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Numbers of values of k for which roots of equation x2 − 3x + k = 0 lie in the interval (0,1) is
- a)only one
- b)no value
- c)finite but more than one
- d)k ≤ 9/4
Correct answer is option 'B'. Can you explain this answer?
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Numbers of values of k for which roots of equation x2 − 3x + k =...
Understanding the Quadratic Equation
The given quadratic equation is x² - 3x + k = 0. We need to determine the values of k for which the roots of this equation lie within the interval (0,1).
Condition for Roots in Interval (0,1)
For the roots of a quadratic equation ax² + bx + c = 0 to lie in an interval (a, b), the following conditions must be satisfied:
1. Sum of Roots: The sum of the roots (r1 + r2) = -b/a should be greater than 0 and less than 2.
2. Product of Roots: The product of the roots (r1 * r2) = c/a should be greater than 0.
In our case:
- Sum of Roots: r1 + r2 = 3
- Product of Roots: r1 * r2 = k
Analyzing the Conditions
1. Sum of Roots: Since 3 is not in the interval (0, 2), the first condition fails. This implies that the roots cannot both be positive and less than 1.
2. Product of Roots: For the product k to be positive, k must be greater than 0. However, this doesn’t help as the sum condition already invalidates the possibility of the roots lying in (0,1).
Conclusion
Given that the sum of the roots is fixed at 3, which does not allow both roots to lie within (0,1), we conclude that:
- There are no values of k that satisfy the conditions for the roots to lie in the interval (0,1).
Thus, the correct answer is option B: no value.
The given quadratic equation is x² - 3x + k = 0. We need to determine the values of k for which the roots of this equation lie within the interval (0,1).
Condition for Roots in Interval (0,1)
For the roots of a quadratic equation ax² + bx + c = 0 to lie in an interval (a, b), the following conditions must be satisfied:
1. Sum of Roots: The sum of the roots (r1 + r2) = -b/a should be greater than 0 and less than 2.
2. Product of Roots: The product of the roots (r1 * r2) = c/a should be greater than 0.
In our case:
- Sum of Roots: r1 + r2 = 3
- Product of Roots: r1 * r2 = k
Analyzing the Conditions
1. Sum of Roots: Since 3 is not in the interval (0, 2), the first condition fails. This implies that the roots cannot both be positive and less than 1.
2. Product of Roots: For the product k to be positive, k must be greater than 0. However, this doesn’t help as the sum condition already invalidates the possibility of the roots lying in (0,1).
Conclusion
Given that the sum of the roots is fixed at 3, which does not allow both roots to lie within (0,1), we conclude that:
- There are no values of k that satisfy the conditions for the roots to lie in the interval (0,1).
Thus, the correct answer is option B: no value.
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Community Answer
Numbers of values of k for which roots of equation x2 − 3x + k =...
As −b/2a = 3/2 ∉ (0,1), so both roots can not lie between 0 and 1 . So no values of k possible.
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