conversations listlengths 2 2 |
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[
{
"from": "human",
"value": "Check whether 6n can end with the digit 0 for any natural number n."
},
{
"from": "gpt",
"value": "To check whether 6n can end with the digit 0:\nStep 1: A number ends in 0 if it is divisible by 10.\nStep 2: 10 = 2 × 5. So, 6n must be divisible by both 2 and 5.\nStep... |
[
{
"from": "human",
"value": "Can a number of the form 4n end in 0 for any natural number n?"
},
{
"from": "gpt",
"value": "Step 1: A number ends in 0 if it is divisible by 10.\nStep 2: 10 = 2 × 5. So the number must be divisible by both 2 and 5.\nStep 3: 4 = 2 × 2, so 4n is divisible by 2.\nStep... |
[
{
"from": "human",
"value": "Can 9n ever end in 0 for any natural number n?"
},
{
"from": "gpt",
"value": "Step 1: For a number to end in 0, it must be divisible by 10 = 2 × 5.\nStep 2: 9 = 3 × 3, so 9n is always divisible by 3.\nStep 3: But 9n is not divisible by 2 or 5 unless n has those facto... |
[
{
"from": "human",
"value": "Can the product of any three consecutive natural numbers be divisible by 6?"
},
{
"from": "gpt",
"value": "Step 1: Let the three consecutive natural numbers be n, n+1, and n+2.\nStep 2: Among any three consecutive numbers:\n- One of them is divisible by 2 (even numbe... |
[
{
"from": "human",
"value": "Can the square of an even number ever end in 5?"
},
{
"from": "gpt",
"value": "Step 1: Let the even number be 2k.\nStep 2: Then, its square is (2k)^2 = 4k^2.\nStep 3: 4k^2 is an even number and always ends in 0, 4, 6, etc.\nStep 4: A number ending in 5 must have 5 in... |
[
{
"from": "human",
"value": "Check whether 6n can end with the digit 0 for any natural number n."
},
{
"from": "gpt",
"value": "Step 1: A number ends in 0 if it is divisible by 10.\nStep 2: 10 = 2 × 5. So, 6n must be divisible by both 2 and 5.\nStep 3: 6 = 2 × 3. So, 6n will always be divisible ... |
[
{
"from": "human",
"value": "Consider the numbers 4n, where n is a natural number. Check whether there is any value of n for which 4n ends with the digit zero."
},
{
"from": "gpt",
"value": "Step 1: A number ends in 0 if it is divisible by 10.\nStep 2: 10 = 2 × 5. So the number must be divisible... |
[
{
"from": "human",
"value": "Check whether 8n can end with the digit 0 for any natural number n."
},
{
"from": "gpt",
"value": "Step 1: A number ends in 0 if it is divisible by 10.\nStep 2: 10 = 2 × 5. So the number must be divisible by both 2 and 5.\nStep 3: 8 = 2 × 2 × 2. So 8n is divisible by... |
[
{
"from": "human",
"value": "Check whether 3n can end with the digit 0 for any natural number n."
},
{
"from": "gpt",
"value": "Step 1: A number ends in 0 if it is divisible by 10 = 2 × 5.\nStep 2: 3 is not divisible by 2 or 5.\nStep 3: 3n will be divisible only by powers of 3.\nStep 4: So, 3n i... |
[
{
"from": "human",
"value": "Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers"
},
{
"from": "gpt",
"value": "To explain why both expressions are composite numbers:\n\n**First Expression: 7 × 11 × 13 + 13**\nStep 1: Consider the expression: 7 × 11 × 13 + 13\nSt... |
[
{
"from": "human",
"value": "Is 15 × 12 + 3 a composite number? Give reason."
},
{
"from": "gpt",
"value": "Step 1: Evaluate the expression: 15 × 12 + 3\nStep 2: 15 × 12 = 180\nStep 3: Add 3 to the result: 180 + 3 = 183\nStep 4: Check divisibility: 183 is divisible by 3 (since 1 + 8 + 3 = 12, an... |
[
{
"from": "human",
"value": "Prove that 6 × 17 + 1 is not a composite number."
