CBSE Class 9 Maths Ganita Manjari Part 2 Ch 09 Propositions and their Converses Short Answer Questions with Answers

Short Answer Questions

Question 1. Write the converse of each of the following propositions, and state whether the proposition and its converse are true. Justify your answers.
(i) If a quadrilateral is a rectangle, then its opposite sides are equal.
(ii) If a number is a multiple of 20, then it is a multiple of 4.
Answer:
(i) Converse: If the opposite sides of a quadrilateral are equal, then it is a rectangle.
Proposition: true, as a rectangle's opposite sides are equal.
Converse: false. A parallelogram with sides 7 cm and 4 cm and angles 60° and 120° has equal opposite sides but no right angle.
(ii) Converse: If a number is a multiple of 4, then it is a multiple of 20.
Proposition: true, since 20k = 4 × 5k. Converse: false, since 8 is a multiple of 4 but not of 20.
In simple words: Both originals are true, but both converses fail, one for a slanted parallelogram and one for the number 8.

Teacher's Note:
1. Justify true statements with a reason, and false ones with a counterexample.
2. The slanted parallelogram is a useful all-purpose counterexample.

Question 2. Frame the converse of 'If two triangles are congruent, then their corresponding angles are equal.' Determine whether the converse is true, and give a counterexample if it is false.
Answer:
Converse: If the corresponding angles of two triangles are equal, then the triangles are congruent.
It is false. Two equilateral triangles of sides 2 cm and 5 cm have all angles 60°, yet they have different sizes, so they are not congruent.
Equal angles fix the shape, but a side length is needed to fix the size.
In simple words: Same angles give the same shape but not necessarily the same size.

Teacher's Note:
1. There is no 'AAA' congruence rule.
2. Equal angles mean the triangles are similar, not congruent.

Question 3. Show that the proposition 'If a number is a multiple of 3, then it is a multiple of 6' is false. Then state which of this proposition and its converse is true, and justify that one.
Answer:
False: 9 = 3 × 3 is a multiple of 3, but 9 ÷ 6 is not a whole number.
The converse, 'If a number is a multiple of 6, then it is a multiple of 3', is true.
Any multiple of 6 is 6k = 3 × 2k, which is 3 times a whole number.
In simple words: 9 breaks the original claim, but every multiple of 6 is certainly a multiple of 3.

Teacher's Note:
1. One counterexample settles the false direction.
2. Algebra (6k = 3 × 2k) proves the true direction for all cases.

Question 4. Let n be a positive integer. Prove that if n is even, then \( n^2 \) is even. Then write the converse of this proposition and decide whether the converse is true.
Answer:
Let n = 2m. Then \( n^2 = 4m^2 = 2(2m^2) \), which is even. So the proposition is true.
Converse: If \( n^2 \) is even, then n is even.
True. If n were odd, n = 2m + 1, then \( n^2 = 4m^2 + 4m + 1 = 2(2m^2 + 2m) + 1 \), which is odd. So an even \( n^2 \) forces n to be even.
In simple words: Even numbers have even squares and odd numbers have odd squares, so both directions hold.

Teacher's Note:
1. The converse is proved by ruling out the other case (n odd).
2. Write n = 2m or n = 2m + 1 to handle evenness.

Question 5. Explain the difference between a proposition and its converse. Illustrate your explanation with a pair in which the proposition is true but the converse is false, justifying both of these truth values.
Answer:
A proposition 'if X then Y' says Y follows whenever X holds. Its converse 'if Y then X' makes the claim in the opposite direction. They are different statements, and one being true tells us nothing about the other.
Example: 'If a quadrilateral is a rhombus, then its diagonals are perpendicular.' True: a rhombus's diagonals always cross at right angles.
Converse: 'If the diagonals of a quadrilateral are perpendicular, then it is a rhombus.' False: a kite with AB = AD = 3 cm and CB = CD = 6 cm has perpendicular diagonals but unequal sides.
In simple words: Reversing the direction of a statement makes a new statement that must be checked on its own.

Teacher's Note:
1. Keep one clear true-but-false-converse example ready (rhombus/kite).
2. Emphasise independence of the two directions.

Question 6. Consider the proposition 'If a quadrilateral has all four sides equal, then it is a square.' Write its converse, and decide whether each of the two statements is true, giving a counterexample wherever one is needed.
Answer:
Converse: If a quadrilateral is a square, then all four of its sides are equal.
Proposition: false. A rhombus of side 5 cm with angles 60° and 120° has four equal sides but is not a square.
Converse: true, as every square has four equal sides.
In simple words: Equal sides alone give a rhombus; a square needs right angles as well.

Teacher's Note:
1. Here the converse is true and the original is false.
2. A square needs both equal sides and equal angles.

Question 7. Let p be a prime number. Show that \( p^2 \) has exactly three factors and name them. Must every positive integer that has exactly three factors be the square of a prime number? Justify your answer.
Answer:
Every factor of \( p^2 \) is made only from the prime p, so the factors are 1, p and \( p^2 \). These are different since p ≥ 2, so there are exactly three.
Yes. If n has exactly three factors, the pairing f ↔ n ÷ f leaves an odd count only if n = f × f.
If f had a factor d with 1 < d < f, then 1, d, f, n ÷ d and n would give at least five factors. So f is prime and n = \( f^2 \).
In simple words: Exactly three factors happens only for the square of a prime, such as 4, 9, 25 or 49.

