A counterexample to a proposition is:

an example that shows the proposition is false

an example that agrees with the proposition

the converse of the proposition

a proof that the proposition is true in every case

Which is the converse of 'If a triangle with sides a, b, c is right-angled, then a² + b² = c²'?

If a² + b² = c², then the triangle is right-angled.

If the triangle is not right-angled, then a² + b² ≠ c².

If a² + b² = c², then the triangle is equilateral.

If the triangle is right-angled, then a + b = c.

Proposition: 'If a quadrilateral is a square, then all its angles are equal.' Which shape is a counterexample to its converse?

A rectangle that is not a square

Which sentence does NOT have the same meaning as 'If a number ends in 0, then it is divisible by 5'?

A number ends in 0 when it is divisible by 5.

A number ending in 0 implies it is divisible by 5.

A number is divisible by 5 when it ends in 0.

Every number that ends in 0 is divisible by 5.

The argument: (i) n has an odd number of factors ⇒ (ii) some factor-partner pair repeats, say (f, f) ⇒ (iii) n = f × f. Which proposition does this chain prove?

If n has an odd number of factors, then n is a perfect square.

If n is a perfect square, then n has an odd number of factors.

Both directions at once, since every step reverses.

If n is a perfect square, then n has exactly 3 factors.

To prove the converse, ∆ABC has sides a, b, c with a² + b² = c². A right triangle ∆XYZ is built with legs a and b. What is the length of XY?

c, because XY² = a² + b² = c²

Which sentence has exactly the same meaning as 'If a number is a perfect square, then it has an odd number of factors'?

A number has an odd number of factors when it is a perfect square.

A number is a perfect square when it has an odd number of factors.

Having an odd number of factors implies a number is a perfect square.

A number has an even number of factors when it is not a perfect square.

Which of these is a TRUE proposition whose converse is FALSE?

If a quadrilateral is a rhombus, then its diagonals are perpendicular.

If two angles of a triangle are equal, then the sides opposite them are equal.

If a number is a multiple of 4, then it is a multiple of 8.

If the sides a, b, c of a triangle satisfy a² + b² = c², then it is right-angled.

Propositions and their Converses — Class 9 MCQs with Answers

Class 9 CBSE Mathematics · Chapter 9

90 practice questions · 30 Easy · 30 Medium · 30 Hard · Updated 1 Oct 2026

Practise the most important Class 9 CBSE Mathematics questions from Chapter 9, "Propositions and their Converses". You get 9 timed quizzes made from 90 NCERT-based MCQs, with answers and explanations. The questions are split into 30 Easy, 30 Medium and 30 Hard. Warm up on the basics, then move on to the exam-level questions that top scorers in CBSE Board exams and the JEE & NEET foundation years get right.

To score well in "Propositions and their Converses", focus on fast problem-solving, formula recall and step-by-step working. Each MCQ here is timed and uses exam-style marking (+4 correct, −1 wrong, 0 skipped). This helps you stay accurate when time is short, just as real papers need. Every question has a short explanation, so a wrong answer becomes a quick lesson. It is the fastest way to fix gaps before a test.

Use this chapter for focused revision. Start with the Easy set to check your basics on Propositions and their Converses, then move to Medium and Hard to practise applying them. Your accuracy, streaks and XP save automatically. This chapter also adds to your overall Class 9 Mathematics mastery score. 14 sample questions are solved in full below, with the answer and a worked explanation. Sign in free to start practising.

Propositions and their Converses — important questions & MCQs with answers (Class 9 Mathematics)

14 solved questions from this chapter's difficulty levels, each with its answer and explanation.

