 1)d.” 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 uₙ = u₁ + (n − 1)d. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Obtains u₂₀ = 5 + 19(3) = 62. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Uses Sₙ = n/2(u₁ + uₙ). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains S₂₀ = 10(5 + 62) = 670. 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 uₙ = u₁ + (n − 1)d.
    • Identifies the missing requirement: Obtains u₂₀ = 5 + 19(3) = 62.
    • Identifies the missing requirement: Uses Sₙ = n/2(u₁ + uₙ).
    • Identifies the missing requirement: Obtains S₂₀ = 10(5 + 62) = 670.

    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.

    A geometric sequence has first term 8 and common ratio 1.5. Find the 6th term and determine whether the sequence converges.

    [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 uₙ = u₁rⁿ⁻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: Obtains u₆ = 8(1.5)⁵ ≈ 60.75. 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 convergence condition |r| < 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: Concludes the sequence diverges because |1.5| > 1. 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 geometric sequence only has a finite sum to infinity when |r| < 1; check this before attempting such a calculation.

    Marking points

    • Uses uₙ = u₁rⁿ⁻¹.
    • Obtains u₆ = 8(1.5)⁵ ≈ 60.75.
    • States the convergence condition |r| < 1.
    • Concludes the sequence diverges because |1.5| > 1.

    Examiner tip: A geometric sequence only has a finite sum to infinity when |r| < 1; check this before attempting such a calculation.

  4. 4.

    Marking analysis: A learner attempts the following task: “A geometric sequence has first term 8 and common ratio 1.5. Find the 6th term and determine whether the sequence converges.” Their response addresses only this point: “Uses uₙ = u₁rⁿ⁻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]

    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 uₙ = u₁rⁿ⁻¹. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Obtains u₆ = 8(1.5)⁵ ≈ 60.75. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: States the convergence condition |r| < 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Concludes the sequence diverges because |1.5| > 1. 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 uₙ = u₁rⁿ⁻¹.
    • Identifies the missing requirement: Obtains u₆ = 8(1.5)⁵ ≈ 60.75.
    • Identifies the missing requirement: States the convergence condition |r| < 1.
    • Identifies the missing requirement: Concludes the sequence diverges because |1.5| > 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.

  5. 5.

    Find the sum to infinity of the geometric series 12 + 6 + 3 + 1.5 + ...

    [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: Identifies the common ratio r = 0.5. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Confirms |r| < 1, so the sum to infinity exists. 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 S∞ = u₁/(1 − r). 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 S∞ = 12/0.5 = 24. 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 verify the convergence condition before applying the infinite-sum formula, not just when explicitly asked.

    Marking points

    • Identifies the common ratio r = 0.5.
    • Confirms |r| < 1, so the sum to infinity exists.
    • Uses S∞ = u₁/(1 − r).
    • Obtains S∞ = 12/0.5 = 24.

    Examiner tip: Always verify the convergence condition before applying the infinite-sum formula, not just when explicitly asked.

  6. 6.

    Marking analysis: A learner attempts the following task: “Find the sum to infinity of the geometric series 12 + 6 + 3 + 1.5 + ...” Their response addresses only this point: “Identifies the common ratio r = 0.5.” 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 common ratio r = 0.5. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Confirms |r| < 1, so the sum to infinity exists. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Uses S∞ = u₁/(1 − r). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains S∞ = 12/0.5 = 24. 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 common ratio r = 0.5.
    • Identifies the missing requirement: Confirms |r| < 1, so the sum to infinity exists.
    • Identifies the missing requirement: Uses S∞ = u₁/(1 − r).
    • Identifies the missing requirement: Obtains S∞ = 12/0.5 = 24.

    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.

    Find the term independent of x in the binomial expansion of (2x + 1/x)⁶.

    [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 general term as ₆Cᵣ(2x)⁶⁻ʳ(1/x)ʳ. 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 power of x to x⁶⁻²ʳ. 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: Sets 6 − 2r = 0 to find the term independent of x. 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 r = 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Work through this mathematical step: Calculates the term: ₆C₃(2)³ = 20 × 8 = 160. 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: Set the exponent of x equal to zero to locate the independent term before calculating its numerical value.