Teacher's Note:
1. Both directions are true here.
2. The factor-pair idea is the key tool.

Question 8. Decide whether 'If a positive integer n has exactly 4 divisors, then n is the product of two different prime numbers' is true. Justify your answer fully.
Answer:
The statement is false.
Take n = 8. Its divisors are 1, 2, 4, 8, which is exactly four. But 8 = \( 2^3 \), a power of one prime, not a product of two different primes.
In fact, a number with exactly 4 divisors is either a product of two different primes (like 15 = 3 × 5) or the cube of a prime (like 8 or 27).
In simple words: 8 has four divisors but is 2 × 2 × 2, not two different primes.

Teacher's Note:
1. Encourage listing all divisors carefully.
2. Mention both types of numbers with exactly 4 divisors.

Question 9. Rewrite each of the following in 'If … then …' form and then write its converse.
(i) A triangle is isosceles when two of its angles are equal.
(ii) A number is divisible by 4 when the number formed by its last two digits is divisible by 4.
Answer:
(i) If two angles of a triangle are equal, then the triangle is isosceles.
Converse: If a triangle is isosceles, then two of its angles are equal.
(ii) If the number formed by the last two digits of a number is divisible by 4, then the number is divisible by 4.
Converse: If a number is divisible by 4, then the number formed by its last two digits is divisible by 4.
In simple words: Move the 'when' part to the front as the 'if' part, then swap to get the converse.

Teacher's Note:
1. In both cases the statement and its converse are true.
2. The 'when' part is always the hypothesis.

Question 10. In the figure, two lines are cut by a transversal and one pair of corresponding angles is marked.
Using the proposition on parallel lines and its converse, explain how this figure can be used (i) to find x when the two lines are known to be parallel, and (ii) to test whether the two lines are parallel when both marked angles are measured.
Answer:
(i) Lines known to be parallel: use the proposition 'If two lines are parallel, then corresponding angles are equal'. The angles x and 68° are in the same position at the two crossings, so they are corresponding angles. Hence x = 68°.
(ii) Lines not known to be parallel: use the converse 'If corresponding angles are equal, then the lines are parallel'. If x measures 68°, the lines are parallel; if x is any other value, they are not.
In simple words: Known parallel lines let us find x; measured equal angles let us prove the lines parallel.

Teacher's Note:
1. Which statement to use depends on what is given.
2. The proposition finds angles; the converse proves parallelism.

Question 11. Write the converse of each of the following, and state whether the converse is true.
(i) If a number is a multiple of 100, then it is a multiple of 10.
(ii) If two angles are vertically opposite angles, then they are equal.
(iii) If a triangle is equilateral, then it is isosceles.
Answer:
(i) If a number is a multiple of 10, then it is a multiple of 100. False: 10.
(ii) If two angles are equal, then they are vertically opposite angles. False: the base angles of an isosceles triangle are equal but not vertically opposite.
(iii) If a triangle is isosceles, then it is equilateral. False: a 5 cm, 5 cm, 8 cm triangle.
All three originals are true, yet every converse is false.
In simple words: Three true statements, three false converses: never reverse a statement without checking.

Teacher's Note:
1. A good warning example against assuming converses.
2. Each false converse needs its own counterexample.

Question 12. Let n be a positive integer. Show that the claim 'Every number of the form \( 3^n + 2 \) is prime' is false by finding the smallest counterexample, and show the working that rules out the smaller values of n.
Answer:
n = 1: 3 + 2 = 5 (prime)
n = 2: 9 + 2 = 11 (prime)
n = 3: 27 + 2 = 29 (prime)
n = 4: 81 + 2 = 83 (prime)
n = 5: 243 + 2 = 245 = 5 × 7 × 7 (composite)
So n = 5 is the smallest counterexample, and the claim is false.
In simple words: The formula gives primes for n = 1 to 4, then 245 at n = 5, which is not prime.

Teacher's Note:
1. Set the work out as a table of n and values.
2. Four successes did not make the claim true.

Question 13. Let x, y and a be real numbers. Consider the proposition 'If x = y, then x + a = y + a' together with its converse. Explain why both are true, and explain how this pair of statements is used while solving an equation.
Answer:
Proposition: if x = y they are the same number, so adding a to each gives the same result. True.
Converse: if x + a = y + a, subtracting a from both sides gives x = y. True.
In solving x − 3 = 8, adding 3 gives x = 11 (one direction). The converse guarantees x = 11 satisfies x − 3 = 8. Because the step works both ways, no solution is gained or lost.
In simple words: Adding the same number to both sides can always be undone, so both directions hold.

Teacher's Note:
1. This explains why 'doing the same to both sides' is safe.
2. Contrast with squaring both sides, which is not reversible.