Q1Easy
Which of these sentences is a proposition, that is, a statement that is either true or false?
A.7 is a prime number.✓ Correct
B.Please close the door.
C.Is 7 a prime number?
D.What a lovely day!
Answer: A. 7 is a prime number.
Explanation: '7 is a prime number' can be judged true or false, so it is a proposition. A request, a question or an exclamation cannot be called true or false.
Q2Easy
The converse of the proposition 'If X then Y' is:
A.If not X then not Y
B.If not Y then not X
C.If X then not Y
D.If Y then X✓ Correct
Answer: D. If Y then X
Explanation: To form the converse, swap the 'if' part and the 'then' part: 'If Y then X'. Adding 'not' to the parts gives a different statement, not the converse.
Q3Easy
A counterexample to a proposition is:
A.an example that agrees with the proposition
B.the converse of the proposition
C.an example that shows the proposition is false✓ Correct
D.a proof that the proposition is true in every case
Answer: C. an example that shows the proposition is false
Explanation: A counterexample is an example that goes against the stated proposition. It is a short and convincing way to show the proposition is false.
Q4Easy
Which list shows all the factor-partner pairs of 12?
A.(1, 12), (2, 6)
B.(1, 12), (2, 6), (3, 4)✓ Correct
C.(2, 6), (3, 4)
D.(1, 12), (3, 4)
Answer: B. (1, 12), (2, 6), (3, 4)
Explanation: Each factor pairs with the factor that multiplies with it to give 12: 1 × 12, 2 × 6 and 3 × 4. Leaving out (1, 12) is a common slip.
Q5Easy
Which is the converse of 'If a triangle with sides a, b, c is right-angled, then a² + b² = c²'?
A.If the triangle is not right-angled, then a² + b² ≠ c².
B.If a² + b² = c², then the triangle is right-angled.✓ Correct
C.If a² + b² = c², then the triangle is equilateral.
D.If the triangle is right-angled, then a + b = c.
Answer: B. If a² + b² = c², then the triangle is right-angled.
Explanation: Keep 'Let a, b, c be the sides of a triangle' and swap the two parts: 'If a² + b² = c², then the triangle is right-angled.' Adding 'not' to both parts gives a different statement.
Q6Easy
Proposition: 'If a quadrilateral is a square, then all its angles are equal.' Which shape is a counterexample to its converse?
A.A rhombus that is not a square
B.A square
C.A kite with unequal angles
D.A rectangle that is not a square✓ Correct
Answer: D. A rectangle that is not a square
Explanation: The converse says 'If all angles of a quadrilateral are equal, then it is a square.' A non-square rectangle has four 90° angles but is not a square. A rhombus has equal sides, not equal angles, so it does not meet the 'if' part.
Q7Medium
Which sentence does NOT have the same meaning as 'If a number ends in 0, then it is divisible by 5'?
A.A number ending in 0 implies it is divisible by 5.
B.A number is divisible by 5 when it ends in 0.
C.Every number that ends in 0 is divisible by 5.
D.A number ends in 0 when it is divisible by 5.✓ Correct
Answer: D. A number ends in 0 when it is divisible by 5.
Explanation: 'A number ends in 0 when it is divisible by 5' means 'If a number is divisible by 5, then it ends in 0', which is the converse, and 15 shows it is false. The other three are just other ways of saying the original.
Q8Medium
P: 'If two triangles have the same area, then they are congruent.' Q: 'If two triangles are congruent, then they have the same area.' Which is correct?
A.Both P and Q are true.
B.P is false and Q is true.✓ Correct
C.P is true and Q is false.
D.Both P and Q are false.
Answer: B. P is false and Q is true.
Explanation: Congruent triangles cover the same region, so Q is true. A triangle with base 6 and height 4 and one with base 8 and height 3 both have area 12 but are not congruent, so P is false.
Q9Medium
Claim: 'All numbers of the form n² + n + 11 are prime' (n = 1, 2, 3, …). What is the smallest n that gives a counterexample?
A.10✓ Correct
B.11
C.9
D.5
Answer: A. 10
Explanation: n = 1 to 9 give 13, 17, 23, 31, 41, 53, 67, 83, 101, all prime. n = 10 gives 100 + 10 + 11 = 121 = 11 × 11. n = 11 gives 143 = 11 × 13, also composite, but it is not the smallest.
Q10Medium
The argument: (i) n has an odd number of factors ⇒ (ii) some factor-partner pair repeats, say (f, f) ⇒ (iii) n = f × f. Which proposition does this chain prove?
A.If n has an odd number of factors, then n is a perfect square.✓ Correct
B.If n is a perfect square, then n has an odd number of factors.
C.Both directions at once, since every step reverses.
D.If n is a perfect square, then n has exactly 3 factors.
Answer: A. If n has an odd number of factors, then n is a perfect square.
Explanation: The chain starts from 'odd number of factors' and ends at 'perfect square', so it proves only that direction. The other direction needs an extra step, because (ii) alone does not give (i).
Q11Medium
To prove the converse, ∆ABC has sides a, b, c with a² + b² = c². A right triangle ∆XYZ is built with legs a and b. What is the length of XY?
A.a + b
B.√(a + b)
C.c², because XY = a² + b²
D.c, because XY² = a² + b² = c²✓ Correct
Answer: D. c, because XY² = a² + b² = c²
Explanation: By the Baudhāyana–Pythagoras theorem in the right triangle XYZ, XY² = a² + b², and this equals c², so XY = c. The square root must be taken, so XY is c, not c².
Q12Hard
Which sentence has exactly the same meaning as 'If a number is a perfect square, then it has an odd number of factors'?
A.A number is a perfect square when it has an odd number of factors.
B.A number has an odd number of factors when it is a perfect square.✓ Correct
C.Having an odd number of factors implies a number is a perfect square.
D.A number has an even number of factors when it is not a perfect square.
Answer: B. A number has an odd number of factors when it is a perfect square.
Explanation: 'Y when X' means 'If X then Y', so the 'when' part must be 'it is a perfect square'. The two look-alikes that start from the factor count state the converse. The 'even … not a perfect square' sentence is a different statement, even though it happens to be true.
Q13Hard
Which of these is a TRUE proposition whose converse is FALSE?
A.If two angles of a triangle are equal, then the sides opposite them are equal.
B.If a quadrilateral is a rhombus, then its diagonals are perpendicular.✓ Correct
C.If a number is a multiple of 4, then it is a multiple of 8.
D.If the sides a, b, c of a triangle satisfy a² + b² = c², then it is right-angled.
Answer: B. If a quadrilateral is a rhombus, then its diagonals are perpendicular.
Explanation: A rhombus always has perpendicular diagonals, but a kite also has perpendicular diagonals without being a rhombus, so the converse fails. The triangle statements have true converses, and the multiple-of-4 statement is itself false (12).
Q14Hard
For n = 0, 1, 2, 3, what are the values of Fermat's numbers 2^(2ⁿ) + 1?
A.2, 5, 17, 65
B.3, 5, 17, 257✓ Correct
C.3, 5, 9, 17
D.3, 5, 17, 65
Answer: B. 3, 5, 17, 257
Explanation: The powers 2ⁿ are 1, 2, 4, 8, so the numbers are 2¹ + 1, 2² + 1, 2⁴ + 1, 2⁸ + 1 = 3, 5, 17, 257, all prime. Using 2^(2n) instead of 2^(2ⁿ) gives 2, 5, 17, 65, and reading 2³ as 2 × 3 in the last one gives 65.

Key concepts: Propositions and their Converses (Class 9 Mathematics)

Proposition: A statement that is either true or false, such as 'If two sides of a triangle are equal, then the angles opposite them are equal'. A question or a request is not a proposition.
'If X then Y' and 'X implies Y': Both mean the same thing. X is the 'if' part (the condition) and Y is the 'then' part (the conclusion).
Converse: The converse of 'If X then Y' is 'If Y then X': swap the two parts. Adding 'not' to both parts does not give the converse.
Equal sides and equal angles in a triangle: 'Equal sides ⇒ equal opposite angles' and its converse 'equal angles ⇒ equal opposite sides' are both true. The converse is proved by drawing the altitude from the third vertex and using AAS.
Counterexample: An example that meets the 'if' part but breaks the 'then' part, so it shows a proposition is false. One counterexample is enough; a wet road from a spilled tanker breaks 'if the road is wet, then it has rained'.
Fermat numbers and Euler: Fermat claimed every 2^(2ⁿ) + 1 is prime (3, 5, 17, 257, …). Euler showed 2^(2⁵) + 1 is composite, one counterexample that ended the claim.
Multiples of 6 and 3: 'A multiple of 6 is a multiple of 3' is true, but its converse fails: 15 is a multiple of 3 and not of 6. A true proposition can have a false converse.
Factor-partner pairs: Each factor f of n pairs with n ÷ f, e.g. (1, 12), (2, 6), (3, 4) for 12. A repeating pair (f, f) is counted once, which is how a factor count can be odd.
Perfect squares and odd factor counts: The chain 'odd number of factors ⇒ a repeating pair (f, f) ⇒ n = f × f' proves only that odd-factor numbers are squares. For the other direction you also need that a number has at most one repeating pair, so every square has an odd number of factors.
Statements that imply each other: When a proposition and its converse are both true, X and Y imply each other ('X if and only if Y'), e.g. 'n is a perfect square' and 'n has an odd number of factors'.
Same area vs congruent: Congruent triangles have the same area (true), but equal area does not mean congruent: base 6, height 4 and base 8, height 3 both give 12.
Four possible outcomes: A proposition and its converse can be true and false, both true, or both false (e.g. 'a multiple of 2 is a multiple of 3'). Each one must be checked separately.
Converse of the Baudhāyana–Pythagoras theorem: If a triangle's sides satisfy a² + b² = c², it is right-angled. Proof: build a right triangle with legs a and b; its third side is c by the theorem, so the two triangles are congruent by SSS.
Other ways to say 'if–then': 'Y when X' means 'If X then Y'. 'A number has an odd number of factors when it is a perfect square' is the same as 'If a number is a perfect square, then it has an odd number of factors'.

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Propositions and their Converses — FAQs

What are the key concepts in Class 9 Mathematics Propositions and their Converses?+

A proposition is a statement that is true or false. This chapter swaps the 'if' and 'then' parts to form the converse, shows that a true statement can have a false converse, uses counterexamples to disprove claims, and proves both directions for perfect squares and for the Baudhāyana–Pythagoras theorem. Key ideas include Proposition, 'If X then Y' and 'X implies Y', Converse, Equal sides and equal angles in a triangle, Counterexample, Fermat numbers and Euler.

What does Class 9 Mathematics Chapter 9 (Propositions and their Converses) cover on XamBaaz?+

It has 90 NCERT-based MCQs on "Propositions and their Converses": 30 Easy, 30 Medium and 30 Hard. Together they make 9 timed quizzes, and you never get the same set twice. Every question has an instant explanation. They help you prepare for CBSE Board exams and the JEE & NEET foundation years.

Are these "Propositions and their Converses" questions free to practise?+

Yes. Sign in with Google to practise "Propositions and their Converses" free. Full unlimited access is ₹999/year. One year from the day you pay. You stay in Class 9 till 31 March; on 1 April your account moves up to Class 10 and the rest of your year carries over. No chapter is charged separately.

How should I revise "Propositions and their Converses" for the exam?+

Start with the Easy quiz to check your basics, then try Medium and Hard to practise applying them. There are 9 timed quizzes on this chapter, so you can come back for a fresh set instead of one you have seen. Read each explanation, retry the questions you miss, and track your accuracy until it stays high.

Are these "Propositions and their Converses" MCQs available with answers?+

Yes. 14 sample questions are shown here in full, each with the correct option and a step-by-step "Why" explanation. Sign in free with Google to start practising, with instant scoring.

Is there negative marking in the "Propositions and their Converses" quizzes?+

Yes. The timed quizzes use exam-style marking: +4 for a right answer, −1 for a wrong one and 0 for a skip. MHT-CET and CBSE board papers have no negative marking. Our mocks for those are scored their way.

What are the important questions from Propositions and their Converses (Class 9 Mathematics)?+

The questions that matter most test Proposition, 'If X then Y' and 'X implies Y', Converse, Equal sides and equal angles in a triangle, Counterexample, Fermat numbers and Euler. This page shows 14 solved important MCQs with answers and explanations. Sign in to practise all 90 questions on the chapter as timed quizzes.

Is there an online quiz for Propositions and their Converses?+

Yes — Class 9 Mathematics Propositions and their Converses has timed online quizzes at Easy, Medium and Hard levels, with instant scoring and a worked explanation on every question. The first quiz on the chapter is free.

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