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
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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 + 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 + Self-assessed Not marked yet
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