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Given that α and γ are the roots of the equation Ax2 - 4x + 1 = 0, and β and δ are the roots of the equation Bx2 - 6x + 1 = 0. Find the value of (B - A)/10, such that α, β, γ, δ are in H.P.
(Answer round off upto 1 decimal place)
Correct answer is '0.5'. Can you explain this answer?
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Most Upvoted Answer
Given that α and γ are the roots of the equation Ax2 - 4x + 1 = 0, an...
Given:
- The equation Ax^2 - 4x + 1 = 0 has roots α and γ.
- The equation Bx^2 - 6x + 1 = 0 has roots β and δ.
- α, β, γ, and δ are in harmonic progression (H.P.).
To find:
The value of (B - A)/10
Solution:
Step 1: Find the value of α + γ and β + δ.
- For the equation Ax^2 - 4x + 1 = 0, the sum of roots is given by: α + γ = -(-4)/A = 4/A.
- For the equation Bx^2 - 6x + 1 = 0, the sum of roots is given by: β + δ = -(-6)/B = 6/B.
Step 2: Express α, β, γ, and δ in terms of the sum of roots.
- Since α, β, γ, and δ are in H.P., we have: 2/(α + γ) = 1/α + 1/γ and 2/(β + δ) = 1/β + 1/δ.
- Simplifying the above expressions, we get: α + γ = 2αγ/(α + γ) and β + δ = 2βδ/(β + δ).
Step 3: Substitute the values of α + γ and β + δ obtained in Step 1 into the expressions obtained in Step 2.
- For α + γ = 4/A, we have: 4/A = 2αγ/(α + γ).
- Simplifying the above expression, we get: 4/A = 2αγA/(4 + A).
- Cross multiplying, we get: 4(4 + A) = 2αγA.
- Expanding and rearranging, we get: 8A - 2αγA = 16.
- Factoring out A, we get: A(8 - 2αγ) = 16.
- Dividing both sides by 8 - 2αγ, we get: A = 16/(8 - 2αγ).
- Similarly, for β + δ = 6/B, we have: B = 36/(12 - 2βδ).
Step 4: Substitute the values of A and B obtained in Step 3 into the expression (B - A)/10.
- (B - A)/10 = (36/(12 - 2βδ) - 16/(8 - 2αγ))/10.
Step 5: Simplify the expression obtained in Step 4.
- (B - A)/10 = (36(8 - 2αγ) - 16(12 - 2βδ))/(10(12 - 2βδ)(8 - 2αγ)).
- (B - A)/10 = (288 - 72αγ - 192 + 32βδ)/(10(12 - 2βδ)(8 - 2αγ)).
- (B - A)/10 = (96 - 72αγ + 32βδ)/(10(12
- The equation Ax^2 - 4x + 1 = 0 has roots α and γ.
- The equation Bx^2 - 6x + 1 = 0 has roots β and δ.
- α, β, γ, and δ are in harmonic progression (H.P.).
To find:
The value of (B - A)/10
Solution:
Step 1: Find the value of α + γ and β + δ.
- For the equation Ax^2 - 4x + 1 = 0, the sum of roots is given by: α + γ = -(-4)/A = 4/A.
- For the equation Bx^2 - 6x + 1 = 0, the sum of roots is given by: β + δ = -(-6)/B = 6/B.
Step 2: Express α, β, γ, and δ in terms of the sum of roots.
- Since α, β, γ, and δ are in H.P., we have: 2/(α + γ) = 1/α + 1/γ and 2/(β + δ) = 1/β + 1/δ.
- Simplifying the above expressions, we get: α + γ = 2αγ/(α + γ) and β + δ = 2βδ/(β + δ).
Step 3: Substitute the values of α + γ and β + δ obtained in Step 1 into the expressions obtained in Step 2.
- For α + γ = 4/A, we have: 4/A = 2αγ/(α + γ).
- Simplifying the above expression, we get: 4/A = 2αγA/(4 + A).
- Cross multiplying, we get: 4(4 + A) = 2αγA.
- Expanding and rearranging, we get: 8A - 2αγA = 16.
- Factoring out A, we get: A(8 - 2αγ) = 16.
- Dividing both sides by 8 - 2αγ, we get: A = 16/(8 - 2αγ).
- Similarly, for β + δ = 6/B, we have: B = 36/(12 - 2βδ).
Step 4: Substitute the values of A and B obtained in Step 3 into the expression (B - A)/10.
- (B - A)/10 = (36/(12 - 2βδ) - 16/(8 - 2αγ))/10.
Step 5: Simplify the expression obtained in Step 4.
- (B - A)/10 = (36(8 - 2αγ) - 16(12 - 2βδ))/(10(12 - 2βδ)(8 - 2αγ)).
- (B - A)/10 = (288 - 72αγ - 192 + 32βδ)/(10(12 - 2βδ)(8 - 2αγ)).
- (B - A)/10 = (96 - 72αγ + 32βδ)/(10(12
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Community Answer
Given that α and γ are the roots of the equation Ax2 - 4x + 1 = 0, an...
As per the given conditions,
Let d be the common difference.
Adding both of the equations, we get
6 - 4 = 2d
or, d = 1
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Given that α and γ are the roots of the equation Ax2 - 4x + 1 = 0, and β and δ are the roots of the equation Bx2 - 6x + 1 = 0. Find the value of (B - A)/10, such that α, β, γ, δ are in H.P.(Answer round off upto 1 decimal place)Correct answer is '0.5'. Can you explain this answer?
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Given that α and γ are the roots of the equation Ax2 - 4x + 1 = 0, and β and δ are the roots of the equation Bx2 - 6x + 1 = 0. Find the value of (B - A)/10, such that α, β, γ, δ are in H.P.(Answer round off upto 1 decimal place)Correct answer is '0.5'. 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 Given that α and γ are the roots of the equation Ax2 - 4x + 1 = 0, and β and δ are the roots of the equation Bx2 - 6x + 1 = 0. Find the value of (B - A)/10, such that α, β, γ, δ are in H.P.(Answer round off upto 1 decimal place)Correct answer is '0.5'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Given that α and γ are the roots of the equation Ax2 - 4x + 1 = 0, and β and δ are the roots of the equation Bx2 - 6x + 1 = 0. Find the value of (B - A)/10, such that α, β, γ, δ are in H.P.(Answer round off upto 1 decimal place)Correct answer is '0.5'. Can you explain this answer?.
Given that α and γ are the roots of the equation Ax2 - 4x + 1 = 0, and β and δ are the roots of the equation Bx2 - 6x + 1 = 0. Find the value of (B - A)/10, such that α, β, γ, δ are in H.P.(Answer round off upto 1 decimal place)Correct answer is '0.5'. 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 Given that α and γ are the roots of the equation Ax2 - 4x + 1 = 0, and β and δ are the roots of the equation Bx2 - 6x + 1 = 0. Find the value of (B - A)/10, such that α, β, γ, δ are in H.P.(Answer round off upto 1 decimal place)Correct answer is '0.5'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Given that α and γ are the roots of the equation Ax2 - 4x + 1 = 0, and β and δ are the roots of the equation Bx2 - 6x + 1 = 0. Find the value of (B - A)/10, such that α, β, γ, δ are in H.P.(Answer round off upto 1 decimal place)Correct answer is '0.5'. Can you explain this answer?.
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