About Logic & Proof — Class 9 IB

Construct logical arguments, use deductive reasoning, and write simple geometric proofs. This topic is part of the IB Class 9 mathematics syllabus (chapter: Unit 11). On this page you can practice 50 questions across three difficulty levels — 20 easy, 20 medium, and 10 hard — each with a visual step-by-step solution, plus a timed 27-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Logic & Proof — solved examples for Class 9 IB

Practice questions on Logic & Proof

1. Q1.easy

   What is the negation of the statement: 'All squares are rectangles'?

   1. A)All rectangles are squares.
   2. B)No squares are rectangles.
   3. C)Some squares are not rectangles.
   4. D)Some rectangles are not squares.
   Show answer

   Answer: Some squares are not rectangles.

   Hint: To negate a universal affirmative statement ('All A are B'), you need to show that there is at least one instance where it is not true.

2. Q2.easy

   In the conditional statement 'If a polygon has three sides, then it is a triangle', identify the conclusion.

   1. A)A polygon has three sides.
   2. B)It is a triangle.
   3. C)If a polygon has three sides.
   4. D)Then it is a triangle.
   Show answer

   Answer: It is a triangle.

   Hint: A conditional statement is structured as 'If P, then Q'. P is the hypothesis and Q is the conclusion.

3. Q3.easy

   Consider the statement: 'If a number is even, then it is divisible by 2'. Which of the following is its converse?

   1. A)If a number is divisible by 2, then it is even.
   2. B)If a number is not even, then it is not divisible by 2.
   3. C)If a number is not divisible by 2, then it is not even.
   4. D)A number is even if and only if it is divisible by 2.
   Show answer

   Answer: If a number is divisible by 2, then it is even.

   Hint: The converse of a conditional statement 'If P, then Q' is formed by swapping the hypothesis and the conclusion to become 'If Q, then P'.

4. Q4.medium

   Given that lines AB and CD are parallel (AB || CD) and transversal line XY intersects them at P and Q respectively. If ∠APX = 75°, what is the measure of ∠CQY?

   1. A)105°
   2. B)75°
   3. C)90°
   4. D)45°
   Show answer

   Answer: 75°

   Hint: Consider the relationships between angles formed when a transversal intersects parallel lines, specifically corresponding angles or vertically opposite angles.

5. Q5.medium

   A statement that is accepted without proof is called a/an _________.

   1. A)Theorem
   2. B)Conjecture
   3. C)Axiom
   4. D)Lemma
   Show answer

   Answer: Axiom

   Hint: Think about the foundational statements in mathematics that are assumed to be true without needing to be proven.

6. Q6.medium

   Rohan states: "All prime numbers are odd." Which of the following is a counterexample to Rohan's statement?

   1. A)3
   2. B)7
   3. C)9
   4. D)2
   Show answer

   Answer: 2

   Hint: A counterexample is a specific case that proves a general statement to be false. Look for a prime number that is not odd.

7. Q7.hard

   Based on the following statements, what can be logically concluded about Rohan?
   1. All students who complete their homework pass the test.
   2. Some students who pass the test also score above 90%.
   3. No student who scores above 90% fails the test.
   4. Rohan completed his homework.

   1. A)Rohan must score above 90%.
   2. B)Rohan must pass the test.
   3. C)Rohan cannot score above 90%.
   4. D)We cannot determine if Rohan passed the test.
   Show answer

   Answer: Rohan must pass the test.

   Hint: Focus on direct deductions from the given premises. Not all 'if-then' statements are reversible or imply 'all' if only 'some' is mentioned.

8. Q8.hard

   Which of the following quadrilaterals serves as a counterexample to the statement: "If a quadrilateral has two pairs of equal adjacent sides, then it is a rhombus"?

   1. A)Square
   2. B)Parallelogram
   3. C)Kite
   4. D)Rectangle
   Show answer

   Answer: Kite

   Hint: A counterexample must satisfy the 'if' part of the statement but not the 'then' part. Think about quadrilaterals with specific side properties.

9. Q9.hard

   Ravi attempts to prove the statement: "For any two integers a and b, if a + b is even, then both a and b must be even." He provides the following proof:
   Step 1: Assume a + b is even.
   Step 2: Assume, for the sake of contradiction, that a and b are NOT both even. This means either (a is even AND b is odd) OR (a is odd AND b is even) OR (a is odd AND b is odd).
   Step 3: Case 1: If (a is even AND b is odd), then a = 2k, b = 2m + 1. So a + b = 2k + 2m + 1 = 2(k+m) + 1, which is odd. This contradicts Step 1.
   Step 4: Case 2: If (a is odd AND b is even), then a = 2k + 1, b = 2m. So a + b = 2k + 1 + 2m = 2(k+m) + 1, which is odd. This contradicts Step 1.
   Step 5: Therefore, since both cases lead to contradiction, our initial assumption in Step 2 must be false. Hence, a and b must both be even.
   Where is the error in Ravi's proof?

   1. A)Step 1: The initial assumption should be that a + b is odd.
   2. B)Step 2: The negation of "both a and b are even" is not correctly stated.
   3. C)The proof correctly shows that Case 1 and Case 2 lead to a contradiction, but it omits a crucial case from Step 2.
   4. D)The conclusion in Step 5 does not logically follow from the preceding steps.
   Show answer

   Answer: The proof correctly shows that Case 1 and Case 2 lead to a contradiction, but it omits a crucial case from Step 2.

   Hint: When proving by contradiction, ensure that the negation of the conclusion covers ALL possibilities. Recheck the cases for 'a and b are NOT both even'.

These are 9 of the 50 questions available for Logic & Proof. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.