Physics Exam > Physics Questions > The system of equationsαx + y + z =&alp...
Start Learning for Free
The system of equations αx + y + z = α - 1, x + αy + z = α - 1, x + y + αz = α - 1 has no solution. Find the maximum value of α
Correct answer is '2'. Can you explain this answer?
Verified Answer
The system of equationsαx + y + z =α - 1, x +αy + z ...
The given system is,
The system won't have any solution if rank (A) ≠ Rank(A : B) i.e.
∴ α = either 2 or –1 but not equals to 0 or 3.
The correct answer is: 2
Most Upvoted Answer
The system of equationsαx + y + z =α - 1, x +αy + z ...
Understanding the System of Equations
The given system of equations is:
1. αx + y + z = α - 1
2. x + αy + z = α - 1
3. x + y + αz = α - 1
To analyze when this system has no solution, we can represent it in matrix form and assess its determinant.
Matrix Representation
The coefficients can be arranged in a matrix:
| α 1 1 |
| 1 α 1 |
| 1 1 α |
The corresponding constants are:
| α - 1 |
| α - 1 |
| α - 1 |
Condition for No Solution
For the system to have no solution, the determinant of the coefficient matrix must be zero. If the determinant is zero, it indicates that the equations are linearly dependent.
Calculating the Determinant
The determinant of the coefficient matrix is calculated as follows:
- Expand using the first row to find the determinant.
Solving this determinant, we find:
Det = α(α^2 - 3) - (1 - α) - (1 - α) = α^3 - 3α - 2
Setting this equal to zero gives us the equation:
α^3 - 3α - 2 = 0
Finding Roots
To find the maximum value of α for which the equation has no solution, we solve for when the discriminant is less than or equal to zero. Through analysis or numerical methods, we find that the roots can be approximated or calculated directly.
The maximum real root of the equation occurs at α = 2.
Conclusion
Thus, the maximum value of α for which the system of equations has no solution is:
α = 2
The given system of equations is:
1. αx + y + z = α - 1
2. x + αy + z = α - 1
3. x + y + αz = α - 1
To analyze when this system has no solution, we can represent it in matrix form and assess its determinant.
Matrix Representation
The coefficients can be arranged in a matrix:
| α 1 1 |
| 1 α 1 |
| 1 1 α |
The corresponding constants are:
| α - 1 |
| α - 1 |
| α - 1 |
Condition for No Solution
For the system to have no solution, the determinant of the coefficient matrix must be zero. If the determinant is zero, it indicates that the equations are linearly dependent.
Calculating the Determinant
The determinant of the coefficient matrix is calculated as follows:
- Expand using the first row to find the determinant.
Solving this determinant, we find:
Det = α(α^2 - 3) - (1 - α) - (1 - α) = α^3 - 3α - 2
Setting this equal to zero gives us the equation:
α^3 - 3α - 2 = 0
Finding Roots
To find the maximum value of α for which the equation has no solution, we solve for when the discriminant is less than or equal to zero. Through analysis or numerical methods, we find that the roots can be approximated or calculated directly.
The maximum real root of the equation occurs at α = 2.
Conclusion
Thus, the maximum value of α for which the system of equations has no solution is:
α = 2
|
Explore Courses for Physics exam
|
|
Similar Physics Doubts
Question Description
The system of equationsαx + y + z =α - 1, x +αy + z =α - 1, x + y +αz =α - 1has no solution. Find the maximum value ofαCorrect answer is '2'. Can you explain this answer? for Physics 2025 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about The system of equationsαx + y + z =α - 1, x +αy + z =α - 1, x + y +αz =α - 1has no solution. Find the maximum value ofαCorrect answer is '2'. Can you explain this answer? covers all topics & solutions for Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The system of equationsαx + y + z =α - 1, x +αy + z =α - 1, x + y +αz =α - 1has no solution. Find the maximum value ofαCorrect answer is '2'. Can you explain this answer?.
The system of equationsαx + y + z =α - 1, x +αy + z =α - 1, x + y +αz =α - 1has no solution. Find the maximum value ofαCorrect answer is '2'. Can you explain this answer? for Physics 2025 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about The system of equationsαx + y + z =α - 1, x +αy + z =α - 1, x + y +αz =α - 1has no solution. Find the maximum value ofαCorrect answer is '2'. Can you explain this answer? covers all topics & solutions for Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The system of equationsαx + y + z =α - 1, x +αy + z =α - 1, x + y +αz =α - 1has no solution. Find the maximum value ofαCorrect answer is '2'. Can you explain this answer?.
Solutions for The system of equationsαx + y + z =α - 1, x +αy + z =α - 1, x + y +αz =α - 1has no solution. Find the maximum value ofαCorrect answer is '2'. Can you explain this answer? in English & in Hindi are available as part of our courses for Physics.
Download more important topics, notes, lectures and mock test series for Physics Exam by signing up for free.
Here you can find the meaning of The system of equationsαx + y + z =α - 1, x +αy + z =α - 1, x + y +αz =α - 1has no solution. Find the maximum value ofαCorrect answer is '2'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The system of equationsαx + y + z =α - 1, x +αy + z =α - 1, x + y +αz =α - 1has no solution. Find the maximum value ofαCorrect answer is '2'. Can you explain this answer?, a detailed solution for The system of equationsαx + y + z =α - 1, x +αy + z =α - 1, x + y +αz =α - 1has no solution. Find the maximum value ofαCorrect answer is '2'. Can you explain this answer? has been provided alongside types of The system of equationsαx + y + z =α - 1, x +αy + z =α - 1, x + y +αz =α - 1has no solution. Find the maximum value ofαCorrect answer is '2'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The system of equationsαx + y + z =α - 1, x +αy + z =α - 1, x + y +αz =α - 1has no solution. Find the maximum value ofαCorrect answer is '2'. Can you explain this answer? tests, examples and also practice Physics tests.
|
|
Explore Courses for Physics exam
|
|
Signup to solve all Doubts
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup on EduRev and stay on top of your study goals
10M+ students crushing their study goals daily