},
{
"from": "gpt",
"value": "Step 1: Evaluate the expression: 6 × 17 + 1 = 102 + 1 = 103\nStep 2: Check whether 103 has any factors other than 1 and itself\nStep 3: 103 is not divisible by 2, 3, 5, 7, or any other nu... |
[
{
"from": "human",
"value": "Is 11 × 19 + 19 a composite number? Explain."
},
{
"from": "gpt",
"value": "Step 1: Consider the expression: 11 × 19 + 19\nStep 2: Take 19 common: = 19 × (11 + 1) = 19 × 12\nStep 3: 19 × 12 = 228\nStep 4: 228 has more than two factors (1, 2, 3, 4, 6, 12, 19, 38, 57, ... |
[
{
"from": "human",
"value": "Is 4 × 5 × 6 + 1 a composite number?"
},
{
"from": "gpt",
"value": "Step 1: Evaluate the expression: 4 × 5 × 6 = 120\nStep 2: Add 1: 120 + 1 = 121\nStep 3: 121 = 11 × 11\nStep 4: 121 has three factors: 1, 11, 121\nConclusion: 121 is a composite number as it has more ... |
[
{
"from": "human",
"value": "Express each number as a product of its prime factors:\n(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429"
},
{
"from": "gpt",
"value": "**(i) 140**\nStep 1: Divide by 2 → 140 ÷ 2 = 70\nStep 2: Divide by 2 → 70 ÷ 2 = 35\nStep 3: Divide by 5 → 35 ÷ 5 = 7\nStep 4: Divide ... |
[
{
"from": "human",
"value": "Express the following numbers as the product of prime factors:\n(i) 90 (ii) 250 (iii) 187 (iv) 729 (v) 825"
},
{
"from": "gpt",
"value": "**(i) 90**\nStep 1: Divide by 2 → 90 ÷ 2 = 45\nStep 2: Divide by 3 → 45 ÷ 3 = 15\nStep 3: Divide by 3 → 15 ÷ 3 = 5\nStep 4: 5 is ... |
[
{
"from": "human",
"value": "Find the HCF and LCM of 6, 72, and 120, using the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 6: 6 = 2 × 3\nStep 2: Prime factorisation of 72: 72 = 2³ × 3²\nStep 3: Prime factorisation of 120: 120 = 2³ × 3 × 5\nStep 4: HC... |
[
{
"from": "human",
"value": "Find the HCF and LCM of 14, 56, and 98 using the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 14: 14 = 2 × 7\nStep 2: Prime factorisation of 56: 56 = 2³ × 7\nStep 3: Prime factorisation of 98: 98 = 2 × 7²\nStep 4: HCF = 7\... |
[
{
"from": "human",
"value": "Find the HCF and LCM of 15, 25, and 35 using the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 15: 15 = 3 × 5\nStep 2: Prime factorisation of 25: 25 = 5²\nStep 3: Prime factorisation of 35: 35 = 5 × 7\nStep 4: HCF = 5\nStep... |
[
{
"from": "human",
"value": "Find the HCF and LCM of 10, 50, and 70 using the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 10: 10 = 2 × 5\nStep 2: Prime factorisation of 50: 50 = 2 × 5²\nStep 3: Prime factorisation of 70: 70 = 2 × 5 × 7\nStep 4: HCF =... |
[
{
"from": "human",
"value": "Find the HCF and LCM of 12, 36, and 72 using the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 12: 12 = 2² × 3\nStep 2: Prime factorisation of 36: 36 = 2² × 3²\nStep 3: Prime factorisation of 72: 72 = 2³ × 3²\nStep 4: HCF =... |
[
{
"from": "human",
"value": "Find the HCF and LCM of 18, 24, and 54 using the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 18: 18 = 2 × 3²\nStep 2: Prime factorisation of 24: 24 = 2³ × 3\nStep 3: Prime factorisation of 54: 54 = 2 × 3³\nStep 4: HCF = 2... |
[
{
"from": "human",
"value": "Find the HCF and LCM of 8, 64, and 128 using the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 8: 8 = 2³\nStep 2: Prime factorisation of 64: 64 = 2⁶\nStep 3: Prime factorisation of 128: 128 = 2⁷\nStep 4: HCF = 2³ = 8\nStep ... |
[
{
"from": "human",
"value": "Find the HCF and LCM of 9, 36, and 81 using the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 9: 9 = 3²\nStep 2: Prime factorisation of 36: 36 = 2² × 3²\nStep 3: Prime factorisation of 81: 81 = 3⁴\nStep 4: HCF = 3² = 9\nSte... |
[
{
"from": "human",
"value": "Find the HCF and LCM of 24, 72, and 120 using the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 24: 24 = 2³ × 3\nStep 2: Prime factorisation of 72: 72 = 2³ × 3²\nStep 3: Prime factorisation of 120: 120 = 2³ × 3 × 5\nStep 4:... |
