Mathematics AA: Higher Level
Number and algebra — Topic 1
- 1.
Given z₁ = 3 + 2i and z₂ = 1 − 4i, calculate z₁ + z₂ and z₁z₂, giving each answer in the form a + bi.
[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.
[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.
[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.
[5 marks] · no calculator - 5.
Solve the quadratic equation z² − 4z + 13 = 0, giving both roots in the form a + bi.
[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.
[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.
[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.
[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°.
[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.
[5 marks] · no calculator - 11.
Prove by mathematical induction that 1 + 2 + 3 + … + n = n(n + 1)/2 for all positive integers n.
[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.
[5 marks] · no calculator - 13.
The polynomial p(x) = x³ − 5x² + 9x − 5 has 2 + i as one root. Find the other two roots.
[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.
[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.
[5 marks] - 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.
[5 marks] - 17.
Find the number of distinct arrangements of the letters in the word STATS.
[4 marks] - 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.
[4 marks] - 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.
[4 marks] - 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.
[4 marks] - 21.
Find the coefficient of x³ in the binomial expansion of (2x − 3)⁵.
[4 marks] - 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.
[4 marks] - 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.
[3 marks] - 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.
[3 marks] - 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.
[3 marks] - 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.
[3 marks] - 27.
Prove by contradiction that √2 is an irrational number.
[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.
[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.
[3 marks] - 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.
[3 marks]