Share & grow the world's knowledge!
We want to connect the people who have knowledge to the people who need it, to bring together people with different perspectives so they can understand each other better, and to empower everyone to share their knowledge.
We have identified different types of quadrilaterals—squares, rectangles, parallelograms, rhombi, kites and trapezia. One can identify more types (e.g., we could create a category of quadrilaterals that have equal-length opposite sides). Suppose we have identified a category of quadrilaterals called Q and we have to construct a quadrilateral of this type. For this, we are to use two thin sticks, put them together as diagonals so that the quadrilateral obtained by joining their endpoints is of type Q (see the Fig. 9.2). (i) Suppose Q satisfies the following property. If a quadrilateral is of type Q, then it has equal-length diagonals. (a) Should the two sticks be of equal length? Why or why not? (b) Will it matter how the two sticks are put together? (ii) Instead of the property mentioned above, suppose Q satisfies the following property. If a quadrilateral has equal diagonals, then it is of type Q. What will be your answers to (a) and (b) now?
For (i): (a) Yes, sticks must be equal because every quadrilateral in Q has equal-length diagonals. (b) Yes, how they are joined matters; equal diagonals are necessary but not sufficient to ensure the shape belongs to Q. For (ii): (a) Yes, sticks must be equal. (b) No, it will not matter how they crRead more
For (i): (a) Yes, sticks must be equal because every quadrilateral in Q has equal-length diagonals. (b) Yes, how they are joined matters; equal diagonals are necessary but not sufficient to ensure the shape belongs to Q. For (ii): (a) Yes, sticks must be equal. (b) No, it will not matter how they cross, because any quadrilateral with equal diagonals automatically belongs to Q.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 9 Propositions and their Converses Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-9/
See lessRecall that a shortcut to check whether a given number is divisible by 3 is to add the digits of the number and check if the sum is a multiple of 3. Express the relationship between ‘a number is divisible by 3’ and ‘sum of the digits is a multiple of 3’ using ‘If-then’ sentences.
The mutual relationship is expressed through both the conditional statement and its converse: Proposition: If a number is divisible by 3, then the sum of its digits is a multiple of 3. Converse: If the sum of the digits of a number is a multiple of 3, then the number is divisible by 3. Both statemenRead more
The mutual relationship is expressed through both the conditional statement and its converse: Proposition: If a number is divisible by 3, then the sum of its digits is a multiple of 3. Converse: If the sum of the digits of a number is a multiple of 3, then the number is divisible by 3. Both statements are mathematically true and form a biconditional equivalence.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 9 Propositions and their Converses Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-9/
See lessFind counterexamples to the following statements: (i) If n is a prime number, then 2ⁿ − 1 is a prime number. (ii) If n is an even number, then 2ⁿ + 1 is a prime number.
Counterexamples disprove both statements: For (i), take the prime n = 11. Then 2¹¹ − 1 = 2048 − 1 = 2047, which factors as 23 × 89, making it composite. For (ii), take the even integer n = 6. Then 2⁶ + 1 = 64 + 1 = 65, which factors as 5 × 13, confirming it is not a prime number. For more NCERead more
Counterexamples disprove both statements:
For (i), take the prime n = 11. Then 2¹¹ − 1 = 2048 − 1 = 2047, which factors as 23 × 89, making it composite.
For (ii), take the even integer n = 6. Then 2⁶ + 1 = 64 + 1 = 65, which factors as 5 × 13, confirming it is not a prime number.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 9 Propositions and their Converses Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-9/
See lessConsider the statement: ‘If a number is divisible by 8, then it is divisible by both 2 and 4’. (i) Justify the statement. (ii) Recall the divisibility shortcuts that we have studied for different numbers. To check whether a given number is divisible by 8, is it enough to check whether it is divisible by 2 and 4? Why or why not?
(i) The statement is true because any multiple of 8 can be written as 8k = 2(4k) = 4(2k), meaning 2 and 4 are factors of 8. (ii) No, checking divisibility by 2 and 4 is not sufficient for 8 because 2 and 4 are not coprime. For example, 12 and 20 are divisible by both 2 and 4, but neither is divisiblRead more
(i) The statement is true because any multiple of 8 can be written as 8k = 2(4k) = 4(2k), meaning 2 and 4 are factors of 8.
(ii) No, checking divisibility by 2 and 4 is not sufficient for 8 because 2 and 4 are not coprime. For example, 12 and 20 are divisible by both 2 and 4, but neither is divisible by 8.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 9 Propositions and their Converses Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-9/
See lessThere are no known ‘neat’ expressions that generate only primes! Find counterexamples to the following claims: (i) All numbers of the form 4n² + 1 are prime. (ii) All numbers of the form n² + n + 11 are prime. (iii) All numbers of the form 4ⁿ + 3 are prime.
Each claim is proven false by counterexamples: For (i), let n = 4; then 4(4)² + 1 = 65, which is composite (5 × 13). For (ii), let n = 10 or 11; taking n = 10 yields 10² + 10 + 11 = 121 = 11², a composite number. For (iii), let n = 4; then 4⁴ + 3 = 256 + 3 = 259 = 7 × 37, which is composite. Read more
Each claim is proven false by counterexamples:
For (i), let n = 4; then 4(4)² + 1 = 65, which is composite (5 × 13).
For (ii), let n = 10 or 11; taking n = 10 yields 10² + 10 + 11 = 121 = 11², a composite number.
For (iii), let n = 4; then 4⁴ + 3 = 256 + 3 = 259 = 7 × 37, which is composite.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 9 Propositions and their Converses Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-9/
See less