Question 14. Prove that the converse of 'If a number is divisible by 6, then it is divisible by both 2 and 3' is true. Then explain why the same reasoning fails for the statement 'If a number is divisible by both 2 and 4, then it is divisible by 8.'
Answer:
Converse: if n is divisible by 2 and 3, then n is divisible by 6.
Write n = 3k. Since 2 divides 3k and 2 shares no factor with 3, 2 divides k: k = 2m. So n = 6m. True.
For 2 and 4: n = 4k is already even, so 'divisible by 2' adds nothing. 2 and 4 share the factor 2, so together they only guarantee divisibility by 4. 20 is divisible by 2 and 4 but not by 8.
In simple words: Coprime divisors (2 and 3) combine; overlapping divisors (2 and 4) do not.

Teacher's Note:
1. The key word is 'coprime'.
2. The LCM of the two divisors is what is really guaranteed.

Question 15. A student claims: 'Every square has equal diagonals, so any quadrilateral with equal diagonals must be a square.' Identify the error in this reasoning and give a counterexample.
Answer:
The student has turned the true proposition 'square ⇒ equal diagonals' into its converse 'equal diagonals ⇒ square' without proof.
The converse is false. An 8 cm × 3 cm rectangle has equal diagonals, each \( \sqrt{8^2 + 3^2} = \sqrt{73} \) cm, but it is not a square.
An isosceles trapezium is another counterexample.
In simple words: The student reversed a true statement, and the reversed version is false.

Teacher's Note:
1. Name the mistake: assuming the converse.
2. Always back a 'false' verdict with a counterexample.

Question 16. Write the converse of the Baudhāyana–Pythagoras theorem, and use it to decide whether a triangle with sides 9 cm, 40 cm and 41 cm is right-angled.
Answer:
Converse: In a triangle with sides a, b, c, if \( a^2 + b^2 = c^2 \), then the triangle is right-angled.
Longest side = 41 cm: \( 9^2 + 40^2 = 81 + 1600 = 1681 \) and \( 41^2 = 1681 \).
They are equal, so the triangle is right-angled, with the right angle between the 9 cm and 40 cm sides.
In simple words: 81 + 1600 = 1681 = 41², so the triangle has a right angle.

Teacher's Note:
1. Always compare with the longest side.
2. State where the right angle is for full marks.

Question 17. Let n be a positive integer. State the converse of 'If n is divisible by 60, then n is divisible by both 5 and 12.' Decide whether the converse is true and justify your decision.
Answer:
Converse: If n is divisible by both 5 and 12, then n is divisible by 60.
True. 5 and 12 have no common factor other than 1.
Write n = 12k. Since 5 divides 12k and shares no factor with 12, 5 divides k: k = 5m. So n = 60m.
In simple words: 5 and 12 are coprime, so a number divisible by both is divisible by 60.

Teacher's Note:
1. Check coprimality first: gcd(5, 12) = 1.
2. Here 5 × 12 = LCM = 60.

Question 18. Consider the proposition 'If a quadrilateral is a parallelogram, then its diagonals bisect each other.' Write its converse. Both statements are true; explain how each of the two is used — one to deduce a property of a figure and the other to identify the figure.
Answer:
Converse: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
The proposition deduces a property: once we know a figure is a parallelogram, we may state that its diagonals bisect each other.
The converse identifies the figure: if we find that the diagonals bisect each other, we may name it a parallelogram and use all its properties.
In simple words: One direction gives facts about a known shape; the other recognises the shape from a fact.

Teacher's Note:
1. Think of it as a two-way street.
2. Proofs often use the converse to identify parallelograms.

Question 19. Let n be a positive integer. Write the converse of 'If n is a perfect square, then the units digit of n is 0, 1, 4, 5, 6 or 9', and give a counterexample to show that the converse is false.
Answer:
Converse: If the units digit of n is 0, 1, 4, 5, 6 or 9, then n is a perfect square.
Counterexample: 11 ends in 1, but \( 3^2 = 9 \) and \( 4^2 = 16 \), so 11 is not a perfect square.
The units-digit rule can rule a number out (ending in 2, 3, 7, 8), but can never rule it in.
In simple words: Ending in 1 doesn't make 11 a square; the rule only works one way.

Teacher's Note:
1. Great example of a one-way test.
2. Other counterexamples: 10, 14, 15, 19, 20.

Question 20. Explain what it means to say that two statements imply each other. Use 'n is a multiple of 4' and 'the number formed by the last two digits of n is a multiple of 4' as your example.
Answer:
Two statements imply each other when 'if the first then the second' and 'if the second then the first' are both true. They then hold in exactly the same cases.
Write n = 100q + t, where t is the number formed by the last two digits. Since 100 = 4 × 25, 100q is always a multiple of 4.
So n is a multiple of 4 exactly when t is. Example: 3716 → 16 is a multiple of 4, so 3716 is too.
In simple words: Each statement guarantees the other, so we can test the easy one (last two digits) instead of the hard one.

Teacher's Note:
1. 'Imply each other' = both directions true.
2. This is why divisibility shortcuts are reliable.