JEE Exam > JEE Questions > The smallest value of k, for which both the ...
Start Learning for Free
The smallest value of k, for which both the roots of the equation x2 - 8kx + 16(k2 - k + 1) = 0 are real, distinct and have values at least 4 , is
Correct answer is '2'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
The smallest value of k, for which both the roots of the equation x2 ...
The equation is x2 - 8kx + 16(k2 - k + 1) = 0
We have,
D = 64(k2 - (k2- k + 1)) = 64(k - 1) > 0,
⇒ k > 1
f(4) ≥ 0,
⇒ 16 - 32k + 16(k2 - k + 1)) ≥ 0
Which yield: k2 - 3k + 2 ≥ 0
⇒ k ≤ 1 or k ≥ 2.
Hence, k = 2
Free Test
FREE
| Start Free Test |
Community Answer
The smallest value of k, for which both the roots of the equation x2 ...
Given equation: x^2 - 8kx + 16(k^2 - k + 1) = 0
To find the smallest value of k for which both roots of the equation are real, distinct, and have values at least 4, we need to analyze the discriminant of the quadratic equation.
Discriminant: The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by Δ = b^2 - 4ac. It helps determine the nature of the roots of the quadratic equation.
If the discriminant is positive (Δ > 0), the roots are real and distinct.
If the discriminant is zero (Δ = 0), the roots are real and equal.
If the discriminant is negative (Δ < 0),="" the="" roots="" are="" />
Analysis:
Let's calculate the discriminant of the given equation and set it greater than zero to ensure real and distinct roots.
Δ = (-8k)^2 - 4(1)(16(k^2 - k + 1))
= 64k^2 - 64(k^2 - k + 1)
= 64k^2 - 64k^2 + 64k - 64
= 64k - 64
We want the roots to be real and distinct, so Δ > 0.
64k - 64 > 0
64k > 64
k > 1
Thus, the value of k must be greater than 1 for the roots to be real and distinct.
Minimum value of k for roots at least 4:
To ensure that both roots have values at least 4, we can set up the following conditions:
Root 1: x ≥ 4
Root 2: x ≥ 4
Since the quadratic equation is symmetric, the sum of the roots is given by:
Sum of roots = -b/a = 8k/1 = 8k
For real and distinct roots, the sum of the roots must be greater than or equal to 8.
8k ≥ 8
k ≥ 1
Combining the conditions k > 1 and k ≥ 1, we find that the smallest value of k that satisfies both conditions is k = 2.
Hence, the correct answer is '2'.
To find the smallest value of k for which both roots of the equation are real, distinct, and have values at least 4, we need to analyze the discriminant of the quadratic equation.
Discriminant: The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by Δ = b^2 - 4ac. It helps determine the nature of the roots of the quadratic equation.
If the discriminant is positive (Δ > 0), the roots are real and distinct.
If the discriminant is zero (Δ = 0), the roots are real and equal.
If the discriminant is negative (Δ < 0),="" the="" roots="" are="" />
Analysis:
Let's calculate the discriminant of the given equation and set it greater than zero to ensure real and distinct roots.
Δ = (-8k)^2 - 4(1)(16(k^2 - k + 1))
= 64k^2 - 64(k^2 - k + 1)
= 64k^2 - 64k^2 + 64k - 64
= 64k - 64
We want the roots to be real and distinct, so Δ > 0.
64k - 64 > 0
64k > 64
k > 1
Thus, the value of k must be greater than 1 for the roots to be real and distinct.
Minimum value of k for roots at least 4:
To ensure that both roots have values at least 4, we can set up the following conditions:
Root 1: x ≥ 4
Root 2: x ≥ 4
Since the quadratic equation is symmetric, the sum of the roots is given by:
Sum of roots = -b/a = 8k/1 = 8k
For real and distinct roots, the sum of the roots must be greater than or equal to 8.
8k ≥ 8
k ≥ 1
Combining the conditions k > 1 and k ≥ 1, we find that the smallest value of k that satisfies both conditions is k = 2.
Hence, the correct answer is '2'.
Attention JEE Students!
To make sure you are not studying endlessly, EduRev has designed JEE study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in JEE.
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
The smallest value of k, for which both the roots of the equation x2 - 8kx + 16(k2 - k + 1) = 0 are real, distinct and have values at least 4 , isCorrect answer is '2'. Can you explain this answer?
Question Description
The smallest value of k, for which both the roots of the equation x2 - 8kx + 16(k2 - k + 1) = 0 are real, distinct and have values at least 4 , isCorrect answer is '2'. 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 smallest value of k, for which both the roots of the equation x2 - 8kx + 16(k2 - k + 1) = 0 are real, distinct and have values at least 4 , isCorrect answer is '2'. 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 smallest value of k, for which both the roots of the equation x2 - 8kx + 16(k2 - k + 1) = 0 are real, distinct and have values at least 4 , isCorrect answer is '2'. Can you explain this answer?.
The smallest value of k, for which both the roots of the equation x2 - 8kx + 16(k2 - k + 1) = 0 are real, distinct and have values at least 4 , isCorrect answer is '2'. 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 smallest value of k, for which both the roots of the equation x2 - 8kx + 16(k2 - k + 1) = 0 are real, distinct and have values at least 4 , isCorrect answer is '2'. 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 smallest value of k, for which both the roots of the equation x2 - 8kx + 16(k2 - k + 1) = 0 are real, distinct and have values at least 4 , isCorrect answer is '2'. Can you explain this answer?.
Solutions for The smallest value of k, for which both the roots of the equation x2 - 8kx + 16(k2 - k + 1) = 0 are real, distinct and have values at least 4 , isCorrect answer is '2'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of The smallest value of k, for which both the roots of the equation x2 - 8kx + 16(k2 - k + 1) = 0 are real, distinct and have values at least 4 , isCorrect answer is '2'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The smallest value of k, for which both the roots of the equation x2 - 8kx + 16(k2 - k + 1) = 0 are real, distinct and have values at least 4 , isCorrect answer is '2'. Can you explain this answer?, a detailed solution for The smallest value of k, for which both the roots of the equation x2 - 8kx + 16(k2 - k + 1) = 0 are real, distinct and have values at least 4 , isCorrect answer is '2'. Can you explain this answer? has been provided alongside types of The smallest value of k, for which both the roots of the equation x2 - 8kx + 16(k2 - k + 1) = 0 are real, distinct and have values at least 4 , isCorrect answer is '2'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The smallest value of k, for which both the roots of the equation x2 - 8kx + 16(k2 - k + 1) = 0 are real, distinct and have values at least 4 , isCorrect answer is '2'. Can you explain this answer? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup