CBSE Class 9 Maths Ganita Manjari Part 2 Ch 09 Propositions and their Converses MCQs with Answers Set 01

Answer: (c) Every multiple of 4 is a multiple of 2.

Explanation:
1. A proposition is a sentence that can be judged either true or false.
2. (c) makes a definite claim about numbers (and it is true), so it is a proposition.
3. (a) is an instruction, (b) is a question and (d) is an exclamation; none of them can be true or false.

Teacher's Note:
1. Ask: 'Can I reply true or false to this sentence?'
2. Instructions, questions and exclamations are never propositions.

Answer: (a) If a number is a multiple of 5, then it is a multiple of 10.

Explanation:
1. The converse of 'if X then Y' is 'if Y then X': the two parts swap places.
2. X = 'multiple of 10', Y = 'multiple of 5', so the converse is option (a).
3. (b) and (c) add 'not' to the parts instead of swapping them, and (d) just repeats the original.

Teacher's Note:
1. Converse means swap, not negate.
2. Students often confuse the converse with 'if not X then not Y'.

Answer: (d) prove that the proposition is false

Explanation:
1. 'If X then Y' claims that Y holds in every case where X holds.
2. One case where X is true but Y is false breaks that claim completely.
3. It says nothing about the converse, and no proof can rescue a statement that has a counterexample.

Teacher's Note:
1. One counterexample disproves; no number of examples proves.
2. A counterexample must satisfy the hypothesis and fail the conclusion.

Answer: (b) the proposition is true and its converse is false

Explanation:
1. A multiple of 9 is 9k = 3 × 3k, so it is a multiple of 3: the proposition is true.
2. The converse says every multiple of 3 is a multiple of 9. But 6 is a multiple of 3 and not of 9.
3. So the proposition is true and the converse is false.

Teacher's Note:
1. Write the multiple as 9k to prove the true direction.
2. Use a small number like 6 as the counterexample.

Answer: (c) 20

Explanation:
1. We need a multiple of 4 that is not a multiple of 8.
2. 20 = 4 × 5, but 20 ÷ 8 = 2.5, so 20 is not a multiple of 8.
3. 8, 16 and 24 are all multiples of 8, so they agree with the claim.

Teacher's Note:
1. A single example (8) can never prove an 'every' statement.
2. Check each option against both the hypothesis and the conclusion.

Answer: (b) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Explanation:
1. 'Y when X' means 'if X then Y'. The part after 'when' is the hypothesis.
2. Here X = 'diagonals bisect each other' and Y = 'it is a parallelogram'.
3. So the sentence is option (b). Option (a) is its converse, a different statement (though also true).

Teacher's Note:
1. Underline the word 'when': what follows it is the 'if' part.
2. Two statements can both be true and still be different statements.

Answer: (d) is true, and its converse is false

Explanation:
1. If x = y they are the same number, so their squares are equal: the proposition is true.
2. Converse: if \( x^2 = y^2 \) then x = y. Take x = 2, y = −2: \( x^2 = y^2 = 4 \) but x ≠ y.
3. So the proposition is true and its converse is false.

Teacher's Note:
1. Squaring loses the sign, so it cannot be 'undone' uniquely.
2. Negative numbers are the usual source of counterexamples here.

Answer: (a) If two triangles have equal areas, then they are congruent.

Explanation:
1. Swap the parts: 'equal areas' becomes the hypothesis and 'congruent' the conclusion.
2. That gives option (a). (b) and (c) negate parts instead of swapping, and (d) talks about perimeters.
3. Note: this converse is actually false, but it is still the correct converse.

Teacher's Note:
1. Writing the converse and deciding its truth are two separate tasks.
2. A converse can be false and still be 'the converse'.

Answer: (c) A right triangle with legs 6 cm and 4 cm, and a right triangle with legs 8 cm and 3 cm

Explanation:
1. We need two triangles with equal areas that are not congruent.
2. In (c): areas are \( \frac{1}{2} \times 6 \times 4 = 12 \) cm² and \( \frac{1}{2} \times 8 \times 3 = 12 \) cm², which are equal.
3. The legs differ, so the triangles are not congruent. In the other options the two triangles are identical.

Teacher's Note:
1. A counterexample needs the hypothesis true (equal area) and the conclusion false (not congruent).
2. Pairs of identical triangles can never be counterexamples here.

Answer: (b) showing that \( 2^{2^5} + 1 \) is a composite number

Explanation:
1. A claim about every n is disproved by one value of n where it fails.
2. For n = 0 to 4 the numbers 3, 5, 17, 257, 65537 are all prime, so they support the claim.
3. Euler took n = 5: \( 2^{32} + 1 = 4294967297 = 641 \times 6700417 \), which is composite.

Teacher's Note:
1. Five successes did not make Fermat's claim true.
2. Checking more examples can support a claim, never prove it.

Answer: (d) true and false

Explanation:
1. A square has four right angles, so all its angles are equal: the proposition is true.
2. Converse: if all angles are equal, the quadrilateral is a square.
3. A rectangle (e.g. 6 cm × 4 cm) has four equal 90° angles but is not a square, so the converse is false.

Teacher's Note:
1. Rectangles are the natural counterexample for 'equal angles ⇒ square'.
2. A square needs equal sides and equal angles.

Answer: (b) if \( a^2 + b^2 = c^2 \), then the triangle is right-angled

Explanation:
1. The theorem: right angle ⇒ \( a^2 + b^2 = c^2 \).
2. The converse swaps these: \( a^2 + b^2 = c^2 \) ⇒ right angle, which is option (b).
3. (a) is the theorem itself, (c) is a different fact and (d) negates both parts.

Teacher's Note:
1. The theorem finds a missing side; the converse tests for a right angle.
2. Remind students that 'c' must be the longest side.

Answer: (a) imply each other

Explanation:
1. Factors pair up as (f, n ÷ f). Pairs use factors two at a time unless a pair repeats, i.e. n = f × f.
2. So a perfect square has exactly one repeated pair, giving an odd count.
3. And an odd count forces a repeated pair, so n is a perfect square. Each implies the other.

Teacher's Note:
1. Use 36: pairs (1,36), (2,18), (3,12), (4,9), (6,6) give 9 factors.
2. Two-way statements are called 'implying each other'.

Answer: (c) n = 10

Explanation:
1. n = 1 gives 13, n = 2 gives 17, n = 5 gives 41, all prime.
2. n = 10 gives 100 + 10 + 11 = 121 = 11 × 11, which is composite.
3. So n = 10 is the counterexample.

Teacher's Note:
1. Pattern: \( n(n + 1) + 11 \) at n = 10 is \( 10 \times 11 + 11 = 11^2 \).
2. Test the options; don't assume the smallest is the answer.

Answer: (a) is true

Explanation:
1. Converse: if a number is divisible by both 3 and 4, then it is divisible by 12.
2. 3 and 4 have no common factor other than 1, so divisibility by both means divisibility by 3 × 4 = 12.
3. So the converse is true. 6 and 18 are not divisible by 4, and 24 is divisible by 12, so none of them is a counterexample.

Teacher's Note:
1. Coprime divisors combine: divisible by 3 and 4 ⇒ divisible by 12.
2. Check that a proposed counterexample really satisfies the hypothesis.

Answer: (d) 20

Explanation:
1. Converse: divisible by 2 and 4 ⇒ divisible by 8.
2. 20 is divisible by 2 and by 4, but 20 ÷ 8 = 2.5.
3. 2 and 4 share the factor 2, so together they only guarantee divisibility by 4.

Teacher's Note:
1. Contrast with Q15: 3 and 4 are coprime, 2 and 4 are not.
2. 16, 24 and 32 are multiples of 8, so they cannot be counterexamples.

Answer: (a) is true, so the two statements together form a pair of results on parallel lines

Explanation:
1. Converse: if corresponding angles are equal, then the lines are parallel.
2. This is a true result and holds for every transversal, not only a perpendicular one.
3. Together, the proposition and the converse give a two-way result, used both to find angles and to test for parallel lines.

Teacher's Note:
1. The converse is the tool for proving lines parallel.
2. The proposition is the tool for finding angles once lines are parallel.

Answer: (c) both are false

Explanation:
1. All four combinations of truth values can happen.
2. Example: 'If n is prime, then n is odd' is false (2), and its converse 'If n is odd, then n is prime' is false (9).
3. So both can be false, and the restrictions in (a), (b) and (d) are wrong.

Teacher's Note:
1. The truth of one direction tells you nothing about the other.
2. Keep one example ready for each of the four combinations.

Answer: (b) If n is a prime number, then n is odd.

Explanation:
1. (b): 2 is prime but even, and 9 is odd but not prime, so both directions fail.
2. (a): true, converse false (3). (c): both true. (d): both true, since 3 and 5 are coprime.

Teacher's Note:
1. The number 2 is the only even prime and a classic counterexample.
2. Test each direction separately before choosing.

Answer: (b) If X then Y.

Explanation:
1. 'X implies Y' is another way of writing 'if X then Y'.
2. (a) is the converse, (c) claims both are true, and (d) describes a two-way link.

Teacher's Note:
1. 'Implies' keeps the order: X first (hypothesis), Y second (conclusion).
2. An implication never asserts that X is actually true.

Answer: (d) the road may be wet for some other reason, such as water spilt from a tanker

Explanation:
1. Converse: if the road is wet, then it has rained.
2. A road can be wet for other reasons, such as a tanker spilling water.
3. That case makes the hypothesis true and the conclusion false, so the converse is false. (b) would attack the original, not the converse.

Teacher's Note:
1. Everyday examples help students feel why converses fail.
2. Make sure the counterexample attacks the right statement.

Answer: (a) true and false

Explanation:
1. If \( a = p^2 \) and \( b = q^2 \), then \( ab = (pq)^2 \): the proposition is true.
2. Converse: if ab is a perfect square, then a and b are. But a = 2, b = 8 gives ab = 16, a square, while 2 and 8 are not squares.
3. So: true and false.

Teacher's Note:
1. Algebra proves the true direction; a small counterexample kills the converse.
2. 2 × 8 = 16 is a favourite counterexample.

Answer: (c) Bisect the given angle using a ruler and a compass.

Explanation:
1. (c) is an instruction: it can be carried out, but it cannot be true or false.
2. (a) and (d) are true claims and (b) is a false claim, but all three are propositions.

Teacher's Note:
1. A false statement is still a proposition.
2. Only true-or-false claims count as propositions.

Answer: (b) is true

Explanation:
1. Converse: if n has exactly 3 factors, then n is the square of a prime.
2. An odd number of factors means n = f × f. If f had a factor other than 1 and itself, n would have more than 3 factors, so f is prime.
3. So the converse is true. 16 has 5 factors and 12 has 6, so they do not satisfy the hypothesis; 4 = 2² fits the converse.

Teacher's Note:
1. Check that a claimed counterexample actually has exactly 3 factors.
2. Numbers with exactly 3 factors: 4, 9, 25, 49, …

Answer: (a) the two sticks must be of equal length

Explanation:
1. The sticks are the diagonals, so the diagonals of the figure have exactly the stick lengths.
2. Equal diagonals therefore means equal sticks.
3. How the sticks are crossed changes the shape, but never the lengths of the diagonals.

Teacher's Note:
1. Link the physical sticks to the geometric diagonals.
2. Placement matters for the shape, not for the diagonal lengths.

Answer: (c) the converse 'If n has an odd number of factors, then n is a perfect square'

Explanation:
1. An argument proves the statement whose hypothesis it starts from and whose conclusion it reaches.
2. Here it starts at 'odd number of factors' and ends at 'perfect square'.
3. So it proves 'If n has an odd number of factors, then n is a perfect square'. The other direction needs its own argument.

Teacher's Note:
1. The direction of the argument decides what is proved.
2. An argument cannot simply be read backwards.

Answer: (d) n = 11

Explanation:
1. \( 2^3 - 1 = 7 \), \( 2^5 - 1 = 31 \), \( 2^7 - 1 = 127 \), all prime, so they support the claim.
2. \( 2^{11} - 1 = 2047 = 23 \times 89 \), which is composite, although 11 is prime.
3. So n = 11 is a counterexample.

Teacher's Note:
1. The first few cases working is not a proof.
2. Encourage checking by division, e.g. 2047 ÷ 23 = 89.

Answer: (b) true, and it states that in a triangle the sides opposite two equal angles are equal

Explanation:
1. Hypothesis: two sides equal. Conclusion: the opposite angles equal.
2. Swapping gives: if two angles are equal, then the sides opposite them are equal.
3. This converse is true and is used to prove a triangle is isosceles.

Teacher's Note:
1. Both directions hold: equal sides ⇔ equal opposite angles.
2. The converse is the test for an isosceles triangle.

Answer: (d) one quadrilateral with equal diagonals that is not a rectangle

Explanation:
1. The claim is about every quadrilateral with equal diagonals.
2. One such quadrilateral that is not a rectangle disproves it, for example an isosceles trapezium.
3. (a) and (c) agree with the claim, and (b) describes something that cannot exist.

Teacher's Note:
1. The isosceles trapezium is the standard counterexample here.
2. Counterexample = hypothesis true, conclusion false.

Answer: (c) is true

Explanation:
1. Suppose n were a multiple of 3, say n = 3k. Then n + 3 = 3(k + 1) is also a multiple of 3.
2. So 3 would be a common factor, which the hypothesis rules out.
3. Hence n cannot be a multiple of 3: the proposition is true. (6 and 9 fail the hypothesis; 4 satisfies both parts.)

Teacher's Note:
1. This is a short proof by contradiction.
2. Always check whether a 'counterexample' satisfies the hypothesis.