School / IB / MATH AA HL / Number and algebra Question practice
Number and algebra About this practice HL-only depth: counting principles, complex numbers in Cartesian and polar forms, proof, and linear systems.
Mathematics AA: Higher Level Number and algebra
View All Incomplete Complete Review mistakes One at a time
Revision Ladder All stages Easy Medium HardSoon
Paper All papers Paper 1 style Paper 2 style General practice (not paper-specific)
About paper groupings Paper labels are an unofficial, independently authored grouping, applied only where a question's own format genuinely matches a real paper convention (such as IB Mathematics AA's non-calculator/calculator split). They do not reproduce any exam board's real paper numbering or mark allocation, and uncertain questions are labelled general practice instead.
1 Given z₁ = 3 + 2i and z₂ = 1 − 4i, calculate z₁ + z₂ and z₁z₂, giving each answer in the form a + bi. Paper 1 style Medium 4 marks No calculator + 2 Marking analysis: A learner attempts the following task: “Given z₁ = 3 + 2i and z₂ = 1 − 4i, calculate z₁ + z₂ and z₁z₂, giving each answer in the form a + bi.” Their response addresses only this point: “Adds real and imaginary parts separately to obtain z₁ + z₂ = 4 − 2i.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks No calculator + 3 Express z = −2 + 2√3 i in the polar form r(cos θ + i sin θ), giving r and θ exactly. Paper 1 style Medium 5 marks No calculator + 4 Marking analysis: A learner attempts the following task: “Express z = −2 + 2√3 i in the polar form r(cos θ + i sin θ), giving r and θ exactly.” Their response addresses only this point: “Calculates r = |z| = √((−2)² + (2√3)²).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 5 Solve the quadratic equation z² − 4z + 13 = 0, giving both roots in the form a + bi. Paper 1 style Medium 4 marks No calculator + 6 Marking analysis: A learner attempts the following task: “Solve the quadratic equation z² − 4z + 13 = 0, giving both roots in the form a + bi.” Their response addresses only this point: “Uses the quadratic formula z = [4 ± √(16 − 52)]/2.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks No calculator + 7 Given z = 2(cos 30° + i sin 30°), use De Moivre's theorem to find z⁶ in the form a + bi. Paper 1 style Medium 5 marks No calculator + 8 Marking analysis: A learner attempts the following task: “Given z = 2(cos 30° + i sin 30°), use De Moivre's theorem to find z⁶ in the form a + bi.” Their response addresses only this point: “States De Moivre's theorem zⁿ = rⁿ(cos nθ + i sin nθ).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 9 Find the three cube roots of z = 8i, giving each in the form r(cos θ + i sin θ) with 0 ≤ θ < 360°. Paper 1 style Medium 5 marks No calculator + 10 Marking analysis: A learner attempts the following task: “Find the three cube roots of z = 8i, giving each in the form r(cos θ + i sin θ) with 0 ≤ θ < 360°.” Their response addresses only this point: “Writes z in polar form: 8i = 8(cos 90° + i sin 90°).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 11 Prove by mathematical induction that 1 + 2 + 3 + … + n = n(n + 1)/2 for all positive integers n. Paper 1 style Medium 5 marks No calculator + 12 Marking analysis: A learner attempts the following task: “Prove by mathematical induction that 1 + 2 + 3 + … + n = n(n + 1)/2 for all positive integers n.” Their response addresses only this point: “States the base case n = 1: LHS = 1, RHS = 1(2)/2 = 1, so the statement holds.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 13 The polynomial p(x) = x³ − 5x² + 9x − 5 has 2 + i as one root. Find the other two roots. Paper 1 style Medium 5 marks No calculator + 14 Marking analysis: A learner attempts the following task: “The polynomial p(x) = x³ − 5x² + 9x − 5 has 2 + i as one root. Find the other two roots.” Their response addresses only this point: “States that since p(x) has real coefficients, complex roots occur in conjugate pairs, so 2 − i is also a root.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 15 Solve the system of linear equations 2x + y − z = 3, x − y + 2z = 4, 3x + 2y − z = 7 using row reduction, or state if it has no unique solution. Paper 2 style Medium 5 marks Calculator + 16 Marking analysis: A learner attempts the following task: “Solve the system of linear equations 2x + y − z = 3, x − y + 2z = 4, 3x + 2y − z = 7 using row reduction, or state if it has no unique solution.” Their response addresses only this point: “Writes the augmented matrix for the system.