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The roots of the equation ax^3 +bx -x+1 =0 are real, distinct and are in H.P. ,then.
a) b £ ( -∞ , 1/3)
b) a £ (-1/27 , ∞)
c) 27a+ 9b =2
d) None of these
Explain which answer is correct and How?
a) b £ ( -∞ , 1/3)
b) a £ (-1/27 , ∞)
c) 27a+ 9b =2
d) None of these
Explain which answer is correct and How?
Most Upvoted Answer
The roots of the equation ax^3 +bx -x+1 =0 are real, distinct and are ...
Understanding the Problem
The equation given is \( ax^3 + (b-1)x + 1 = 0 \). The roots are real, distinct, and in Harmonic Progression (H.P.).
Characteristics of Roots in H.P.
For roots \( r_1, r_2, r_3 \) in H.P., their reciprocals \( \frac{1}{r_1}, \frac{1}{r_2}, \frac{1}{r_3} \) must be in A.P. If we denote the roots as \( a, a+d, a+2d \), the roots in H.P. are given by:
- \( r_1 = \frac{1}{a} \)
- \( r_2 = \frac{1}{a+d} \)
- \( r_3 = \frac{1}{a+2d} \)
Conditions for Distinct Roots
The discriminant of the cubic polynomial must be positive for distinct real roots. The necessary conditions include:
- The first derivative, \( f'(x) = 3ax^2 + (b - 1) \), should have two distinct roots, implying that \( b - 1 < 0="" \)="" or="" \(="" b="" />< 1="" />
Analyzing Options
1. Option a: \( b \leq (-\infty, \frac{1}{3}) \)
- This is incorrect, as \( b < 1="" \)="" does="" not="" imply="" \(="" b="" \leq="" \frac{1}{3}="" />
2. Option b: \( a \leq (-\frac{1}{27}, \infty) \)
- This statement does not hold as it doesn’t provide a meaningful constraint on \( a \).
3. Option c: \( 27a + 9b = 2 \)
- This equation can represent a specific relationship, but it’s not a general condition that holds for any \( a \) and \( b \).
4. Option d: None of these
- Given the analysis above, this seems to be the most viable choice.
Conclusion
Thus, the correct answer is d) None of these, as the other options do not adequately capture the necessary relationships for the roots of the cubic equation to be real, distinct, and in H.P.
The equation given is \( ax^3 + (b-1)x + 1 = 0 \). The roots are real, distinct, and in Harmonic Progression (H.P.).
Characteristics of Roots in H.P.
For roots \( r_1, r_2, r_3 \) in H.P., their reciprocals \( \frac{1}{r_1}, \frac{1}{r_2}, \frac{1}{r_3} \) must be in A.P. If we denote the roots as \( a, a+d, a+2d \), the roots in H.P. are given by:
- \( r_1 = \frac{1}{a} \)
- \( r_2 = \frac{1}{a+d} \)
- \( r_3 = \frac{1}{a+2d} \)
Conditions for Distinct Roots
The discriminant of the cubic polynomial must be positive for distinct real roots. The necessary conditions include:
- The first derivative, \( f'(x) = 3ax^2 + (b - 1) \), should have two distinct roots, implying that \( b - 1 < 0="" \)="" or="" \(="" b="" />< 1="" />
Analyzing Options
1. Option a: \( b \leq (-\infty, \frac{1}{3}) \)
- This is incorrect, as \( b < 1="" \)="" does="" not="" imply="" \(="" b="" \leq="" \frac{1}{3}="" />
2. Option b: \( a \leq (-\frac{1}{27}, \infty) \)
- This statement does not hold as it doesn’t provide a meaningful constraint on \( a \).
3. Option c: \( 27a + 9b = 2 \)
- This equation can represent a specific relationship, but it’s not a general condition that holds for any \( a \) and \( b \).
4. Option d: None of these
- Given the analysis above, this seems to be the most viable choice.
Conclusion
Thus, the correct answer is d) None of these, as the other options do not adequately capture the necessary relationships for the roots of the cubic equation to be real, distinct, and in H.P.
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The roots of the equation ax^3 +bx -x+1 =0 are real, distinct and are in H.P. ,then.a) b £ ( -∞ , 1/3) b) a £ (-1/27 , ∞)c) 27a+ 9b =2 d) None of these Explain which answer is correct and How?
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The roots of the equation ax^3 +bx -x+1 =0 are real, distinct and are in H.P. ,then.a) b £ ( -∞ , 1/3) b) a £ (-1/27 , ∞)c) 27a+ 9b =2 d) None of these Explain which answer is correct and How? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about The roots of the equation ax^3 +bx -x+1 =0 are real, distinct and are in H.P. ,then.a) b £ ( -∞ , 1/3) b) a £ (-1/27 , ∞)c) 27a+ 9b =2 d) None of these Explain which answer is correct and How? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The roots of the equation ax^3 +bx -x+1 =0 are real, distinct and are in H.P. ,then.a) b £ ( -∞ , 1/3) b) a £ (-1/27 , ∞)c) 27a+ 9b =2 d) None of these Explain which answer is correct and How?.
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