    Marking points

    • Writes the general term as ₆Cᵣ(2x)⁶⁻ʳ(1/x)ʳ.
    • Simplifies the power of x to x⁶⁻²ʳ.
    • Sets 6 − 2r = 0 to find the term independent of x.
    • Obtains r = 3.
    • Calculates the term: ₆C₃(2)³ = 20 × 8 = 160.

    Examiner tip: Set the exponent of x equal to zero to locate the independent term before calculating its numerical value.

  8. 8.

    Marking analysis: A learner attempts the following task: “Find the term independent of x in the binomial expansion of (2x + 1/x)⁶.” Their response addresses only this point: “Writes the general term as ₆Cᵣ(2x)⁶⁻ʳ(1/x)ʳ.” 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 general term as ₆Cᵣ(2x)⁶⁻ʳ(1/x)ʳ. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Simplifies the power of x to x⁶⁻²ʳ. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Sets 6 − 2r = 0 to find the term independent of x. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains r = 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: Calculates the term: ₆C₃(2)³ = 20 × 8 = 160. 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 general term as ₆Cᵣ(2x)⁶⁻ʳ(1/x)ʳ.
    • Identifies the missing requirement: Simplifies the power of x to x⁶⁻²ʳ.
    • Identifies the missing requirement: Sets 6 − 2r = 0 to find the term independent of x.
    • Identifies the missing requirement: Obtains r = 3.
    • Identifies the missing requirement: Calculates the term: ₆C₃(2)³ = 20 × 8 = 160.

    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.

    Solve for x: log₃(x) = 2 log₃(5) − log₃(4).

    [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 power law: 2 log₃(5) = log₃(25). 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 quotient law: log₃(25) − log₃(4) = log₃(25/4). 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: Equates log₃(x) = log₃(25/4). 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 x = 25/4 = 6.25. 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: Combine the right-hand side into a single logarithm before equating arguments; do not equate coefficients directly.

    Marking points

    • Uses the power law: 2 log₃(5) = log₃(25).
    • Uses the quotient law: log₃(25) − log₃(4) = log₃(25/4).
    • Equates log₃(x) = log₃(25/4).
    • States x = 25/4 = 6.25.

    Examiner tip: Combine the right-hand side into a single logarithm before equating arguments; do not equate coefficients directly.

  10. 10.

    Marking analysis: A learner attempts the following task: “Solve for x: log₃(x) = 2 log₃(5) − log₃(4).” Their response addresses only this point: “Uses the power law: 2 log₃(5) = log₃(25).” 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 power law: 2 log₃(5) = log₃(25). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Uses the quotient law: log₃(25) − log₃(4) = log₃(25/4). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Equates log₃(x) = log₃(25/4). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: States x = 25/4 = 6.25. 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 power law: 2 log₃(5) = log₃(25).
    • Identifies the missing requirement: Uses the quotient law: log₃(25) − log₃(4) = log₃(25/4).
    • Identifies the missing requirement: Equates log₃(x) = log₃(25/4).
    • Identifies the missing requirement: States x = 25/4 = 6.25.

    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.

    $3000 is invested at a nominal annual interest rate of 6%, compounded monthly. Find the value of the investment after 4 years.

    [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 compound interest formula FV = PV(1 + r/n)^(nt). 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 PV = 3000, r = 0.06, n = 12, t = 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 (1 + 0.06/12)^48 ≈ 1.2705. 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 FV ≈ $3811.47. 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 interest rate and time must both be adjusted to match the compounding period (monthly, here), not left as annual figures.

    Marking points

    • Uses the compound interest formula FV = PV(1 + r/n)^(nt).
    • Substitutes PV = 3000, r = 0.06, n = 12, t = 4.
    • Calculates (1 + 0.06/12)^48 ≈ 1.2705.
    • Obtains FV ≈ $3811.47.

    Examiner tip: The interest rate and time must both be adjusted to match the compounding period (monthly, here), not left as annual figures.

  12. 12.

    Marking analysis: A learner attempts the following task: “$3000 is invested at a nominal annual interest rate of 6%, compounded monthly. Find the value of the investment after 4 years.” Their response addresses only this point: “Uses the compound interest formula FV = PV(1 + r/n)^(nt).” 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 compound interest formula FV = PV(1 + r/n)^(nt). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Substitutes PV = 3000, r = 0.06, n = 12, t = 4. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Calculates (1 + 0.06/12)^48 ≈ 1.2705. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains FV ≈ $3811.47. 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 compound interest formula FV = PV(1 + r/n)^(nt).
    • Identifies the missing requirement: Substitutes PV = 3000, r = 0.06, n = 12, t = 4.
    • Identifies the missing requirement: Calculates (1 + 0.06/12)^48 ≈ 1.2705.
    • Identifies the missing requirement: Obtains FV ≈ $3811.47.

