26. Simplify the left-hand side to complete the proof: (x + 2)(x - 2) + 4 equiv
- x^2
- x^2 + 4
- x^2 + 8
- x^2 - 8
Answer: A) x^2
Explanation: Expanding and collecting like terms gives (x + 2)(x - 2) + 4 = x^2. SLO M-09-B-15 · Knowledge
MCQs / Mathematics / Logic
Logic MCQs — Mathematics (page 2)
64 multiple-choice questions on Logic from Mathematics, each with the correct answer and, where available, a teacher's explanation. Read through them to revise the chapter, then practise the same questions one by one to test yourself.
27. Which expression is identical to (x - 2)^2 - 1?
- x^2+4x+3
- x^2-4x+3
- x^2-4x+1
- x^2-4x-5
Answer: B) x^2-4x+3
Explanation: (x - 2)^2 - 1 = x^2 - 4x + 4 - 1 = x^2-4x+3, so LHS = RHS. SLO M-09-B-15 · Knowledge
28. Simplify the left-hand side to complete the proof: (2x + 3)^2 - 12x equiv
- 4x^2 - 9
- 4x^2 + 24x + 9
- 4x^2 + 9
- 2x^2 + 9
Answer: C) 4x^2 + 9
Explanation: Expanding and collecting like terms gives (2x + 3)^2 - 12x = 4x^2 + 9. SLO M-09-B-15 · Knowledge
29. Checking that a rule works for 100 examples:
- makes it an axiom
- proves it is always true
- makes it a theorem
- does not prove it is always true
Answer: D) does not prove it is always true
Explanation: Examples can support a statement, but a proof must cover all cases. SLO M-09-B-13 · Knowledge
30. To show x^2 - 6x + 10 0 for all x, we complete the square:
- (x - 3)^2 + 1
- (x + 3)^2 + 1
- (x - 3)^2 - 1
- (x - 6)^2 + 10
Answer: A) (x - 3)^2 + 1
Explanation: (x - 3)^2 ge 0, so (x - 3)^2 + 1 ge 1 0. SLO M-09-B-15 · Knowledge
31. To prove “every multiple of 4 is even”, we can write a multiple of 4 as 4k and note that:
- 4k = 4 + k
- 4k = 2(2k)
- 4k = 2 + 2k
- 4k = k^4
Answer: B) 4k = 2(2k)
Explanation: 4k = 2(2k) is 2 times an integer, so it is even. SLO M-09-B-13 · Knowledge
32. “Things which are equal to the same thing are equal to one another” is an example of:
- a theorem
- an axiom
- a conjecture
- a counterexample
Answer: B) an axiom
Explanation: This is one of Euclid's common notions (axioms). SLO M-09-B-14 · Knowledge
33. Show that the square of any odd number 2n + 1 is odd:
- (2n + 1)^2 = 2(2n^2 + 1)
- (2n + 1)^2 = 4n^2 + 1
- (2n + 1)^2 = 2(2n^2 + 2n) + 1
- (2n + 1)^2 = 2n^2 + 1
Answer: C) (2n + 1)^2 = 2(2n^2 + 2n) + 1
Explanation: (2n + 1)^2 = 4n^2 + 4n + 1 = 2(2n^2 + 2n) + 1: twice an integer, plus one. SLO M-09-B-15 · Knowledge
34. “Through two distinct points there is exactly one straight line” is usually taken as:
- a counterexample
- a conjecture
- an axiom (postulate)
- a theorem
Answer: C) an axiom (postulate)
