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

Very Short Answer Questions

Question 1. Explain what is meant by a proposition. Give one sentence that is a proposition and one sentence that is not.
Answer:
A proposition is a sentence that is either true or false, even if we need some work to decide which.
Proposition: 'Every multiple of 6 is a multiple of 3.' (true)
Not a proposition: 'Draw a circle of radius 4 cm.' It is an instruction and cannot be true or false.

Teacher's Note:
1. A false sentence like '7 is even' is still a proposition.
2. Questions, orders and exclamations are not propositions.

Question 2. Write the converse of 'If a number is a multiple of 15, then it is a multiple of 5.'
Answer:
Converse: If a number is a multiple of 5, then it is a multiple of 15.
The original is true (15k = 5 × 3k), but the converse is false: 10 is a multiple of 5 and not of 15.

Teacher's Note:
1. Swap the two parts; do not add 'not'.
2. Mentioning truth values earns extra credit.

Question 3. Write the converse of 'If a quadrilateral is a rhombus, then its diagonals are perpendicular to each other.'
Answer:
Hypothesis: the quadrilateral is a rhombus. Conclusion: its diagonals are perpendicular.
Converse: If the diagonals of a quadrilateral are perpendicular to each other, then the quadrilateral is a rhombus.

Teacher's Note:
1. Identify hypothesis and conclusion first, then swap.
2. This converse is false (a kite is a counterexample).

Question 4. Rewrite 'If a triangle is equilateral, then it is isosceles' in the form 'X implies Y', stating clearly what X and Y are.
Answer:
X: the triangle is equilateral.
Y: the triangle is isosceles.
'X implies Y': A triangle being equilateral implies that it is isosceles.

Teacher's Note:
1. 'X implies Y' keeps the same order as 'if X then Y'.
2. Every equilateral triangle has at least two equal sides.

Question 5. Rewrite 'A number is even when it is a multiple of 4' as an 'If … then …' sentence, and then write the converse of that sentence.
Answer:
The part after 'when' is the hypothesis.
If a number is a multiple of 4, then it is even.
Converse: If a number is even, then it is a multiple of 4. (False: 6 is even but not a multiple of 4.)

Teacher's Note:
1. 'Y when X' = 'if X then Y'.
2. Students often put the 'when' part in the conclusion by mistake.

Question 6. Give a counterexample to the claim 'Every odd number is a prime number.'
Answer:
9 is odd, but 9 = 3 × 3, so it is not prime.
Hence 9 is a counterexample and the claim is false.

Teacher's Note:
1. Other counterexamples: 15, 21, 25.
2. 1 is odd and not prime, and also works.

Question 7. Let n be a positive integer. Give a counterexample to the claim 'Every number of the form \( n^2 + n + 3 \) is prime.'
Answer:
Take n = 2: \( 4 + 2 + 3 = 9 = 3 \times 3 \), which is composite.
So n = 2 is a counterexample. (n = 1 gives 5, which is prime, so one success proves nothing.)

Teacher's Note:
1. Try small values of n one by one.
2. Show the factorisation to prove the value is composite.

Question 8. State whether the proposition 'If a number is divisible by 6, then it is divisible by 2' is true, and whether its converse is true. Give a brief reason for each answer.
Answer:
Proposition: true. A multiple of 6 is 6k = 2 × 3k, a multiple of 2.
Converse: false. 'If divisible by 2, then divisible by 6' fails for 4.

Teacher's Note:
1. Prove the true direction with algebra (6k).
2. Disprove the false one with a single number.

Question 9. For real numbers x and y, is the converse of 'If x = y, then \( x^3 = y^3 \)' true? Justify your answer.
Answer:
Yes. The converse is 'If \( x^3 = y^3 \), then x = y'.
Cubing keeps the sign and keeps order: if x < y then \( x^3 < y^3 \). So two different real numbers never have equal cubes.
Hence equal cubes force x = y.

Teacher's Note:
1. Contrast with squares: \( (-2)^2 = 2^2 \) but \( (-2)^3 \neq 2^3 \).
2. Cubing is 'reversible', squaring is not.

Question 10. Write the converse of 'If two angles of a triangle are equal, then the sides opposite them are equal', and state whether the converse is true.
Answer:
Converse: If two sides of a triangle are equal, then the angles opposite them are equal.
It is true; it is the basic property of an isosceles triangle.

Teacher's Note:
1. Here both the statement and its converse are true.
2. Together they say: equal sides ⇔ equal opposite angles.

