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Logical Reasoning · Form 4

Logical Reasoning: Practice Questions

Original SPM-style practice questions for Logical Reasoning, each with full worked solutions, Paper 1 multiple-choice and Paper 2 structured.

Written by the spmmath.com.my editorial team.

Original practice questions for Logical Reasoning, in the style of Mathematics Paper 1 (Objective) and Mathematics Paper 2 (Subjective). Each answer is worked so you can check your method, not just the result.

Multiple-choice (Paper 1 style)

Question 1

Which of the following sentences is a statement?

  1. A. Is 7 a prime number?
  2. B. Please close the window.
  3. C. 9 is a multiple of 3.
  4. D. x + 2 = 5

Question 2

Which of the following statements is false?

  1. A. 2³ = 8
  2. B. −5 < −2
  3. C. 0.5 is a rational number
  4. D. 7 is a factor of 30

Question 3

What is the negation of the statement '18 is divisible by 5'?

  1. A. 18 is divisible by 5
  2. B. 18 is not divisible by 5
  3. C. 5 is divisible by 18
  4. D. 18 is divisible by 3

Question 4

Which quantifier makes the statement '____ multiples of 10 are even numbers' true?

  1. A. All
  2. B. Some ... are not
  3. C. No
  4. D. Only one

Question 5

Given two statements p: '15 > 9' and q: '15 is a multiple of 4', determine the truth value of 'p and q'.

  1. A. True, because p is true
  2. B. False, because q is false
  3. C. True, because at least one is true
  4. D. False, because both are false

Question 6

Given p: '4 is a factor of 20' and q: '4 is a factor of 14', determine the truth value of 'p or q'.

  1. A. True
  2. B. False
  3. C. Cannot be determined
  4. D. True only if both parts are true

Question 7

Consider the implication 'If a number is a multiple of 9, then it is a multiple of 3.' What is the antecedent?

  1. A. a number is a multiple of 9
  2. B. it is a multiple of 3
  3. C. a number is a multiple of 3
  4. D. it is not a multiple of 9

Question 8

What is the converse of the implication 'If x is a multiple of 6, then x is a multiple of 2'?

  1. A. If x is a multiple of 2, then x is a multiple of 6
  2. B. If x is not a multiple of 6, then x is not a multiple of 2
  3. C. If x is a multiple of 6, then x is a multiple of 2
  4. D. x is a multiple of 6 and a multiple of 2

Question 9

The two implications 'If m is even, then m² is even' and 'If m² is even, then m is even' can be combined into a single statement. Which statement is it?

  1. A. m is even if and only if m² is even
  2. B. If m is even, then m² is odd
  3. C. m is even and m² is even
  4. D. If m is odd, then m² is even

Question 10

Premise 1: If a number is a multiple of 12, then it is a multiple of 4. Premise 2: The number 18 is not a multiple of 4.

Which is the valid conclusion?

  1. A. 18 is not a multiple of 12
  2. B. 18 is a multiple of 12
  3. C. 18 is a multiple of 4
  4. D. 18 is a multiple of 3

Structured (Paper 2 style)

Question 1 (5 marks)

Consider the statement: 'If a number n is a multiple of 8, then n is a multiple of 4.' (a) State the antecedent and the consequent of the implication.

(b) Write the converse of the implication. (c) Determine whether the converse is true or false.

Give one example to support your answer.

  1. The implication has the form 'if p, then q', so the antecedent is p = 'n is a multiple of 8' and the consequent is q = 'n is a multiple of 4'.
  2. The converse of 'if p, then q' is 'if q, then p', obtained by interchanging p and q.
  3. Converse: 'If n is a multiple of 4, then n is a multiple of 8.'
  4. Test the converse with n = 12: 12 is a multiple of 4 (12 ÷ 4 = 3) but 12 is not a multiple of 8 (12 ÷ 8 = 1.5).
  5. Since a counterexample exists, the converse is false.

Question 2 (4 marks)

Complete each argument by writing its conclusion, and state the argument form (Form I, II or III). (a) Premise 1: All multiples of 10 are divisible by 5.

Premise 2: 80 is a multiple of 10. Conclusion: ______ (b) Premise 1: If a quadrilateral is a square, then all four of its sides are equal.

Premise 2: Quadrilateral ABCD is a square. Conclusion: ______

  1. (a) uses the pattern 'All A are B; C is A; therefore C is B', which is Argument Form I.
  2. Applying it: since 80 is a multiple of 10 and all multiples of 10 are divisible by 5, the conclusion is '80 is divisible by 5'.
  3. (b) uses the pattern 'If p, then q; p is true; therefore q is true', which is Argument Form II (modus ponens).
  4. Applying it with p = 'ABCD is a square' (true): the conclusion is 'all four sides of ABCD are equal'.

Question 3 (4 marks)

Study the following number pattern: 1 = 3(1) − 2, 4 = 3(2) − 2, 7 = 3(3) − 2, 10 = 3(4) − 2. (a) Make a general conclusion by induction for the number in the pattern in terms of n, where n = 1, 2, 3, ...

(b) Hence, determine the 15th number in the pattern.

  1. In each line the number equals 3 × (line number) − 2, and the line number is n.
  2. By induction, the general conclusion is: number = 3n − 2, for n = 1, 2, 3, ...
  3. For the 15th number, substitute n = 15: 3(15) − 2 = 45 − 2 = 43.

Question 4 (6 marks)

Two statements are given: p: 6 is a factor of 24. q: 6 is a prime number.

(a) State, with a reason, whether p is true or false and whether q is true or false. (b) Write the compound statement 'p and q' and determine its truth value.

(c) Write the compound statement 'p or q' and determine its truth value.

  1. p: 24 ÷ 6 = 4, a whole number, so 6 is a factor of 24; p is true.
  2. q: 6 = 2 × 3 has factors other than 1 and itself, so 6 is not a prime number; q is false.
  3. 'p and q': '6 is a factor of 24 and 6 is a prime number.' A statement joined by 'and' is true only if both parts are true; since q is false, 'p and q' is false.
  4. 'p or q': '6 is a factor of 24 or 6 is a prime number.' A statement joined by 'or' is true if at least one part is true; since p is true, 'p or q' is true.

Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)

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Frequently asked questions

How do the Paper 1 and Paper 2 questions in this logical reasoning set differ?

Paper 1 questions are multiple-choice, testing statements, implications, and quantifiers quickly. Paper 2 questions ask you to write out reasoning in full, forming compound statements, checking validity of arguments, and completing truth tables with clear steps, matching how Paper 2 rewards method.

Why should I try each question myself before looking at the worked solution?

Working through logical reasoning problems on your own builds the habit of identifying quantifiers and implications correctly under exam conditions. Checking the worked solution first can make an argument's validity look obvious in hindsight, hiding the reasoning gaps you would face in a real Paper 1449/2 question.

What mistakes do students commonly make with logical reasoning questions?

Common errors include confusing a statement's converse with its contrapositive, misreading 'and' versus 'or' in compound statements, and treating one true example as proof instead of checking for a counter-example. This practice set is designed to expose those slips before the real exam.

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