TMUA Mathematical Reasoning & Logical Thinking 2 — Questions and Answers
Question 1: Which of the following is the contrapositive of the statement 'If n is divisible by 6, then n is divisible by 2'?
- If n is not divisible by 2, then n is not divisible by 6 (Correct answer)
- If n is divisible by 2, then n is divisible by 6
- If n is not divisible by 6, then n is not divisible by 2
- If n is divisible by 6, then n is not divisible by 2
Correct answer: If n is not divisible by 2, then n is not divisible by 6
The contrapositive of 'If P then Q' is 'If not Q then not P', so 'If n is not divisible by 2, then n is not divisible by 6'.
Question 2: A sequence is defined by a₁ = 3 and aₙ₊₁ = 2aₙ − 1. What is a₄?
- 15
- 17 (Correct answer)
- 13
- 19
Correct answer: 17
a₂ = 2(3)−1 = 5, a₃ = 2(5)−1 = 9, a₄ = 2(9)−1 = 17.
Question 3: For integers m and n, which statement is always true if m and n are both odd?
- mn is even
- m + n is odd
- m − n is even (Correct answer)
- m² + n² is odd
Correct answer: m − n is even
Odd minus odd equals even, since (2j+1)−(2k+1) = 2(j−k), which is always even.
Question 4: How many integers between 1 and 100 inclusive are divisible by 3 but not by 9?
- 22 (Correct answer)
- 24
- 11
- 33
Correct answer: 22
Divisible by 3: floor(100/3)=33; divisible by 9: floor(100/9)=11; answer = 33−11 = 22.
Question 5: If p → q and q → r are both true, which conclusion must follow by the chain rule?
- r → p
- p → r (Correct answer)
- ¬p → ¬r
- p ↔ r
Correct answer: p → r
Hypothetical syllogism (chain rule) states: if p→q and q→r, then p→r.
Question 6: A fair six-sided die is rolled twice. What is the probability that the product of the two results is even?
- 1/4
- 3/4 (Correct answer)
- 1/2
- 2/3
Correct answer: 3/4
P(product odd) = P(both odd) = (1/2)² = 1/4, so P(product even) = 1 − 1/4 = 3/4.
Question 7: Which of the following is a necessary condition for a natural number n to be prime (other than n = 2)?
- n is divisible by 3
- n is odd (Correct answer)
- n ends in 1, 3, 7, or 9
- n has exactly two factors
Correct answer: n is odd
Every prime greater than 2 must be odd; being odd is therefore a necessary (but not sufficient) condition.
TMUA Mathematical Reasoning & Logical Thinking 2
Which of the following is the contrapositive of the statement 'If n is divisible by 6, then n is divisible by 2'?