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)
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)
(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)
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)
(i) Weighting each friend's average by their sample count (Yusuf: 6, Srikanth: 4, Kashvi: 5, Prasanna: 4, Gracy: 4; total: 23) gives the combined average expression: [(6aᵧ) + (4aₛ) + (5aₖ) + (4aₚ) + (4a₉)] / 23. (ii) Counting items by type gives 13 nylon and 10 feather shuttles, producing the expresRead more
(i) Weighting each friend’s average by their sample count (Yusuf: 6, Srikanth: 4, Kashvi: 5, Prasanna: 4, Gracy: 4; total: 23) gives the combined average expression: [(6aᵧ) + (4aₛ) + (5aₖ) + (4aₚ) + (4a₉)] / 23.
(ii) Counting items by type gives 13 nylon and 10 feather shuttles, producing the expression [(13aₙ) + (10a𝒻)] / 23. Yes, this value equals the previous method because both represent the identical total cost divided by 23.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 10 How Quantities Combine Understanding Data Question Answer (2026-27)
Recall 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 lessFive friends who play badminton surveyed the city and collected the price of shuttlecocks across brands and types (Nylon and Feather). The following is the list of the prices, in rupees (₹), that each one compiled. Yusuf: 40(N), 105(N), 383(F), 108(N), 165(F), 116(F) Srikanth: 194(N), 85(N), 93(N), 121(N) Kashvi: 49(N), 297(F), 105(N), 275(F), 40(N) Prasanna: 124(N), 333(F), 182(N), 258(F) Gracy: 220(F), 458(F), 129(F), 183(N) Can you describe a way they can work together to find the average price of a shuttlecock across types? (i) Each one calculated the average of the prices they had gathered. Suppose these are aᵧ, aₛ, aₖ, aₚ, a₉ respectively. Write an expression that gives the combined average. (ii) Suppose aₙ, a𝒻 are the average prices of the nylon shuttlecocks and the feather shuttlecocks, respectively. Write an expression that gives the average price of a shuttlecock. Will this be equal to the answer we got using the method in the previous part?
(i) Weighting each friend's average by their sample count (Yusuf: 6, Srikanth: 4, Kashvi: 5, Prasanna: 4, Gracy: 4; total: 23) gives the combined average expression: [(6aᵧ) + (4aₛ) + (5aₖ) + (4aₚ) + (4a₉)] / 23. (ii) Counting items by type gives 13 nylon and 10 feather shuttles, producing the expresRead more
(i) Weighting each friend’s average by their sample count (Yusuf: 6, Srikanth: 4, Kashvi: 5, Prasanna: 4, Gracy: 4; total: 23) gives the combined average expression: [(6aᵧ) + (4aₛ) + (5aₖ) + (4aₚ) + (4a₉)] / 23.
(ii) Counting items by type gives 13 nylon and 10 feather shuttles, producing the expression [(13aₙ) + (10a𝒻)] / 23. Yes, this value equals the previous method because both represent the identical total cost divided by 23.
For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 10 How Quantities Combine Understanding Data Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-10/
See less