    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 sum of the first n terms of an arithmetic sequence is given by Sₙ = n² + 3n. Find the first term and the common difference of the sequence.

    [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: Finds the first term using u₁ = S₁ = 1 + 3 = 4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Finds S₂ = 4 + 6 = 10. 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 u₂ = S₂ − S₁ = 10 − 4 = 6. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: Calculates the common difference d = u₂ − u₁ = 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 first term as 4 and the common difference as 2. 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 term of a sequence can be recovered from its sum function using uₙ = Sₙ − Sₙ₋₁ for n ≥ 2.

    Marking points

    • Finds the first term using u₁ = S₁ = 1 + 3 = 4.
    • Finds S₂ = 4 + 6 = 10.
    • Calculates u₂ = S₂ − S₁ = 10 − 4 = 6.
    • Calculates the common difference d = u₂ − u₁ = 2.
    • States the first term as 4 and the common difference as 2.

    Examiner tip: A term of a sequence can be recovered from its sum function using uₙ = Sₙ − Sₙ₋₁ for n ≥ 2.

  14. 14.

    Marking analysis: A learner attempts the following task: “The sum of the first n terms of an arithmetic sequence is given by Sₙ = n² + 3n. Find the first term and the common difference of the sequence.” Their response addresses only this point: “Finds the first term using u₁ = S₁ = 1 + 3 = 4.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [5 marks]

    Answer explanation

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

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: Finds the first term using u₁ = S₁ = 1 + 3 = 4. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Finds S₂ = 4 + 6 = 10. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Calculates u₂ = S₂ − S₁ = 10 − 4 = 6. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Calculates the common difference d = u₂ − u₁ = 2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: States the first term as 4 and the common difference as 2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: Finds the first term using u₁ = S₁ = 1 + 3 = 4.
    • Identifies the missing requirement: Finds S₂ = 4 + 6 = 10.
    • Identifies the missing requirement: Calculates u₂ = S₂ − S₁ = 10 − 4 = 6.
    • Identifies the missing requirement: Calculates the common difference d = u₂ − u₁ = 2.
    • Identifies the missing requirement: States the first term as 4 and the common difference as 2.

    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.

    Simplify (2⁵ × 2⁻²)/2³ without using a calculator, giving your answer as a single power of 2.

    [3 marks] · no calculator

    Answer explanation

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

    1. Keep each expression equivalent to the previous one. Expand brackets with their signs intact, combine only like terms, or take out a common factor as requested. Check an algebraic result by expanding it back or substituting a permitted value.
    2. Work through this mathematical step: Combines the numerator using the multiplication law: 2⁵⁻² = 2³. 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: Applies the division law: 2³ ÷ 2³. 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 result as 2⁰ = 1. 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: Simplify fully to a single power of 2 before evaluating; leaving the answer as 2⁰ instead of 1 may lose the final mark depending on the command.

    Marking points

    • Combines the numerator using the multiplication law: 2⁵⁻² = 2³.
    • Applies the division law: 2³ ÷ 2³.
    • States the result as 2⁰ = 1.

    Examiner tip: Simplify fully to a single power of 2 before evaluating; leaving the answer as 2⁰ instead of 1 may lose the final mark depending on the command.

  16. 16.

    Marking analysis: A learner attempts the following task: “Simplify (2⁵ × 2⁻²)/2³ without using a calculator, giving your answer as a single power of 2.” Their response addresses only this point: “Combines the numerator using the multiplication law: 2⁵⁻² = 2³.” 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: Combines the numerator using the multiplication law: 2⁵⁻² = 2³. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Applies the division law: 2³ ÷ 2³. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: States the result as 2⁰ = 1. 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: Combines the numerator using the multiplication law: 2⁵⁻² = 2³.
    • Identifies the missing requirement: Applies the division law: 2³ ÷ 2³.
    • Identifies the missing requirement: States the result as 2⁰ = 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.

  17. 17.

