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If the roots of the equation x3 - 12x2 + 39x - 28 = 0 are in A.P., then their common difference will be
- a)+1
- b)+2
- c)+3
- d)+4
Correct answer is option 'C'. Can you explain this answer?
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If the roots of the equation x3 - 12x2 + 39x - 28 = 0 are in A.P., the...
Given:
The equation is x^3 - 12x^2 + 39x - 28 = 0.
To find:
The common difference of the roots if they are in arithmetic progression (A.P.).
Solution:
Let the roots of the equation be a - d, a, and a + d, where a is the middle term and d is the common difference.
Sum of the roots:
According to Vieta's formulas, the sum of the roots of a cubic equation is given by the negative coefficient of the quadratic term divided by the coefficient of the cubic term.
In this case, the sum of the roots is -(-12)/1 = 12.
Using the sum of roots:
The sum of the roots of the equation is given by (a - d) + a + (a + d) = 12.
Simplifying, we get 3a = 12, which gives a = 4.
Product of the roots:
According to Vieta's formulas, the product of the roots of a cubic equation is given by the constant term divided by the coefficient of the cubic term.
In this case, the product of the roots is -(constant term)/(coefficient of cubic term) = -(-28)/1 = 28.
Using the product of roots:
The product of the roots of the equation is given by (a - d)(a)(a + d) = 28.
Expanding, we get a^3 - d^2a = 28.
Using the values of a:
Substituting a = 4 in the equation a^3 - d^2a = 28, we get 64 - 16d^2 = 28.
Simplifying, we get 16d^2 = 36, which gives d^2 = 9.
Taking the square root of both sides, we get d = ±3.
Conclusion:
The common difference of the roots is ±3. However, since the options only include positive integers, the common difference is 3.
Therefore, the correct answer is option C: 3.
The equation is x^3 - 12x^2 + 39x - 28 = 0.
To find:
The common difference of the roots if they are in arithmetic progression (A.P.).
Solution:
Let the roots of the equation be a - d, a, and a + d, where a is the middle term and d is the common difference.
Sum of the roots:
According to Vieta's formulas, the sum of the roots of a cubic equation is given by the negative coefficient of the quadratic term divided by the coefficient of the cubic term.
In this case, the sum of the roots is -(-12)/1 = 12.
Using the sum of roots:
The sum of the roots of the equation is given by (a - d) + a + (a + d) = 12.
Simplifying, we get 3a = 12, which gives a = 4.
Product of the roots:
According to Vieta's formulas, the product of the roots of a cubic equation is given by the constant term divided by the coefficient of the cubic term.
In this case, the product of the roots is -(constant term)/(coefficient of cubic term) = -(-28)/1 = 28.
Using the product of roots:
The product of the roots of the equation is given by (a - d)(a)(a + d) = 28.
Expanding, we get a^3 - d^2a = 28.
Using the values of a:
Substituting a = 4 in the equation a^3 - d^2a = 28, we get 64 - 16d^2 = 28.
Simplifying, we get 16d^2 = 36, which gives d^2 = 9.
Taking the square root of both sides, we get d = ±3.
Conclusion:
The common difference of the roots is ±3. However, since the options only include positive integers, the common difference is 3.
Therefore, the correct answer is option C: 3.
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If the roots of the equation x3 - 12x2 + 39x - 28 = 0 are in A.P., then their common difference will bea)+1b)+2c)+3d)+4Correct answer is option 'C'. Can you explain this answer?
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If the roots of the equation x3 - 12x2 + 39x - 28 = 0 are in A.P., then their common difference will bea)+1b)+2c)+3d)+4Correct answer is option 'C'. Can you explain this answer? for SSC 2024 is part of SSC preparation. The Question and answers have been prepared according to the SSC exam syllabus. Information about If the roots of the equation x3 - 12x2 + 39x - 28 = 0 are in A.P., then their common difference will bea)+1b)+2c)+3d)+4Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for SSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the roots of the equation x3 - 12x2 + 39x - 28 = 0 are in A.P., then their common difference will bea)+1b)+2c)+3d)+4Correct answer is option 'C'. Can you explain this answer?.
If the roots of the equation x3 - 12x2 + 39x - 28 = 0 are in A.P., then their common difference will bea)+1b)+2c)+3d)+4Correct answer is option 'C'. Can you explain this answer? for SSC 2024 is part of SSC preparation. The Question and answers have been prepared according to the SSC exam syllabus. Information about If the roots of the equation x3 - 12x2 + 39x - 28 = 0 are in A.P., then their common difference will bea)+1b)+2c)+3d)+4Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for SSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the roots of the equation x3 - 12x2 + 39x - 28 = 0 are in A.P., then their common difference will bea)+1b)+2c)+3d)+4Correct answer is option 'C'. Can you explain this answer?.
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