WACE Mathematics Specialist Practice Questions
The 64 practice questions inside the WACE Mathematics Specialist Mastery Pack, grouped by area of study. Every question comes with a full worked solution.
- Topic 3.1: Complex numbers
- Multiple choice × 11
- Show that × 1
- Sketch × 1
- Determine × 1
- Solve × 1
- Topic 3.2: Functions and sketching graphs
- Multiple choice × 7
- Determine × 1
- Sketch × 1
- Topic 3.3: Vectors in three dimensions
- Multiple choice × 10
- Prove × 1
- Determine × 1
- Explain × 1
- Calculate × 1
- Topic 4.1: Integration and applications of integration
- Multiple choice × 6
- Evaluate × 1
- Show that × 1
- Topic 4.2: Rates of change and differential equations
- Multiple choice × 9
- Calculate × 1
- Show that × 1
- Determine × 1
- Topic 4.3: Statistical inference
- Multiple choice × 5
- Justify × 1
Show the worked answer
Answer: Worked solution
(a) |z| = √(3 + 1) = 2. The point (√3, −1) is in the fourth quadrant with reference angle tan⁻¹(1/√3) = π/6, so Arg z = −π/6 and z = 2 cis(−π/6) (1 mark).
(b) By de Moivre's theorem z⁶ = 2⁶ cis(6 × (−π/6)) = 64 cis(−π) (1 mark). Since cos(−π) = −1 and sin(−π) = 0, z⁶ = 64(−1 + 0i) = −64, which is real (1 mark).
(c) zⁿ = 2ⁿ cis(−nπ/6) is purely imaginary when −nπ/6 is an odd multiple of π/2, i.e. n/6 = 1/2, 3/2, … so n = 3, 9, … The smallest is n = 3, with z³ = 8 cis(−π/2) = −8i (1 mark).
Marking key: 1 mark exact modulus and argument; 1 mark correct use of de Moivre; 1 mark showing the imaginary part is zero and the value −64; 1 mark n = 3 with justification. A bare 'z⁶ = −64' from expanding without de Moivre does not meet the 'use de Moivre' instruction.
20 full-length practice exams with worked solutions, 20 revision notes, 64 practice questions and 200 flashcards.
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