[
{
"from": "human",
"value": "Find the HCF and LCM of 16, 32, and 48 using the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 16: 16 = 2⁴\nStep 2: Prime factorisation of 32: 32 = 2⁵\nStep 3: Prime factorisation of 48: 48 = 2⁴ × 3\nStep 4: HCF = 2⁴ = 16\n... |
[
{
"from": "human",
"value": "Find the HCF and LCM of 20, 50, and 80 using the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 20: 20 = 2² × 5\nStep 2: Prime factorisation of 50: 50 = 2 × 5²\nStep 3: Prime factorisation of 80: 80 = 2⁴ × 5\nStep 4: HCF = 2... |
[
{
"from": "human",
"value": "Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 96: 96 = 2⁵ × 3\nStep 2: Prime factorisation of 404: 404 = 2² × 101\nStep 3: HCF: The common factors are 2². Therefore, ... |
[
{
"from": "human",
"value": "Find the HCF of 144 and 180 by the prime factorisation method. Hence, find their LCM."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 144: 144 = 2⁴ × 3²\nStep 2: Prime factorisation of 180: 180 = 2² × 3² × 5\nStep 3: HCF: The common factors are 2² × 3².... |
[
{
"from": "human",
"value": "Find the HCF of 56 and 84 by the prime factorisation method. Hence, find their LCM."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 56: 56 = 2³ × 7\nStep 2: Prime factorisation of 84: 84 = 2² × 3 × 7\nStep 3: HCF: The common factors are 2² × 7. Therefor... |
[
{
"from": "human",
"value": "Find the HCF of 30 and 45 by the prime factorisation method. Hence, find their LCM."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 30: 30 = 2 × 3 × 5\nStep 2: Prime factorisation of 45: 45 = 3² × 5\nStep 3: HCF: The common factors are 3 × 5. Therefore,... |
[
{
"from": "human",
"value": "Find the HCF of 18 and 48 by the prime factorisation method. Hence, find their LCM."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 18: 18 = 2 × 3²\nStep 2: Prime factorisation of 48: 48 = 2⁴ × 3\nStep 3: HCF: The common factors are 2 × 3. Therefore, HC... |
[
{
"from": "human",
"value": "Find the HCF of 84 and 105 by the prime factorisation method. Hence, find their LCM."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 84: 84 = 2² × 3 × 7\nStep 2: Prime factorisation of 105: 105 = 3 × 5 × 7\nStep 3: HCF: The common factors are 3 × 7. The... |
[
{
"from": "human",
"value": "Find the HCF of 20 and 50 by the prime factorisation method. Hence, find their LCM."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 20: 20 = 2² × 5\nStep 2: Prime factorisation of 50: 50 = 2 × 5²\nStep 3: HCF: The common factors are 2 × 5. Therefore, HC... |
[
{
"from": "human",
"value": "Find the HCF of 72 and 120 by the prime factorisation method. Hence, find their LCM."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 72: 72 = 2³ × 3²\nStep 2: Prime factorisation of 120: 120 = 2³ × 3 × 5\nStep 3: HCF: The common factors are 2³ × 3. Ther... |
[
{
"from": "human",
"value": "Find the HCF of 28 and 35 by the prime factorisation method. Hence, find their LCM."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 28: 28 = 2² × 7\nStep 2: Prime factorisation of 35: 35 = 5 × 7\nStep 3: HCF: The common factor is 7. Therefore, HCF = 7\n... |
[
{
"from": "human",
"value": "Find the HCF of 18 and 30 by the prime factorisation method. Hence, find their LCM."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 18: 18 = 2 × 3²\nStep 2: Prime factorisation of 30: 30 = 2 × 3 × 5\nStep 3: HCF: The common factors are 2 × 3. Therefore,... |
[
{
"from": "human",
"value": "Find the LCM and HCF of 6 and 20 by the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 6: 6 = 2 × 3\nStep 2: Prime factorisation of 20: 20 = 2² × 5\nStep 3: HCF: The common factor is 2. Therefore, HCF = 2\nStep 4: LCM: LCM i... |
[
{
"from": "human",