    Use proof by induction to show that 1 + 2 + 3 + ... + n = n(n+1)/2 for all positive integers n. Show the base case and the inductive step.

    [6 marks] · no calculator

    Answer explanation

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

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: 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. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: Simplifies the right-hand side to 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.
    6. Work through this mathematical step: Factorises to obtain (k+1)(k+2)/2, matching the required form for n = k+1. 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: Concludes that since the statement holds for n = 1 and n = k implies n = k+1, it is true for all positive integers n by induction. 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 complete induction proof needs all four parts: base case, hypothesis, inductive step, and a concluding statement — missing the conclusion loses marks even with correct 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.
    • Simplifies the right-hand side to k(k+1)/2 + (k+1).
    • Factorises to obtain (k+1)(k+2)/2, matching the required form for n = k+1.
    • Concludes that since the statement holds for n = 1 and n = k implies n = k+1, it is true for all positive integers n by induction.

    Examiner tip: A complete induction proof needs all four parts: base case, hypothesis, inductive step, and a concluding statement — missing the conclusion loses marks even with correct algebra.

  18. 18.

    Marking analysis: A learner attempts the following task: “Use proof by induction to show that 1 + 2 + 3 + ... + n = n(n+1)/2 for all positive integers n. Show the base case and the inductive step.” 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 6-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [6 marks] · no calculator

    Answer explanation

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

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: 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. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Simplifies the right-hand side to k(k+1)/2 + (k+1). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: Factorises to obtain (k+1)(k+2)/2, matching the required form for n = k+1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    7. Requirement 6: Identifies the missing requirement: Concludes that since the statement holds for n = 1 and n = k implies n = k+1, it is true for all positive integers n by induction. 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: 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.
    • Identifies the missing requirement: Simplifies the right-hand side to k(k+1)/2 + (k+1).
    • Identifies the missing requirement: Factorises to obtain (k+1)(k+2)/2, matching the required form for n = k+1.
    • Identifies the missing requirement: Concludes that since the statement holds for n = 1 and n = k implies n = k+1, it is true for all positive integers n by induction.

    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.

    The first three terms of an arithmetic sequence are 2k, k + 8 and 3k − 4. Find the value of k.

    [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 constant-difference property: (k + 8) − 2k = (3k − 4) − (k + 8). 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 left side to 8 − k. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Simplifies the right side to 2k − 12. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: Solves 8 − k = 2k − 12 to obtain k = 20/3. 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: In an arithmetic sequence, the difference between any two consecutive terms must be equal; set this up as a single equation.

    Marking points

    • Uses the constant-difference property: (k + 8) − 2k = (3k − 4) − (k + 8).
    • Simplifies the left side to 8 − k.
    • Simplifies the right side to 2k − 12.
    • Solves 8 − k = 2k − 12 to obtain k = 20/3.

    Examiner tip: In an arithmetic sequence, the difference between any two consecutive terms must be equal; set this up as a single equation.

  20. 20.

    Marking analysis: A learner attempts the following task: “The first three terms of an arithmetic sequence are 2k, k + 8 and 3k − 4. Find the value of k.” Their response addresses only this point: “Uses the constant-difference property: (k + 8) − 2k = (3k − 4) − (k + 8).” 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 constant-difference property: (k + 8) − 2k = (3k − 4) − (k + 8). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Simplifies the left side to 8 − k. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Simplifies the right side to 2k − 12. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Solves 8 − k = 2k − 12 to obtain k = 20/3. 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 constant-difference property: (k + 8) − 2k = (3k − 4) − (k + 8).
    • Identifies the missing requirement: Simplifies the left side to 8 − k.
    • Identifies the missing requirement: Simplifies the right side to 2k − 12.
    • Identifies the missing requirement: Solves 8 − k = 2k − 12 to obtain k = 20/3.

    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.

  21. 21.

    A worker's salary increases by $500 each year, starting at $30 000 in the first year. Calculate the worker's total earnings over the first 10 years of employment.

    [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 this as an arithmetic sequence with u₁ = 30 000 and d = 500. 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 Sₙ = n/2(2u₁ + (n − 1)d). 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 S₁₀ = 10/2(2(30 000) + 9(500)). 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 S₁₀ = $322 500. 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 steadily increasing (or decreasing) real-world quantity by a fixed amount each period is always an arithmetic sequence — recognising this lets you apply the standard sum formula directly instead of adding terms one by one.

