1. “Every even number greater than 2 is the sum of two primes” (Goldbach) is:
- a conjecture
- false
- a theorem
- an axiom
Answer: A) a conjecture
Explanation: It has been checked for enormous numbers but never proved. SLO M-09-B-14 · Knowledge
MCQs / Mathematics / Logic
Logic MCQs — Mathematics
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.
2. The product of two consecutive integers n(n + 1) is always:
- prime
- odd
- even
- a perfect square
Answer: C) even
Explanation: One of n and n + 1 is even, so the product is even. SLO M-09-B-15 · Knowledge
3. “The whole is greater than the part” is:
- an axiom
- a conjecture
- a theorem
- a corollary
Answer: A) an axiom
Explanation: Another of Euclid's common notions. SLO M-09-B-14 · Knowledge
4. Which of these is a theorem, not an axiom?
- The base angles of an isosceles triangle are equal.
- Things equal to the same thing are equal to each other.
- A straight line can be drawn between any two points.
- All right angles are equal.
Answer: A) The base angles of an isosceles triangle are equal.
Explanation: The isosceles-triangle property is proved; the others are Euclid's axioms or postulates. SLO M-09-B-14 · Knowledge
5. Which value is a counterexample to “x^2 x for every number x”?
- 2
- -3
- 5
- 12
Answer: D) 12
Explanation: ( 12 )^2 = 14, which is less than 12. SLO M-09-B-13 · Knowledge
6. In “If n is even, then n^2 is even”, the hypothesis is:
- n^2 is even
- n is even
- n^2 is odd
- n is odd
Answer: B) n is even
Explanation: The hypothesis is the “if” part; the conclusion is the “then” part. SLO M-09-B-13 · Knowledge
7. Which of the following is a mathematical statement?
- 7 is a prime number.
- Draw a triangle.
- Is 15 divisible by 4?
- Find the value of x.
Answer: A) 7 is a prime number.
Explanation: Only “7 is a prime number” can be judged true or false (it is true). SLO M-09-B-13 · Knowledge
8. Euclid's Elements (about 300 BC) is famous for:
- building geometry from a few axioms and postulates
- inventing logarithms
- discovering zero
- introducing the decimal point
Answer: A) building geometry from a few axioms and postulates
Explanation: Euclid proved hundreds of theorems from a small set of axioms. SLO M-09-B-14 · Knowledge
9. A theorem is a statement that:
- is accepted without proof
- has been proved true
- is true for one example
- is only a guess
Answer: B) has been proved true
Explanation: Theorems are proved from axioms and earlier theorems. SLO M-09-B-14 · Knowledge
10. A direct proof starts from:
- a diagram only
- the hypothesis and reasons step by step to the conclusion
- a counterexample
- the conclusion and works backwards
Answer: B) the hypothesis and reasons step by step to the conclusion
Explanation: Direct proof: assume the “if” part and deduce the “then” part. SLO M-09-B-13 · Knowledge
11. Which expression is identical to (x - 4)^2 + 7?
- x^2+8x+23
- x^2-8x+23
- x^2-8x+11
- x^2-8x-9
Answer: B) x^2-8x+23
Explanation: (x - 4)^2 + 7 = x^2 - 8x + 16 + 7 = x^2-8x+23, so LHS = RHS. SLO M-09-B-15 · Knowledge
12. The Four Colour Theorem was proved in 1976 with the help of:
- a computer
- an axiom
- Euclid's Elements
- a counterexample
Answer: A) a computer
Explanation: Appel and Haken used a computer to check many cases. SLO M-09-B-14 · Knowledge
13. A mathematical proof is:
- a guess based on a pattern
- a drawing of the result
- a list of examples that work
- a logical argument showing that a statement is always true
Answer: D) a logical argument showing that a statement is always true
Explanation: A proof uses definitions, axioms and earlier results to show the statement holds in every case. SLO M-09-B-13 · Knowledge
14. A proof by contradiction begins by assuming:
- the converse
- a special case
- that the statement is true
- that the statement is false
Answer: D) that the statement is false
Explanation: Assuming it false leads to something impossible, so the statement must be true. SLO M-09-B-13 · Knowledge
15. Prove that (n + 1)^2 - n^2 is always odd. Simplifying gives:
- 2n - 1
- 2n + 1
- 2n
- 1
Answer: B) 2n + 1
Explanation: (n + 1)^2 - n^2 = n^2 + 2n + 1 - n^2 = 2n + 1, which is odd. SLO M-09-B-15 · Knowledge