"value": "Find the LCM and HCF of 12 and 18 by the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 12: 12 = 2² × 3\nStep 2: Prime factorisation of 18: 18 = 2 × 3²\nStep 3: HCF: The common factors are 2 × 3. Therefore, HCF = 2 × 3 = 6\n... |
[
{
"from": "human",
"value": "Find the LCM and HCF of 8 and 20 by the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 8: 8 = 2³\nStep 2: Prime factorisation of 20: 20 = 2² × 5\nStep 3: HCF: The common factor is 2². Therefore, HCF = 2² = 4\nStep 4: LCM: LC... |
[
{
"from": "human",
"value": "Find the LCM and HCF of 14 and 35 by the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 14: 14 = 2 × 7\nStep 2: Prime factorisation of 35: 35 = 5 × 7\nStep 3: HCF: The common factor is 7. Therefore, HCF = 7\nStep 4: LCM: LCM... |
[
{
"from": "human",
"value": "Find the LCM and HCF of 24 and 30 by the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 24: 24 = 2³ × 3\nStep 2: Prime factorisation of 30: 30 = 2 × 3 × 5\nStep 3: HCF: The common factors are 2 × 3. Therefore, HCF = 2 × 3 = ... |
[
{
"from": "human",
"value": "Find the LCM and HCF of 15 and 25 by the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 15: 15 = 3 × 5\nStep 2: Prime factorisation of 25: 25 = 5²\nStep 3: HCF: The common factor is 5. Therefore, HCF = 5\nStep 4: LCM: LCM is... |
[
{
"from": "human",
"value": "Find the LCM and HCF of 18 and 42 by the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 18: 18 = 2 × 3²\nStep 2: Prime factorisation of 42: 42 = 2 × 3 × 7\nStep 3: HCF: The common factors are 2 × 3. Therefore, HCF = 2 × 3 = ... |
[
{
"from": "human",
"value": "Find the LCM and HCF of 40 and 60 by the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 40: 40 = 2³ × 5\nStep 2: Prime factorisation of 60: 60 = 2² × 3 × 5\nStep 3: HCF: The common factors are 2² × 5. Therefore, HCF = 2² × 5... |
[
{
"from": "human",
"value": "Find the LCM and HCF of 56 and 72 by the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 56: 56 = 2³ × 7\nStep 2: Prime factorisation of 72: 72 = 2³ × 3²\nStep 3: HCF: The common factors are 2³. Therefore, HCF = 2³ = 8\nStep ... |
[
{
"from": "human",
"value": "Find the LCM and HCF of 9 and 12 by the prime factorisation method."
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 9: 9 = 3²\nStep 2: Prime factorisation of 12: 12 = 2² × 3\nStep 3: HCF: The common factor is 3. Therefore, HCF = 3\nStep 4: LCM: LCM is t... |
[
{
"from": "human",
"value": "Find the LCM and HCF of the following integers by applying the prime factorisation method: (i) 12, 15 and 21"
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 12: 12 = 2² × 3\nStep 2: Prime factorisation of 15: 15 = 3 × 5\nStep 3: Prime factorisation of 2... |
[
{
"from": "human",
"value": "Find the LCM and HCF of the following integers by applying the prime factorisation method: (ii) 17, 23 and 29"
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 17: 17 = 17\nStep 2: Prime factorisation of 23: 23 = 23\nStep 3: Prime factorisation of 29: 29 ... |
[
{
"from": "human",
"value": "Find the LCM and HCF of the following integers by applying the prime factorisation method: (iii) 8, 9 and 25"
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 8: 8 = 2³\nStep 2: Prime factorisation of 9: 9 = 3²\nStep 3: Prime factorisation of 25: 25 = 5²\... |
[
{
"from": "human",
"value": "Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) 26 and 91"
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 26: 26 = 2 × 13\nStep 2: Prime factorisation of 91: 91 = 7 × 13\nStep 3: HCF: T... |
[
{
"from": "human",
"value": "Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (ii) 510 and 92"
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 510: 510 = 2 × 3 × 5 × 17\nStep 2: Prime factorisation of 92: 92 = 2² × 23\nS... |
[
{
"from": "human",
"value": "Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (iii) 336 and 54"