    Marking points

    • Identifies this as an arithmetic sequence with u₁ = 30 000 and d = 500.
    • Uses Sₙ = n/2(2u₁ + (n − 1)d).
    • Substitutes S₁₀ = 10/2(2(30 000) + 9(500)).
    • Obtains S₁₀ = $322 500.

    Examiner tip: A steadily increasing (or decreasing) real-world quantity by a fixed amount each period is always an arithmetic sequence — recognising this lets you apply the standard sum formula directly instead of adding terms one by one.

  22. 22.

    Marking analysis: A learner attempts the following task: “A worker's salary increases by $500 each year, starting at $30 000 in the first year. Calculate the worker's total earnings over the first 10 years of employment.” Their response addresses only this point: “Identifies this as an arithmetic sequence with u₁ = 30 000 and d = 500.” 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 this as an arithmetic sequence with u₁ = 30 000 and d = 500. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Uses Sₙ = n/2(2u₁ + (n − 1)d). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Substitutes S₁₀ = 10/2(2(30 000) + 9(500)). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains S₁₀ = $322 500. 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 this as an arithmetic sequence with u₁ = 30 000 and d = 500.
    • Identifies the missing requirement: Uses Sₙ = n/2(2u₁ + (n − 1)d).
    • Identifies the missing requirement: Substitutes S₁₀ = 10/2(2(30 000) + 9(500)).
    • Identifies the missing requirement: Obtains S₁₀ = $322 500.

    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.

    Find the sum of the first 8 terms of the geometric series 3 + 6 + 12 + 24 + ...

    [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 u₁ = 3 and r = 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: Uses Sₙ = u₁(rⁿ − 1)/(r − 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: Substitutes S₈ = 3(2⁸ − 1)/(2 − 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 S₈ = 765. 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: Use the finite geometric sum formula Sₙ = u₁(rⁿ − 1)/(r − 1) whenever a specific, finite number of terms is required — the infinite-sum formula only applies when |r| < 1 and the series continues forever.

    Marking points

    • Identifies u₁ = 3 and r = 2.
    • Uses Sₙ = u₁(rⁿ − 1)/(r − 1).
    • Substitutes S₈ = 3(2⁸ − 1)/(2 − 1).
    • Obtains S₈ = 765.

    Examiner tip: Use the finite geometric sum formula Sₙ = u₁(rⁿ − 1)/(r − 1) whenever a specific, finite number of terms is required — the infinite-sum formula only applies when |r| < 1 and the series continues forever.

  24. 24.

    Marking analysis: A learner attempts the following task: “Find the sum of the first 8 terms of the geometric series 3 + 6 + 12 + 24 + ...” Their response addresses only this point: “Identifies u₁ = 3 and r = 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]

    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 u₁ = 3 and r = 2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Uses Sₙ = u₁(rⁿ − 1)/(r − 1). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Substitutes S₈ = 3(2⁸ − 1)/(2 − 1). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Obtains S₈ = 765. 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 u₁ = 3 and r = 2.
    • Identifies the missing requirement: Uses Sₙ = u₁(rⁿ − 1)/(r − 1).
    • Identifies the missing requirement: Substitutes S₈ = 3(2⁸ − 1)/(2 − 1).
    • Identifies the missing requirement: Obtains S₈ = 765.

    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.

    Simplify 8^(2/3) × 8^(−1/3) without using a calculator, giving your answer as an integer.

    [4 marks] · no calculator

    Answer explanation

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

    1. Keep each expression equivalent to the previous one. Expand brackets with their signs intact, combine only like terms, or take out a common factor as requested. Check an algebraic result by expanding it back or substituting a permitted value.
    2. Work through this mathematical step: Uses the law aᵐ × aⁿ = aᵐ⁺ⁿ. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Simplifies the exponent: 2/3 − 1/3 = 1/3. 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: Obtains 8^(1/3). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: States the value as 2, since 2³ = 8. 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 fractional exponent 1/n represents the nth root — recognising a number as a perfect cube, square, or other power avoids needing a calculator entirely.

    Marking points

    • Uses the law aᵐ × aⁿ = aᵐ⁺ⁿ.
    • Simplifies the exponent: 2/3 − 1/3 = 1/3.
    • Obtains 8^(1/3).
    • States the value as 2, since 2³ = 8.

