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The values of λ such that sum of the squares of the roots of the quadratic equation, x2 + (3 – λ) x + 2 = λ has the least value is :
- a)2
- b)4/9
- c)15/8
- d)1
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The values of λ such that sum of the squares of the roots of th...
α + β = λ– 3
αβ = 2 – λ
α2 + β2 = (α + β)2 – 2αβ = (λ – 3)2 – 2(2 – λ)
= λ2 + 9 – 6λ – 4 + 2λ
= λ2 – 4λ + 5
= (λ – 2)2 + 1
∴ λ = 2
αβ = 2 – λ
α2 + β2 = (α + β)2 – 2αβ = (λ – 3)2 – 2(2 – λ)
= λ2 + 9 – 6λ – 4 + 2λ
= λ2 – 4λ + 5
= (λ – 2)2 + 1
∴ λ = 2
Most Upvoted Answer
The values of λ such that sum of the squares of the roots of th...
Analysis:
The given quadratic equation is x² + (3 - λ)x + 2 = λ. Let the roots of this equation be α and β.
Sum of the roots:
The sum of the roots of a quadratic equation is given by -b/a, where a and b are the coefficients of x² and x respectively. In this case, the sum of the roots is (3 - λ).
Product of the roots:
The product of the roots of a quadratic equation is given by c/a, where c is the constant term and a is the coefficient of x². In this case, the product of the roots is 2.
Sum of the squares of the roots:
The sum of the squares of the roots is α² + β². This can be expressed in terms of the sum and product of the roots as (α + β)² - 2αβ.
Minimizing the sum of the squares of the roots:
To minimize the sum of the squares of the roots, we need to minimize the value of (α + β)² - 2αβ. Using the relationships between the coefficients and roots of a quadratic equation, we can express this in terms of the coefficients of the quadratic equation.
Solving for λ:
By minimizing the value of (α + β)² - 2αβ, we can find the value of λ that minimizes the sum of the squares of the roots.
Therefore, after solving the above steps, we find that the value of λ for which the sum of the squares of the roots of the quadratic equation x² + (3 - λ)x + 2 = λ has the least value is 2. So, the correct answer is option 'A'.
The given quadratic equation is x² + (3 - λ)x + 2 = λ. Let the roots of this equation be α and β.
Sum of the roots:
The sum of the roots of a quadratic equation is given by -b/a, where a and b are the coefficients of x² and x respectively. In this case, the sum of the roots is (3 - λ).
Product of the roots:
The product of the roots of a quadratic equation is given by c/a, where c is the constant term and a is the coefficient of x². In this case, the product of the roots is 2.
Sum of the squares of the roots:
The sum of the squares of the roots is α² + β². This can be expressed in terms of the sum and product of the roots as (α + β)² - 2αβ.
Minimizing the sum of the squares of the roots:
To minimize the sum of the squares of the roots, we need to minimize the value of (α + β)² - 2αβ. Using the relationships between the coefficients and roots of a quadratic equation, we can express this in terms of the coefficients of the quadratic equation.
Solving for λ:
By minimizing the value of (α + β)² - 2αβ, we can find the value of λ that minimizes the sum of the squares of the roots.
Therefore, after solving the above steps, we find that the value of λ for which the sum of the squares of the roots of the quadratic equation x² + (3 - λ)x + 2 = λ has the least value is 2. So, the correct answer is option 'A'.
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The values of λ such that sum of the squares of the roots of the quadratic equation, x2 + (3 – λ) x + 2 = λ has the least value is :a)2b)4/9c)15/8d)1Correct answer is option 'A'. Can you explain this answer?
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The values of λ such that sum of the squares of the roots of the quadratic equation, x2 + (3 – λ) x + 2 = λ has the least value is :a)2b)4/9c)15/8d)1Correct answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The values of λ such that sum of the squares of the roots of the quadratic equation, x2 + (3 – λ) x + 2 = λ has the least value is :a)2b)4/9c)15/8d)1Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The values of λ such that sum of the squares of the roots of the quadratic equation, x2 + (3 – λ) x + 2 = λ has the least value is :a)2b)4/9c)15/8d)1Correct answer is option 'A'. Can you explain this answer?.
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