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 336: 336 = 2⁴ × 3 × 7\nStep 2: Prime factorisation of 54: 54 = 2 × 3³\nStep ... |
[
{
"from": "human",
"value": "Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (iv) 45 and 60"
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 45: 45 = 3² × 5\nStep 2: Prime factorisation of 60: 60 = 2² × 3 × 5\nStep 3: H... |
[
{
"from": "human",
"value": "Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (v) 35 and 56"
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 35: 35 = 5 × 7\nStep 2: Prime factorisation of 56: 56 = 2³ × 7\nStep 3: HCF: Th... |
[
{
"from": "human",
"value": "Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (vi) 18 and 30"
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 18: 18 = 2 × 3²\nStep 2: Prime factorisation of 30: 30 = 2 × 3 × 5\nStep 3: HC... |
[
{
"from": "human",
"value": "Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (vii) 48 and 72"
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 48: 48 = 2⁴ × 3\nStep 2: Prime factorisation of 72: 72 = 2³ × 3²\nStep 3: HCF... |
[
{
"from": "human",
"value": "Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (viii) 16 and 20"
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 16: 16 = 2⁴\nStep 2: Prime factorisation of 20: 20 = 2² × 5\nStep 3: HCF: Th... |
[
{
"from": "human",
"value": "Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (ix) 24 and 40"
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 24: 24 = 2³ × 3\nStep 2: Prime factorisation of 40: 40 = 2³ × 5\nStep 3: HCF: ... |
[
{
"from": "human",
"value": "Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (x) 12 and 48"
},
{
"from": "gpt",
"value": "Step 1: Prime factorisation of 12: 12 = 2² × 3\nStep 2: Prime factorisation of 48: 48 = 2⁴ × 3\nStep 3: HCF: T... |
[
{
"from": "human",
"value": "There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet a... |
[
{
"from": "human",
"value": "Find the LCM of 14 and 20 using prime factorisation."
},
{
"from": "gpt",
"value": "To find the LCM of 14 and 20 using prime factorisation:\nStep 1: Prime factorisation of 14 = 2 × 7\nStep 2: Prime factorisation of 20 = 2² × 5\nStep 3: Take the highest power of all p... |
[
{
"from": "human",
"value": "Find the HCF of 42 and 56 using prime factorisation."
},
{
"from": "gpt",
"value": "To find the HCF of 42 and 56 using prime factorisation:\nStep 1: Prime factorisation of 42 = 2 × 3 × 7\nStep 2: Prime factorisation of 56 = 2³ × 7\nStep 3: Common prime factors with l... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the given numbers represent the sum and product of its zeroes respectively. (i) 1/4 , -1"
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 1/4 and product = -1\nStep 2: Use the identity: x² - (sum of zeroes)x + product =... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the given numbers represent the sum and product of its zeroes respectively. (ii) √2 , 1/3"
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = √2 and product = 1/3\nStep 2: Use the identity: x² - (sum of zeroes)x + product ... |
[
{
"from": "human",
"value": "Find the zeroes of the quadratic polynomial x² - 2x - 8 and verify the relationship between the zeroes and the coefficients."
},
{
"from": "gpt",
"value": "Step 1: Factor the polynomial x² - 2x - 8\nStep 2: x² - 2x - 8 = (x - 4)(x + 2)\nStep 3: Zeroes are x = 4 and x... |
[
{
"from": "human",
"value": "Find the zeroes of the polynomial x² – 3 and verify the relationship between the zeroes and the coefficients."
},
{
"from": "gpt",
"value": "Step 1: x² - 3 = 0\nStep 2: x² = 3 → x = ±√3\nStep 3: Zeroes are √3 and -√3\nVerification:\nSum = √3 + (-√3) = 0 → equals -b/a... |
[
{
"from": "human",
"value": "Verify that 3, -1, -1/3 are the zeroes of the cubic polynomial p(x) = 3x³ - 5x² - 11x - 3 and then verify the relationship between the zeroes and the coefficients."