    Examiner tip: A fractional exponent 1/n represents the nth root — recognising a number as a perfect cube, square, or other power avoids needing a calculator entirely.

  26. 26.

    Marking analysis: A learner attempts the following task: “Simplify 8^(2/3) × 8^(−1/3) without using a calculator, giving your answer as an integer.” Their response addresses only this point: “Uses the law aᵐ × aⁿ = aᵐ⁺ⁿ.” 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 law aᵐ × aⁿ = aᵐ⁺ⁿ. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Simplifies the exponent: 2/3 − 1/3 = 1/3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Obtains 8^(1/3). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: States the value as 2, since 2³ = 8. 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 law aᵐ × aⁿ = aᵐ⁺ⁿ.
    • Identifies the missing requirement: Simplifies the exponent: 2/3 − 1/3 = 1/3.
    • Identifies the missing requirement: Obtains 8^(1/3).
    • Identifies the missing requirement: States the value as 2, since 2³ = 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.

  27. 27.

    Expand (x − 2)⁴ fully, giving your answer in the form ax⁴ + bx³ + cx² + dx + e.

    [5 marks] · no calculator

    Answer explanation

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

    1. Keep each expression equivalent to the previous one. Expand brackets with their signs intact, combine only like terms, or take out a common factor as requested. Check an algebraic result by expanding it back or substituting a permitted value.
    2. Work through this mathematical step: States the general term of the expansion: ⁴Cᵣ x⁴⁻ʳ(−2)ʳ. 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 terms for r = 0 and r = 1: x⁴ and 4x³(−2) = −8x³. 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 terms for r = 2 and r = 3: 6x²(−2)² = 24x² and 4x(−2)³ = −32x. 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 term for r = 4: (−2)⁴ = 16. 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 full expansion x⁴ − 8x³ + 24x² − 32x + 16. 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 signs of the terms alternate whenever the second term of the binomial is negative — always track the sign of (−2)ʳ carefully for each value of r.

    Marking points

    • States the general term of the expansion: ⁴Cᵣ x⁴⁻ʳ(−2)ʳ.
    • Calculates the terms for r = 0 and r = 1: x⁴ and 4x³(−2) = −8x³.
    • Calculates the terms for r = 2 and r = 3: 6x²(−2)² = 24x² and 4x(−2)³ = −32x.
    • Calculates the term for r = 4: (−2)⁴ = 16.
    • States the full expansion x⁴ − 8x³ + 24x² − 32x + 16.

    Examiner tip: The signs of the terms alternate whenever the second term of the binomial is negative — always track the sign of (−2)ʳ carefully for each value of r.

  28. 28.

    Marking analysis: A learner attempts the following task: “Expand (x − 2)⁴ fully, giving your answer in the form ax⁴ + bx³ + cx² + dx + e.” Their response addresses only this point: “States the general term of the expansion: ⁴Cᵣ x⁴⁻ʳ(−2)ʳ.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [5 marks] · no calculator

    Answer explanation

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

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: States the general term of the expansion: ⁴Cᵣ x⁴⁻ʳ(−2)ʳ. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Calculates the terms for r = 0 and r = 1: x⁴ and 4x³(−2) = −8x³. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Calculates the terms for r = 2 and r = 3: 6x²(−2)² = 24x² and 4x(−2)³ = −32x. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Calculates the term for r = 4: (−2)⁴ = 16. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: States the full expansion x⁴ − 8x³ + 24x² − 32x + 16. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: States the general term of the expansion: ⁴Cᵣ x⁴⁻ʳ(−2)ʳ.
    • Identifies the missing requirement: Calculates the terms for r = 0 and r = 1: x⁴ and 4x³(−2) = −8x³.
    • Identifies the missing requirement: Calculates the terms for r = 2 and r = 3: 6x²(−2)² = 24x² and 4x(−2)³ = −32x.
    • Identifies the missing requirement: Calculates the term for r = 4: (−2)⁴ = 16.
    • Identifies the missing requirement: States the full expansion x⁴ − 8x³ + 24x² − 32x + 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.

  29. 29.

    Write the series 2 + 5 + 8 + 11 + ... + 29 using sigma notation, and hence calculate its sum.

