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If λ ϵ R is such that the sum of the cubes of the roots of the equation, x2 + (2 - λ)x + (10 - λ) = 0 is minimum, then the magnitude of the difference of the roots of this equation is:
- a)2√5
- b)20
- c)2√7
- d)4√7
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
If λ ϵ R is such that the sum of the cubes of the roots o...
α3 + β3 = (α + β)(α2 + β2 − αβ)
= −(2 − λ)((λ + β)2 − 3αβ)
= (λ − 2)((λ − 2)2 + 3(λ − 10))
= (λ − 2)(b2 − 4λ + 4 + 3λ − 30)
= −(2 − λ)((λ + β)2 − 3αβ)
= (λ − 2)((λ − 2)2 + 3(λ − 10))
= (λ − 2)(b2 − 4λ + 4 + 3λ − 30)
Most Upvoted Answer
If λ ϵ R is such that the sum of the cubes of the roots o...
Minimizing the Sum of Cubes of Roots:
To find the minimum sum of cubes of roots of the equation x^2 + (2 - λ)x + (10 - λ) = 0, we first need to determine the roots of the equation.
Finding the Roots:
Using the quadratic formula, we have:
x = [-b ± √(b^2 - 4ac)] / 2a
Plugging in the values a = 1, b = 2 - λ, and c = 10 - λ, we get:
x = [-(2 - λ) ± √((2 - λ)^2 - 4(10 - λ))] / 2
Minimizing the Sum of Cubes:
Let the roots be α and β. We know that α^3 + β^3 = (α + β)^3 - 3αβ(α + β).
Since α + β = -(2 - λ) and αβ = 10 - λ, we have:
α^3 + β^3 = (2 - λ)^3 - 3(10 - λ)(2 - λ)
Finding the Minimum Value:
To minimize the sum of cubes, we take the derivative of the expression and set it to zero:
d/dλ [(2 - λ)^3 - 3(10 - λ)(2 - λ)] = 0
Solving this equation gives λ = 2.
Calculating the Difference of Roots:
Substitute λ = 2 back into the roots equation:
x = [-(2 - 2) ± √((2 - 2)^2 - 4(10 - 2))] / 2
This simplifies to x = ±√20
Therefore, the magnitude of the difference of the roots is |√20 - (-√20)| = 2√5, which corresponds to option 'A'.
To find the minimum sum of cubes of roots of the equation x^2 + (2 - λ)x + (10 - λ) = 0, we first need to determine the roots of the equation.
Finding the Roots:
Using the quadratic formula, we have:
x = [-b ± √(b^2 - 4ac)] / 2a
Plugging in the values a = 1, b = 2 - λ, and c = 10 - λ, we get:
x = [-(2 - λ) ± √((2 - λ)^2 - 4(10 - λ))] / 2
Minimizing the Sum of Cubes:
Let the roots be α and β. We know that α^3 + β^3 = (α + β)^3 - 3αβ(α + β).
Since α + β = -(2 - λ) and αβ = 10 - λ, we have:
α^3 + β^3 = (2 - λ)^3 - 3(10 - λ)(2 - λ)
Finding the Minimum Value:
To minimize the sum of cubes, we take the derivative of the expression and set it to zero:
d/dλ [(2 - λ)^3 - 3(10 - λ)(2 - λ)] = 0
Solving this equation gives λ = 2.
Calculating the Difference of Roots:
Substitute λ = 2 back into the roots equation:
x = [-(2 - 2) ± √((2 - 2)^2 - 4(10 - 2))] / 2
This simplifies to x = ±√20
Therefore, the magnitude of the difference of the roots is |√20 - (-√20)| = 2√5, which corresponds to option 'A'.
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If λ ϵ R is such that the sum of the cubes of the roots of the equation, x2 + (2 - λ)x + (10 - λ) = 0 is minimum, then the magnitude of the difference of the roots of this equation is: a)2√5b)20c)2√7d)4√7Correct answer is option 'A'. Can you explain this answer?
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If λ ϵ R is such that the sum of the cubes of the roots of the equation, x2 + (2 - λ)x + (10 - λ) = 0 is minimum, then the magnitude of the difference of the roots of this equation is: a)2√5b)20c)2√7d)4√7Correct answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If λ ϵ R is such that the sum of the cubes of the roots of the equation, x2 + (2 - λ)x + (10 - λ) = 0 is minimum, then the magnitude of the difference of the roots of this equation is: a)2√5b)20c)2√7d)4√7Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If λ ϵ R is such that the sum of the cubes of the roots of the equation, x2 + (2 - λ)x + (10 - λ) = 0 is minimum, then the magnitude of the difference of the roots of this equation is: a)2√5b)20c)2√7d)4√7Correct answer is option 'A'. Can you explain this answer?.
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