16. The twin prime conjecture (there are infinitely many primes p with p + 2 also prime) is:
- known to be false
- a proved theorem
- an axiom
- still unproved
Answer: D) still unproved
Explanation: Nobody has proved or disproved it yet. SLO M-09-B-14 · Knowledge
17. Which of the following is not a statement?
- Solve x + 4 = 9.
- Every square is a rectangle.
- 2 + 3 = 6
- 2 is irrational.
Answer: A) Solve x + 4 = 9.
Explanation: “Solve …” is an instruction; it is neither true nor false. (2 + 3 = 6 is a false statement.) SLO M-09-B-13 · Knowledge
18. Simplify the left-hand side to complete the proof: (a + b)^2 - (a - b)^2 equiv
- 2a^2 + 2b^2
- 0
- 4ab
- 2ab
Answer: C) 4ab
Explanation: Expanding and collecting like terms gives (a + b)^2 - (a - b)^2 = 4ab. SLO M-09-B-15 · Knowledge
19. Which expression is identical to (x - 3)^2 + 5?
- x^2-6x-4
- x^2+6x+14
- x^2-6x+14
- x^2-6x+8
Answer: C) x^2-6x+14
Explanation: (x - 3)^2 + 5 = x^2 - 6x + 9 + 5 = x^2-6x+14, so LHS = RHS. SLO M-09-B-15 · Knowledge
20. Which number is a counterexample to “all prime numbers are odd”?
- 1
- 9
- 3
- 2
Answer: D) 2
Explanation: 2 is prime and even. SLO M-09-B-13 · Knowledge
21. The converse of “If a figure is a square, then it has four sides” is:
- If a figure has four sides, then it is a square.
- A square has four sides.
- If a figure is not a square, then it does not have four sides.
- If a figure does not have four sides, then it is not a square.
Answer: A) If a figure has four sides, then it is a square.
Explanation: The converse swaps hypothesis and conclusion. (Here the converse is false — a rectangle has four sides.) SLO M-09-B-13 · Knowledge
22. To prove an identity LHS equiv RHS, a good method is to:
- assume it is false
- substitute x = 1 only
- simplify one side until it becomes the other side
- multiply both sides by 0
Answer: C) simplify one side until it becomes the other side
Explanation: Work on one side (usually the more complicated one) using algebra until it matches the other. SLO M-09-B-15 · Knowledge
23. In the proof of (x - 3)^2 + 5 = x^2 - 6x + 14, the first step is:
- (x - 3)^2 = x^2 - 9
- x^2 - 6x + 14 = (x - 3)^2
- (x - 3)^2 = x^2 + 9
- (x - 3)^2 = x^2 - 6x + 9
Answer: D) (x - 3)^2 = x^2 - 6x + 9
Explanation: Expand the square, then add 5: x^2 - 6x + 9 + 5 = x^2 - 6x + 14. SLO M-09-B-15 · Knowledge
24. Which expression is identical to (x - 1)^2 + 9?
- x^2-2x+10
- x^2-x+10
- x^2-2x+8
- x^2+2x+10
Answer: A) x^2-2x+10
Explanation: (x - 1)^2 + 9 = x^2 - 2x + 1 + 9 = x^2-2x+10, so LHS = RHS. SLO M-09-B-15 · Knowledge
25. Simplify the left-hand side to complete the proof: (a - b)(a^2 + ab + b^2) equiv
- a^3 - b^3
- a^3 + b^3
- (a - b)^3
- a^3 - 3a^2b + b^3
Answer: A) a^3 - b^3
Explanation: Expanding and collecting like terms gives (a - b)(a^2 + ab + b^2) = a^3 - b^3. SLO M-09-B-15 · Knowledge
Page 1 of 3 Next