    [5 marks]

    Answer explanation

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

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Identifies the series as arithmetic with first term 2 and common difference 3, giving general term 3n − 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: Finds the number of terms by solving 3n − 1 = 29 to obtain n = 10. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Writes the sigma notation Σₙ₌₁¹⁰ (3n − 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: Uses Sₙ = n/2(2a + (n − 1)d) to substitute S₁₀ = 10/2(2(2) + 9(3)). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Work through this mathematical step: Obtains the sum = 155. 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: To write a series in sigma notation, first find the general term as a function of n, then find the range of n values (start and end) that produce every term in the series.

    Marking points

    • Identifies the series as arithmetic with first term 2 and common difference 3, giving general term 3n − 1.
    • Finds the number of terms by solving 3n − 1 = 29 to obtain n = 10.
    • Writes the sigma notation Σₙ₌₁¹⁰ (3n − 1).
    • Uses Sₙ = n/2(2a + (n − 1)d) to substitute S₁₀ = 10/2(2(2) + 9(3)).
    • Obtains the sum = 155.

    Examiner tip: To write a series in sigma notation, first find the general term as a function of n, then find the range of n values (start and end) that produce every term in the series.

  30. 30.

    Marking analysis: A learner attempts the following task: “Write the series 2 + 5 + 8 + 11 + ... + 29 using sigma notation, and hence calculate its sum.” Their response addresses only this point: “Identifies the series as arithmetic with first term 2 and common difference 3, giving general term 3n − 1.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.

    [5 marks]

    Answer explanation

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

    1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
    2. Requirement 1: Recognises credit for the stated point: Identifies the series as arithmetic with first term 2 and common difference 3, giving general term 3n − 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: Finds the number of terms by solving 3n − 1 = 29 to obtain n = 10. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: Writes the sigma notation Σₙ₌₁¹⁰ (3n − 1). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    5. Requirement 4: Identifies the missing requirement: Uses Sₙ = n/2(2a + (n − 1)d) to substitute S₁₀ = 10/2(2(2) + 9(3)). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    6. Requirement 5: Identifies the missing requirement: Obtains the sum = 155. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.

    Marking points

    • Recognises credit for the stated point: Identifies the series as arithmetic with first term 2 and common difference 3, giving general term 3n − 1.
    • Identifies the missing requirement: Finds the number of terms by solving 3n − 1 = 29 to obtain n = 10.
    • Identifies the missing requirement: Writes the sigma notation Σₙ₌₁¹⁰ (3n − 1).
    • Identifies the missing requirement: Uses Sₙ = n/2(2a + (n − 1)d) to substitute S₁₀ = 10/2(2(2) + 9(3)).
    • Identifies the missing requirement: Obtains the sum = 155.

    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.

    A geometric sequence has first term 6 and common ratio 0.8. Find the sum of the first four terms and the sum to infinity. Explain why the latter exists.

    [3 marks]

    Answer explanation

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

    1. The first four terms are 6, 4.8, 3.84 and 3.072; their direct sum verifies the finite-series result.
    2. The infinite sum follows by taking the limit in the finite-sum formula as 0.8^n tends to zero.

    Marking points

    • S4 = 6(1 - 0.8^4)/(1 - 0.8) = 17.712.
    • S infinity = 6/(1 - 0.8) = 30.
    • It converges because |0.8| < 1, so powers of the ratio approach zero.

    Examiner tip: Convergence depends on the absolute value of the common ratio.

  32. 32.

    Marking analysis: A learner attempts the following task: “A geometric sequence has first term 6 and common ratio 0.8. Find the sum of the first four terms and the sum to infinity. Explain why the latter exists.” Their response addresses only this point: “S4 = 6(1 - 0.8^4)/(1 - 0.8) = 17.712.” 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: S4 = 6(1 - 0.8^4)/(1 - 0.8) = 17.712. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    3. Requirement 2: Identifies the missing requirement: S infinity = 6/(1 - 0.8) = 30. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
    4. Requirement 3: Identifies the missing requirement: It converges because |0.8| < 1, so powers of the ratio approach zero. 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: S4 = 6(1 - 0.8^4)/(1 - 0.8) = 17.712.
    • Identifies the missing requirement: S infinity = 6/(1 - 0.8) = 30.
    • Identifies the missing requirement: It converges because |0.8| < 1, so powers of the ratio approach zero.

    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.