Question 11. Is the sentence '119 is a prime number' a proposition? If it is, state whether it is true or false and give the reason.
Answer:
Yes, it is a proposition, because it makes a definite claim that is either true or false.
It is false: 119 = 7 × 17, so 119 is composite.

Teacher's Note:
1. False claims are still propositions.
2. 119 looks prime to many students; always test small primes.

Question 12. Write the converse of 'If a number ends in the digit 0, then it is divisible by 5.' Is the converse true? Give a reason.
Answer:
Converse: If a number is divisible by 5, then it ends in the digit 0.
False. 15 is divisible by 5 but ends in 5.

Teacher's Note:
1. Multiples of 5 end in 0 or 5.
2. One number is enough to disprove the converse.

Question 13. Give one example, taken from geometry, of a proposition that is true but whose converse is false.
Answer:
Proposition: If a quadrilateral is a square, then its diagonals are equal. (True)
Converse: If the diagonals of a quadrilateral are equal, then it is a square. (False). A non-square rectangle has equal diagonals.

Teacher's Note:
1. Other examples: 'square ⇒ all sides equal' (converse fails for a rhombus).
2. Always give the counterexample for the false part.

Question 14. Write the hypothesis and the conclusion of the converse of 'If a number is a multiple of 7, then it is a multiple of 14', and state which of the two statements is true.
Answer:
Converse: If a number is a multiple of 14, then it is a multiple of 7.
Hypothesis: the number is a multiple of 14. Conclusion: it is a multiple of 7.
The converse is true (14k = 7 × 2k). The original is false: 7 is a multiple of 7 but not of 14.

Teacher's Note:
1. Here the converse is true while the original is false.
2. This shows the two directions are independent.

Question 15. Let n be a positive integer. Give a counterexample to 'If \( n^2 \) is a multiple of 4, then n is a multiple of 4.'
Answer:
Take n = 2: \( n^2 = 4 \) is a multiple of 4, but 2 is not a multiple of 4.
So the statement is false.

Teacher's Note:
1. Any n = 2 × (odd number), such as 6 or 10, also works.
2. Check both parts: hypothesis true, conclusion false.

Question 16. Write the converse of 'If n is divisible by 18, then n is divisible by both 2 and 9', and state whether the converse is true.
Answer:
Converse: If n is divisible by both 2 and 9, then n is divisible by 18.
True. 2 and 9 have no common factor other than 1, so n must be divisible by 2 × 9 = 18.

Teacher's Note:
1. Coprime divisors combine to their product.
2. Compare with 2 and 4, which do not.

Question 17. The proposition 'If a number is divisible by both 4 and 6, then it is divisible by 24' is false. Give a counterexample.
Answer:
12 is divisible by 4 (12 = 4 × 3) and by 6 (12 = 6 × 2), but not by 24.
So 12 is a counterexample.

Teacher's Note:
1. 4 and 6 share the factor 2, so together they only guarantee divisibility by 12 (their LCM).
2. Other counterexamples: 36, 60.

Question 18. Write the converse of 'If a triangle is right-angled, then the square on its longest side is equal to the sum of the squares on the other two sides.' Name the result which says that this converse is true.
Answer:
Converse: If in a triangle the square on the longest side equals the sum of the squares on the other two sides, then the triangle is right-angled.
This is the converse of the Baudhāyana–Pythagoras theorem.

Teacher's Note:
1. The theorem calculates a side; the converse tests for a right angle.
2. Both directions are true here.

Question 19. Give an example of a proposition and its converse in which both statements are false.
Answer:
Proposition: If a number is a perfect cube, then it is a perfect square. False, since 8 is a cube but not a square.
Converse: If a number is a perfect square, then it is a perfect cube. False, since 4 is a square but not a cube.

Teacher's Note:
1. Another pair: 'prime ⇒ odd' and 'odd ⇒ prime'.
2. Give a counterexample for each statement separately.

Question 20. Identify the hypothesis and the conclusion in 'If the diagonals of a quadrilateral are equal and bisect each other, then the quadrilateral is a rectangle.'
Answer:
Hypothesis: the diagonals of the quadrilateral are equal and bisect each other.
Conclusion: the quadrilateral is a rectangle.

Teacher's Note:
1. The hypothesis follows 'if'; the conclusion follows 'then'.
2. A hypothesis can contain two conditions joined by 'and'.

Question 21. Show that the statement 'If \( 2^n + 1 \) is a prime number, then n is a prime number' is false, where n is a positive integer.
Answer:
Take n = 4: \( 2^4 + 1 = 17 \), which is prime.
But 4 = 2 × 2 is not prime. The hypothesis holds and the conclusion fails, so the statement is false.