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + 17 Find the number of distinct arrangements of the letters in the word STATS. Paper 2 style Medium 4 marks Calculator + 18 Marking analysis: A learner attempts the following task: “Find the number of distinct arrangements of the letters in the word STATS.” Their response addresses only this point: “Identifies that STATS has 5 letters, with S repeated twice and T repeated twice.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks Calculator + 19 A committee of 4 people is chosen from a group of 6 men and 5 women. Find the number of ways of forming a committee containing exactly 2 men and 2 women. Paper 2 style Medium 4 marks Calculator + 20 Marking analysis: A learner attempts the following task: “A committee of 4 people is chosen from a group of 6 men and 5 women. Find the number of ways of forming a committee containing exactly 2 men and 2 women.” Their response addresses only this point: “Recognises that order does not matter, so combinations are used for each group.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks Calculator + 21 Find the coefficient of x³ in the binomial expansion of (2x − 3)⁵. Paper 2 style Medium 4 marks Calculator + 22 Marking analysis: A learner attempts the following task: “Find the coefficient of x³ in the binomial expansion of (2x − 3)⁵.” Their response addresses only this point: “States the general term of the binomial expansion: ⁿCᵣ aⁿ⁻ʳ bʳ.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks Calculator + 23 An infinite geometric series has first term 8 and common ratio 0.5. Calculate the sum to infinity, stating the condition required for this sum to exist. Paper 2 style Easy 3 marks Calculator + 24 Marking analysis: A learner attempts the following task: “An infinite geometric series has first term 8 and common ratio 0.5. Calculate the sum to infinity, stating the condition required for this sum to exist.” Their response addresses only this point: “States that the condition |r| < 1 is required for the sum to infinity to exist.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Easy 3 marks Calculator + 25 The quadratic equation 2x² − 7x + 3 = 0 has roots α and β. Without solving for the roots directly, calculate the value of α + β and αβ. Paper 2 style Easy 3 marks Calculator + 26 Marking analysis: A learner attempts the following task: “The quadratic equation 2x² − 7x + 3 = 0 has roots α and β. Without solving for the roots directly, calculate the value of α + β and αβ.” Their response addresses only this point: “States that for ax² + bx + c = 0, the sum of roots α + β = −b/a.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Easy 3 marks Calculator + 27 Prove by contradiction that √2 is an irrational number. Paper 1 style Medium 5 marks No calculator + 28 Marking analysis: A learner attempts the following task: “Prove by contradiction that √2 is an irrational number.” Their response addresses only this point: “Assumes, for contradiction, that √2 is rational, so √2 = p/q where p and q are integers with no common factors (in lowest terms) and q ≠ 0.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 29 Find the number of ways in which 5 people can be seated around a circular table, where rotations of the same arrangement are considered identical. Paper 2 style Easy 3 marks Calculator + 30 Marking analysis: A learner attempts the following task: “Find the number of ways in which 5 people can be seated around a circular table, where rotations of the same arrangement are considered identical.” Their response addresses only this point: “States that for circular arrangements, one person's position can be fixed to remove rotational duplicates, leaving (n − 1)! distinct arrangements.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Easy 3 marks Calculator + Self-assessed Not marked yet
Set total 128
This is a self-study tool, not a certified answer key or a predicted IB grade.
Loading progress Reset progress
Before you practise
Questions learners ask about this practice Are these official IB questions? No. These are original SubjectScout practice questions for Number and algebra. They are not official past-paper questions or endorsed material.
What does this Mathematics AA: Higher Level practice page include? It includes selected topic questions, marking points, and feedback prompts designed to help learners practise before requesting teacher support.
Can I request a teacher for this exact topic? Yes. Tell SubjectScout the subject, exact topic, level and deadline, and the team will try to match you with a suitable verified teacher.
Unofficial content under accuracy, provenance, and rights review. Not affiliated with or endorsed by the International Baccalaureate Organization. Official past-paper questions are not reproduced here.
