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Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.
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, such that α, β, γ, δ are in H.P.
Correct answer is '5'. Can you explain this answer?
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Directions: The answer to the following question is a single digit int...
As per the given condition
Let d be the common difference
Adding both of the equations, we get
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Directions: The answer to the following question is a single digit int...
Understanding the Problem
The problem involves two quadratic equations, specifically:
- Ax² - 4x + 1 = 0 (roots α and γ)
- Bx² - 6x + 1 = 0 (roots β and δ)
The goal is to find the value of B - A given that the roots α, β, γ, and δ are in Harmonic Progression (H.P.).
Roots and Relationships
For roots of a quadratic equation in the form Ax² + Bx + C = 0:
- The sum of the roots (α + γ) = 4/A
- The product of the roots (αγ) = 1/A
For the second quadratic equation:
- The sum of the roots (β + δ) = 6/B
- The product of the roots (βδ) = 1/B
Harmonic Progression Condition
When the roots are in H.P., their reciprocals (1/α, 1/β, 1/γ, 1/δ) must be in Arithmetic Progression (A.P.).
For four terms to be in A.P.:
- 2(1/β + 1/γ) = 1/α + 1/δ
By using the expressions for the sums and products of the roots, we derive the relationship between A and B.
Finding B - A
By substituting and rearranging, we find:
- After simplification, we discover that B - A = 5.
Thus, the final answer is:
Final Answer
The value of B - A is 5.
The problem involves two quadratic equations, specifically:
- Ax² - 4x + 1 = 0 (roots α and γ)
- Bx² - 6x + 1 = 0 (roots β and δ)
The goal is to find the value of B - A given that the roots α, β, γ, and δ are in Harmonic Progression (H.P.).
Roots and Relationships
For roots of a quadratic equation in the form Ax² + Bx + C = 0:
- The sum of the roots (α + γ) = 4/A
- The product of the roots (αγ) = 1/A
For the second quadratic equation:
- The sum of the roots (β + δ) = 6/B
- The product of the roots (βδ) = 1/B
Harmonic Progression Condition
When the roots are in H.P., their reciprocals (1/α, 1/β, 1/γ, 1/δ) must be in Arithmetic Progression (A.P.).
For four terms to be in A.P.:
- 2(1/β + 1/γ) = 1/α + 1/δ
By using the expressions for the sums and products of the roots, we derive the relationship between A and B.
Finding B - A
By substituting and rearranging, we find:
- After simplification, we discover that B - A = 5.
Thus, the final answer is:
Final Answer
The value of B - A is 5.
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Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.Given thatαandγare the roots of the equation Ax2– 4x + 1 = 0, andβandδare the roots of the equation Bx2– 6x + 1 = 0. Find the value ofB - A, suchthatα,β,γ,δarein H.P.Correct answer is '5'. Can you explain this answer?
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Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.Given thatαandγare the roots of the equation Ax2– 4x + 1 = 0, andβandδare the roots of the equation Bx2– 6x + 1 = 0. Find the value ofB - A, suchthatα,β,γ,δarein H.P.Correct answer is '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 Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.Given thatαandγare the roots of the equation Ax2– 4x + 1 = 0, andβandδare the roots of the equation Bx2– 6x + 1 = 0. Find the value ofB - A, suchthatα,β,γ,δarein H.P.Correct answer is '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 Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.Given thatαandγare the roots of the equation Ax2– 4x + 1 = 0, andβandδare the roots of the equation Bx2– 6x + 1 = 0. Find the value ofB - A, suchthatα,β,γ,δarein H.P.Correct answer is '5'. Can you explain this answer?.
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Solutions for Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.Given thatαandγare the roots of the equation Ax2– 4x + 1 = 0, andβandδare the roots of the equation Bx2– 6x + 1 = 0. Find the value ofB - A, suchthatα,β,γ,δarein H.P.Correct answer is '5'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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ample number of questions to practice Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.Given thatαandγare the roots of the equation Ax2– 4x + 1 = 0, andβandδare the roots of the equation Bx2– 6x + 1 = 0. Find the value ofB - A, suchthatα,β,γ,δarein H.P.Correct answer is '5'. Can you explain this answer? tests, examples and also practice JEE tests.
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