Teacher's Note:
1. Look for a composite n that makes \( 2^n + 1 \) prime.
2. n = 4 is the smallest such value.

Question 22. A student says, 'The converse of a true statement is always true.' Give one example that shows the student is wrong.
Answer:
'If a quadrilateral is a square, then all its angles are equal' is true.
Its converse, 'If all the angles of a quadrilateral are equal, then it is a square', is false: a 6 cm × 4 cm rectangle has four 90° angles but is not a square.

Teacher's Note:
1. One true statement with a false converse is enough.
2. Truth never transfers automatically to the converse.

Question 23. Write 'A number has an odd number of factors when it is a perfect square' as an 'If … then …' sentence.
Answer:
The part after 'when' is the hypothesis.
If a number is a perfect square, then it has an odd number of factors.

Teacher's Note:
1. 'Y when X' = 'if X then Y'.
2. Check with 9: factors 1, 3, 9 (three factors).

Question 24. Write the converse of 'If two lines are parallel, then a pair of alternate interior angles formed by a transversal are equal', and state whether the converse is true.
Answer:
Converse: If a pair of alternate interior angles formed by a transversal are equal, then the two lines are parallel.
It is true; it is used as a test for parallel lines.

Teacher's Note:
1. Both directions hold for alternate angles, as for corresponding angles.
2. The converse proves lines parallel.

Question 25. Express the relationship between 'a number is divisible by 9' and 'the sum of the digits of the number is a multiple of 9' using two 'If … then …' sentences, and state whether both of them are true.
Answer:
If a number is divisible by 9, then the sum of its digits is a multiple of 9.
If the sum of the digits of a number is a multiple of 9, then the number is divisible by 9.
Both are true, which is why the digit-sum rule is a complete test for 9.

Teacher's Note:
1. A test works only when both directions are true.
2. Example: 729 → 7 + 2 + 9 = 18, and 729 = 9 × 81.

Question 26. Is the claim 'Every quadrilateral that has a pair of parallel sides is a parallelogram' true? If not, give a counterexample.
Answer:
No.
A trapezium ABCD with AB ∥ DC, AB = 8 cm and DC = 5 cm has one pair of parallel sides, but AD and BC are not parallel, so it is not a parallelogram.

Teacher's Note:
1. A parallelogram needs two pairs of parallel sides.
2. The trapezium is the natural counterexample.

Question 27. Let n be a positive integer. Decide whether 'If n is a multiple of 3, then \( n^2 \) is a multiple of 9' is true, and whether its converse is true.
Answer:
Both are true.
If n = 3k, then \( n^2 = 9k^2 \), a multiple of 9.
Converse: if n is not a multiple of 3, then \( n^2 \) leaves remainder 1 when divided by 3, so it cannot be a multiple of 9. So \( n^2 \) being a multiple of 9 forces n to be a multiple of 3.

Teacher's Note:
1. Compare with 4: \( n^2 \) a multiple of 4 does not force n to be a multiple of 4.
2. 3 is prime, which is why this converse works.

Question 28. Write the converse of 'If a triangle is right-angled, then it has an obtuse angle.' State whether each of the two statements is true, giving a counterexample wherever one is needed.
Answer:
Converse: If a triangle has an obtuse angle, then it is right-angled.
Both are false.
Original: a 90°–60°–30° triangle is right-angled with no obtuse angle.
Converse: a 100°–40°–40° triangle has an obtuse angle but no right angle.

Teacher's Note:
1. A triangle cannot have both a right angle and an obtuse angle.
2. Give a separate counterexample for each false statement.

Question 29. What do you get when you take the converse of the converse of a proposition? Explain with one example.
Answer:
You get the original proposition back: swapping the parts twice returns them to their places.
Example: 'If a number is a multiple of 10, then it is even' → converse 'If a number is even, then it is a multiple of 10' → its converse is the original statement again.

Teacher's Note:
1. Swapping twice undoes the swap.
2. So a statement and its converse are each other's converse.

Question 30. Two thin sticks of unequal lengths are placed as the diagonals of a quadrilateral and their endpoints are joined. Can the quadrilateral so obtained be a rectangle? Give a reason.
Answer:
No. The sticks are the diagonals, so the diagonals are unequal.
Every rectangle has equal diagonals, so this quadrilateral cannot be a rectangle.

Teacher's Note:
1. Uses 'rectangle ⇒ equal diagonals' in the form 'unequal diagonals ⇒ not a rectangle'.
2. Link sticks to diagonals clearly.