},
{
"from": "gpt",
"value": "Step 1: Let’s verify p(3), p(-1), p(-1/3)\n→ p(3) = 3(27) - 5(9) - 11(3)... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the given numbers represent the sum and product of its zeroes respectively. (i) 1/4 , -1"
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 1/4 and product = -1\nStep 2: Use the identity: x² - (sum of zeroes)x + product =... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the given numbers represent the sum and product of its zeroes respectively. (ii) √2 , 1/3"
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = √2 and product = 1/3\nStep 2: Use the identity: x² - (sum of zeroes)x + product ... |
[
{
"from": "human",
"value": "Find the zeroes of the quadratic polynomial x² - 2x - 8 and verify the relationship between the zeroes and the coefficients."
},
{
"from": "gpt",
"value": "Step 1: Factor the polynomial x² - 2x - 8\nStep 2: x² - 2x - 8 = (x - 4)(x + 2)\nStep 3: Zeroes are x = 4 and x... |
[
{
"from": "human",
"value": "Find the zeroes of the polynomial x² – 3 and verify the relationship between the zeroes and the coefficients."
},
{
"from": "gpt",
"value": "Step 1: x² - 3 = 0\nStep 2: x² = 3 → x = ±√3\nStep 3: Zeroes are √3 and -√3\nVerification:\nSum = √3 + (-√3) = 0 → equals -b/a... |
[
{
"from": "human",
"value": "Verify that 3, -1, -1/3 are the zeroes of the cubic polynomial p(x) = 3x³ - 5x² - 11x - 3 and then verify the relationship between the zeroes and the coefficients."
},
{
"from": "gpt",
"value": "Step 1: Let’s verify p(3), p(-1), p(-1/3)\n→ p(3) = 3(27) - 5(9) - 11(3)... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 0.2 and -0.1, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 0.2 and product = -0.1\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial be... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 0.4 and 0.2, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 0.4 and product = 0.2\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial beco... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 0.6 and -0.3, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 0.6 and product = -0.3\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial be... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 0.8 and 0.4, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 0.8 and product = 0.4\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial beco... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 1.0 and -0.5, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 1.0 and product = -0.5\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial be... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 1.2 and 0.6, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 1.2 and product = 0.6\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial beco... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 1.4 and -0.7, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 1.4 and product = -0.7\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial be... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 1.6 and 0.8, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 1.6 and product = 0.8\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial beco... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 1.8 and -0.9, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 1.8 and product = -0.9\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial be... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 2.0 and 1.0, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 2.0 and product = 1.0\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial beco... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 2.2 and -1.1, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 2.2 and product = -1.1\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial be... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 2.4 and 1.2, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 2.4 and product = 1.2\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial beco... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 2.6 and -1.3, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 2.6 and product = -1.3\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial be... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 2.8 and 1.4, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 2.8 and product = 1.4\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial beco... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 3.0 and -1.5, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 3.0 and product = -1.5\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial be... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 3.2 and 1.6, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 3.2 and product = 1.6\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial beco... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 3.4 and -1.7, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 3.4 and product = -1.7\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial be... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 3.6 and 1.8, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 3.6 and product = 1.8\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial beco... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 3.8 and -1.9, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 3.8 and product = -1.9\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial be... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 4.0 and 2.0, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 4.0 and product = 2.0\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial beco... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 4.2 and -2.1, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 4.2 and product = -2.1\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial be... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 4.4 and 2.2, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 4.4 and product = 2.2\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial beco... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 4.6 and -2.3, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 4.6 and product = -2.3\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial be... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 4.8 and 2.4, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 4.8 and product = 2.4\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial beco... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 5.0 and -2.5, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 5.0 and product = -2.5\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial be... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 5.2 and 2.6, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 5.2 and product = 2.6\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial beco... |
[
{
"from": "human",
"value": "Find a quadratic polynomial where the sum and product of zeroes are 5.4 and -2.7, respectively."
},
{
"from": "gpt",
"value": "Step 1: Let the sum of zeroes = 5.4 and product = -2.7\nStep 2: Use the identity: x² - (sum of zeroes)x + product = 0\nStep 3: Polynomial be... |
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