TMUA Mathematical Reasoning & Logical Thinking 3 — Questions and Answers
Question 1: Let P = 'x > 0' and Q = 'x² > 0'. Which relationship correctly describes P and Q for real x?
- P is necessary and sufficient for Q
- P is sufficient but not necessary for Q (Correct answer)
- P is necessary but not sufficient for Q
- P and Q are logically independent
Correct answer: P is sufficient but not necessary for Q
P→Q (if x>0 then x²>0) but Q does not imply P since x=−2 gives x²=4>0; so P is sufficient but not necessary.
Question 2: A function f satisfies f(x + y) = f(x) + f(y) for all real x, y, and f(1) = 3. What is f(7)?
- 21 (Correct answer)
- 128
- 18
- 2187
Correct answer: 21
By the Cauchy functional equation with f(1)=3, we get f(n)=3n for integers, so f(7)=21.
Question 3: In a group of 30 students, 18 study Maths, 15 study Physics, and 5 study neither. How many study both?
- 8 (Correct answer)
- 10
- 7
- 6
Correct answer: 8
By inclusion-exclusion: |M∪P| = 30−5 = 25, so |M∩P| = 18+15−25 = 8.
Question 4: Which of the following correctly negates the statement '∀x ∈ ℝ, x² ≥ 0'?
- ∀x ∈ ℝ, x² < 0
- ∃x ∈ ℝ such that x² < 0 (Correct answer)
- ∀x ∈ ℝ, x² > 0
- ∃x ∈ ℝ such that x² ≥ 0
Correct answer: ∃x ∈ ℝ such that x² < 0
Negating a universal statement gives an existential statement with the negated predicate: ∃x such that x² < 0.
Question 5: If the sum of three consecutive even integers is 78, what is the largest of the three?
- 24
- 26
- 28 (Correct answer)
- 30
Correct answer: 28
Let the integers be n, n+2, n+4; then 3n+6=78, so n=24 and the largest is 28.
Question 6: A statement and its converse are both true. What can be concluded?
- The statement is a tautology
- The statement and its converse form a biconditional (Correct answer)
- The inverse is false
- The contrapositive is not equivalent to the original
Correct answer: The statement and its converse form a biconditional
When both 'P→Q' and 'Q→P' are true, together they constitute the biconditional 'P↔Q'.
Question 7: How many ways can 3 distinct books be arranged on a shelf from a collection of 7 distinct books?
- 35
- 210 (Correct answer)
- 21
- 840
Correct answer: 210
This is a permutation: P(7,3) = 7 × 6 × 5 = 210.
TMUA Mathematical Reasoning & Logical Thinking 3
Let P = 'x > 0' and Q = 'x² > 0'.
Which relationship correctly describes P and Q for real x?