ATARMAxxing · Specialist Mathematics
VCE Specialist Mathematics Practice Questions
64 exam-style questions · full worked solutions
The 64 practice questions inside the VCE Specialist Mathematics Mastery Pack, grouped by area of study. Every question comes with a full worked solution.
- AOS1
- Multiple choice × 5
- Prove × 4
- AOS2
- Multiple choice × 5
- Analyse × 1
- Sketch × 1
- Express × 1
- AOS3
- Multiple choice × 5
- Solve × 2
- Find × 1
- AOS5
- Multiple choice × 4
- Prove × 1
- Find × 2
- AOS4a
- Multiple choice × 5
- Integrate by parts × 1
- Use partial fractions × 1
- Arc length of a parametric curve × 1
- AOS4b
- Multiple choice × 4
- Form and solve a differential equation × 1
- Separation of variables × 1
- Logistic model application × 1
- AOS4c
- Multiple choice × 3
- Rectilinear motion with calculus × 1
- Acceleration as a function of position × 1
- Vector calculus: projectile motion × 1
- AOS6a
- Multiple choice × 3
- Linear combination of random variables × 1
- Difference of two independent normals × 1
- AOS6b
- Multiple choice × 3
- Construct a confidence interval × 1
- Two-tailed hypothesis test × 1
- One-tailed test with sample standard deviation × 1
Sample question
Prove by mathematical induction that for all integers n ≥ 1, 1·1! + 2·2! + 3·3! + … + n·n! = (n+1)! − 1. (4 marks)
Show the worked answer
Answer: Worked solution
Let P(n) be the statement 1·1! + 2·2! + … + n·n! = (n+1)! − 1.
Base step (n = 1): LHS = 1·1! = 1; RHS = 2! − 1 = 2 − 1 = 1. So LHS = RHS and P(1) is true.
Inductive step: assume P(k) holds for some integer k ≥ 1, i.e. 1·1! + … + k·k! = (k+1)! − 1.
Consider the sum to k+1 terms:
1·1! + … + k·k! + (k+1)(k+1)! = [(k+1)! − 1] + (k+1)(k+1)! (by the assumption)
= (k+1)!·[1 + (k+1)] − 1 = (k+1)!·(k+2) − 1 = (k+2)! − 1 = ((k+1)+1)! − 1.
This is exactly P(k+1). Hence P(k) ⇒ P(k+1). Since P(1) is true and the implication holds, by the principle of mathematical induction P(n) is true for all integers n ≥ 1. ∎
Included in the VCE 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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