ATARMAxxing · Specialist Mathematics
SACE Specialist Mathematics Practice Questions
64 exam-style questions · full worked solutions
The 64 practice questions inside the SACE Specialist Mathematics Mastery Pack, grouped by area of study. Every question comes with a full worked solution.
- Subtopic 1.1: Proof by mathematical induction
- Multiple choice × 3
- Prove × 1
- Subtopic 2.2: The complex (Argand) plane
- Multiple choice × 3
- Sketch × 1
- Subtopic 3.1: Composition of functions
- Multiple choice × 2
- Subtopic 4.1: The algebra of vectors in three dimensions
- Multiple choice × 2
- Subtopic 4.3: Systems of linear equations
- Multiple choice × 2
- Solve × 1
- Subtopic 5.2: Applications of integral calculus
- Multiple choice × 3
- Find × 1
- Subtopic 6.2: Differential equations
- Multiple choice × 4
- Solve × 1
- Subtopic 2.3: Roots of complex numbers
- Multiple choice × 2
- Find × 1
- Subtopic 3.2: One-to-one functions
- Multiple choice × 2
- Determine × 1
- Subtopic 4.2: Vector and Cartesian equations
- Multiple choice × 5
- Show that × 1
- Find × 1
- Subtopic 5.1: Integration techniques
- Multiple choice × 5
- Show that × 1
- Hence × 1
- Subtopic 6.3: Pairs of varying quantities — polynomials
- Multiple choice × 1
- Subtopic 2.1: Cartesian and polar forms
- Multiple choice × 3
- Show that × 1
- Subtopic 6.1: Implicit differentiation
- Multiple choice × 2
- Show that × 1
- Subtopic 6.4: Related rates, velocity, and tangents
- Multiple choice × 2
- Determine × 1
- Subtopic 2.4: Factorisation of polynomials
- Multiple choice × 3
- Hence × 1
- Subtopic 3.3: Sketching graphs
- Multiple choice × 3
- Draw × 1
- Subtopic 6.5: Trigonometric parameterisations
- Multiple choice × 1
Sample question
Written task 49: Prove by mathematical induction that 1+3+⋯+(2n−1)=n² for every positive integer n.
Show the worked answer
Answer: Worked solution
At n=1, the left side is1 and right side1²=1. Assume 1+3+⋯+(2k−1)=k². The next odd term is2(k+1)−1=2k+1. Adding it gives k²+2k+1=(k+1)², which is exactly the right side for n=k+1. The base and implication establish the statement for all positive integers by induction.
Included in the SACE Specialist Mathematics Mastery Pack
20 full-length practice exams with worked solutions, 20 revision notes, 64 practice questions and 200 flashcards.
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