Mathematics AA: Standard Level
Number and algebra — Topic 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] - 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] - 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] - 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] - 5.
Find the sum to infinity of the geometric series 12 + 6 + 3 + 1.5 + ...
[4 marks] - 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] - 7.
Find the term independent of x in the binomial expansion of (2x + 1/x)⁶.
[5 marks] - 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] - 9.
Solve for x: log₃(x) = 2 log₃(5) − log₃(4).
[4 marks] · no calculator - 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 - 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] - 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] - 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] - 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] - 15.
Simplify (2⁵ × 2⁻²)/2³ without using a calculator, giving your answer as a single power of 2.
[3 marks] · no calculator - 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 - 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 - 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 - 19.
The first three terms of an arithmetic sequence are 2k, k + 8 and 3k − 4. Find the value of k.
[4 marks] - 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] - 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] - 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] - 23.
Find the sum of the first 8 terms of the geometric series 3 + 6 + 12 + 24 + ...
[4 marks] - 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] - 25.
Simplify 8^(2/3) × 8^(−1/3) without using a calculator, giving your answer as an integer.
[4 marks] · no calculator - 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 - 27.
Expand (x − 2)⁴ fully, giving your answer in the form ax⁴ + bx³ + cx² + dx + e.
[5 marks] · no calculator - 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 - 29.
Write the series 2 + 5 + 8 + 11 + ... + 29 using sigma notation, and hence calculate its sum.
[5 marks] - 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] - 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] - 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]