School / IB / MATH AA SL / Number and algebra Question practice
Number and algebra About this practice Sequences, series, the binomial theorem and exponent laws.
Mathematics AA: Standard Level Number and algebra
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1 An arithmetic sequence has first term 5 and common difference 3. Find the 20th term and the sum of the first 20 terms. Paper 2 style Medium 4 marks Calculator + 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. Marking analysis Medium 4 marks Calculator + 3 A geometric sequence has first term 8 and common ratio 1.5. Find the 6th term and determine whether the sequence converges. Paper 2 style Medium 4 marks Calculator + 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. Marking analysis Medium 4 marks Calculator + 5 Find the sum to infinity of the geometric series 12 + 6 + 3 + 1.5 + ... Paper 2 style Medium 4 marks Calculator + 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. Marking analysis Medium 4 marks Calculator + 7 Find the term independent of x in the binomial expansion of (2x + 1/x)⁶. Paper 2 style Medium 5 marks Calculator + 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. Marking analysis Medium 5 marks Calculator + 9 Solve for x: log₃(x) = 2 log₃(5) − log₃(4). Paper 1 style Medium 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. Marking analysis Medium 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. Paper 2 style Medium 4 marks Calculator + 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. 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. Marking analysis Medium 4 marks Calculator + 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. Paper 2 style Medium 5 marks Calculator + 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. Marking analysis Medium 5 marks Calculator + 15 Simplify (2⁵ × 2⁻²)/2³ without using a calculator, giving your answer as a single power of 2. Paper 1 style Easy 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. Marking analysis Easy 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. Paper 1 style Hard 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. Marking analysis Hard 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. Paper 2 style Medium 4 marks Calculator + 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. Marking analysis Medium 4 marks Calculator + 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. Paper 2 style Medium 4 marks Calculator + 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. Marking analysis Medium 4 marks Calculator + 23 Find the sum of the first 8 terms of the geometric series 3 + 6 + 12 + 24 + ... Paper 2 style Medium 4 marks Calculator + 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. Marking analysis Medium 4 marks Calculator + 25 Simplify 8^(2/3) × 8^(−1/3) without using a calculator, giving your answer as an integer. Paper 1 style Medium 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. Marking analysis Medium 4 marks No calculator + 27 Expand (x − 2)⁴ fully, giving your answer in the form ax⁴ + bx³ + cx² + dx + e. Paper 1 style Medium 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. Marking analysis Medium 5 marks No calculator + 29 Write the series 2 + 5 + 8 + 11 + ... + 29 using sigma notation, and hence calculate its sum. Paper 2 style Medium 5 marks Calculator + 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. Marking analysis Medium 5 marks Calculator + 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. Paper 2 style Easy 3 marks Calculator + 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. 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 - 0.84 )/(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. Marking analysis Easy 3 marks Calculator + Self-assessed Not marked yet
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