Explanation: It is a basic postulate of geometry. SLO M-09-B-14 · Knowledge
35. The main difference between a conjecture and a theorem is that a theorem:
- is accepted without proof
- is older
- has a proof
- is about numbers only
Answer: C) has a proof
Explanation: Proof turns a conjecture into a theorem. SLO M-09-B-14 · Knowledge
36. Which expression is identical to (x - 5)^2 + 3?
- x^2+10x+28
- x^2-10x+8
- x^2-10x+28
- x^2-10x-22
Answer: C) x^2-10x+28
Explanation: (x - 5)^2 + 3 = x^2 - 10x + 25 + 3 = x^2-10x+28, so LHS = RHS. SLO M-09-B-15 · Knowledge
37. x + 3 = 5 is called an open sentence because:
- it is always true
- it is always false
- its truth depends on the value of x
- it has no variable
Answer: C) its truth depends on the value of x
Explanation: It is true only when x = 2. SLO M-09-B-13 · Knowledge
38. Simplify the left-hand side to complete the proof: (x + 1)^2 - (x - 1)^2 equiv
- 2
- 0
- 4x
- 2x
Answer: C) 4x
Explanation: Expanding and collecting like terms gives (x + 1)^2 - (x - 1)^2 = 4x. SLO M-09-B-15 · Knowledge
39. Simplify the left-hand side to complete the proof: (a + b)^2 + (a - b)^2 equiv
- 4ab
- a^2 + b^2
- 2a^2
- 2(a^2 + b^2)
Answer: D) 2(a^2 + b^2)
Explanation: Expanding and collecting like terms gives (a + b)^2 + (a - b)^2 = 2(a^2 + b^2). SLO M-09-B-15 · Knowledge
40. Which of these is a conjecture?
- The angles of a triangle add up to 180^circ.
- There are infinitely many prime numbers.
- There are infinitely many prime numbers p such that p + 2 is also prime.
- a^2 + b^2 = c^2 in a right triangle.
Answer: C) There are infinitely many prime numbers p such that p + 2 is also prime.
Explanation: The last option was proved by Euclid; the twin-prime statement has never been proved. SLO M-09-B-14 · Knowledge
41. The negation of “All students passed the test” is:
- At least one student did not pass the test.
- Some students passed the test.
- No student passed the test.
- All students failed the test.
Answer: A) At least one student did not pass the test.
Explanation: To make “all … passed” false, one student failing is enough. SLO M-09-B-13 · Knowledge
42. A lemma is:
- an axiom
- a smaller result proved to help prove a theorem
- a counterexample
- an unproved guess
Answer: B) a smaller result proved to help prove a theorem
Explanation: A lemma is a stepping-stone theorem. SLO M-09-B-14 · Knowledge
43. The truth value of the statement “ 9 is irrational” is:
- both true and false
- true
- it is not a statement
- false
Answer: D) false
Explanation: 9 = 3 = 31 is rational, so the statement is false. SLO M-09-B-13 · Knowledge
44. A mathematical statement is a sentence that is:
- always true
- a question
- a command
- either true or false, but not both
Answer: D) either true or false, but not both
Explanation: A statement has a definite truth value: true or false. SLO M-09-B-13 · Knowledge
45. In a proof, each step must follow from:
- a few examples
- the teacher's opinion
- definitions, axioms or results already proved
- what looks right in a diagram
Answer: C) definitions, axioms or results already proved
Explanation: Every step needs a logical reason. SLO M-09-B-13 · Knowledge
46. Fermat's Last Theorem (x^n + y^n = z^n has no positive whole-number solutions for n 2) was a conjecture for about 350 years until it was proved by:
- Pythagoras
- Andrew Wiles (1994)
- Euclid
- Al-Khwarizmi
Answer: B) Andrew Wiles (1994)
Explanation: Andrew Wiles completed the proof in 1994–95. SLO M-09-B-14 · Knowledge
47. A conjecture that is shown to be false by a counterexample is:
- disproved
- proved
- an axiom
- a lemma
Answer: A) disproved
Explanation: One counterexample disproves it. SLO M-09-B-14 · Knowledge
48. Simplify the left-hand side to complete the proof: n^2 - (n - 1)^2 equiv
- 2n + 1
- n - 1
- 2n - 1
- 1
Answer: C) 2n - 1
Explanation: Expanding and collecting like terms gives n^2 - (n - 1)^2 = 2n - 1. SLO M-09-B-15 · Knowledge
49. The Collatz conjecture (halve if even, 3n + 1 if odd, always reaching 1) is:
- proved false
- a conjecture — not yet proved
- a theorem
- an axiom
Answer: B) a conjecture — not yet proved
Explanation: It works for every number tested but has not been proved. SLO M-09-B-14 · Knowledge
50. To prove that the sum of two odd numbers is even, we write them as:
- 2m and 2n
- m and n
- m + 1 and n + 1
- 2m + 1 and 2n + 1
Answer: D) 2m + 1 and 2n + 1
Explanation: (2m + 1) + (2n + 1) = 2(m + n + 1), which is a multiple of 2. SLO M-09-B-13 · Knowledge