Mechanical Engineering Exam > Mechanical Engineering Questions > For the set of equationsx1 + 2x+ x3+ 4x4 = 03...
Start Learning for Free
For the set of equations
x1 + 2x + x3 + 4x4 = 0
3x1 + 6x2 + 3x3 + 12x4 = 0
x1 + 2x + x3 + 4x4 = 0
3x1 + 6x2 + 3x3 + 12x4 = 0
- a)Only the trivial solution x1 = x2 = x3 = x4 = 0 exists.
- b)There are no solutions.
- c)A unique non-trivial solution exists.
- d)Multiple non-trivial solutions exist
Correct answer is option 'D'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
For the set of equationsx1 + 2x+ x3+ 4x4 = 03x1 + 6x2 + 3x3 + 12x4 = 0...
(d)
Because number of unknowns more them no. of equation.
Because number of unknowns more them no. of equation.
Most Upvoted Answer
For the set of equationsx1 + 2x+ x3+ 4x4 = 03x1 + 6x2 + 3x3 + 12x4 = 0...
Solution:
To determine the solution of the given set of equations, we can write them in matrix form as follows:
Ax = 0
where
A = [1 2 0 4; 3 6 3 12]
x = [x1; x2; x3; x4]
Rank of Matrix A:
The rank of matrix A can be determined using row reduction as follows:
A = [1 2 0 4; 3 6 3 12]
R2 = R2 - 3R1
A = [1 2 0 4; 0 0 3 0]
The matrix A has rank 2, which is less than the number of unknowns (4). Therefore, there are infinitely many solutions or no solutions.
Homogeneous System of Equations:
The given set of equations is homogeneous, i.e., the right-hand side of each equation is zero. Therefore, the trivial solution x1 = x2 = x3 = x4 = 0 is always a solution.
Non-Trivial Solutions:
Since the rank of matrix A is less than 4, there are infinitely many solutions or no solutions. To determine if there are non-trivial solutions, we can use the reduced row echelon form of matrix A as follows:
A = [1 2 0 4; 0 0 3 0]
R1 = R1 - 2R2
A = [1 2 0 4; 0 0 1 0]
R1 = R1 - 4R2
A = [1 2 0 0; 0 0 1 0]
The last row corresponds to the equation x3 = 0. Substituting this into the first row, we get x1 + 2x2 = 0 or x1 = -2x2. Therefore, the general solution can be written as:
x1 = -2t
x2 = t
x3 = 0
x4 = 0
where t is any scalar. This shows that there are infinitely many non-trivial solutions.
Conclusion:
The correct option is (D) Multiple non-trivial solutions exist.
To determine the solution of the given set of equations, we can write them in matrix form as follows:
Ax = 0
where
A = [1 2 0 4; 3 6 3 12]
x = [x1; x2; x3; x4]
Rank of Matrix A:
The rank of matrix A can be determined using row reduction as follows:
A = [1 2 0 4; 3 6 3 12]
R2 = R2 - 3R1
A = [1 2 0 4; 0 0 3 0]
The matrix A has rank 2, which is less than the number of unknowns (4). Therefore, there are infinitely many solutions or no solutions.
Homogeneous System of Equations:
The given set of equations is homogeneous, i.e., the right-hand side of each equation is zero. Therefore, the trivial solution x1 = x2 = x3 = x4 = 0 is always a solution.
Non-Trivial Solutions:
Since the rank of matrix A is less than 4, there are infinitely many solutions or no solutions. To determine if there are non-trivial solutions, we can use the reduced row echelon form of matrix A as follows:
A = [1 2 0 4; 0 0 3 0]
R1 = R1 - 2R2
A = [1 2 0 4; 0 0 1 0]
R1 = R1 - 4R2
A = [1 2 0 0; 0 0 1 0]
The last row corresponds to the equation x3 = 0. Substituting this into the first row, we get x1 + 2x2 = 0 or x1 = -2x2. Therefore, the general solution can be written as:
x1 = -2t
x2 = t
x3 = 0
x4 = 0
where t is any scalar. This shows that there are infinitely many non-trivial solutions.
Conclusion:
The correct option is (D) Multiple non-trivial solutions exist.
Attention Mechanical Engineering Students!
To make sure you are not studying endlessly, EduRev has designed Mechanical Engineering study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Mechanical Engineering.
|
Explore Courses for Mechanical Engineering exam
|
|
Similar Mechanical Engineering Doubts
Top Courses for Mechanical EngineeringView all
For the set of equationsx1 + 2x+ x3+ 4x4 = 03x1 + 6x2 + 3x3 + 12x4 = 0a)Only the trivial solution x1 = x2 = x3 = x4 = 0 exists.b)There are no solutions.c)A unique non-trivial solution exists.d)Multiple non-trivial solutions existCorrect answer is option 'D'. Can you explain this answer?
Question Description
For the set of equationsx1 + 2x+ x3+ 4x4 = 03x1 + 6x2 + 3x3 + 12x4 = 0a)Only the trivial solution x1 = x2 = x3 = x4 = 0 exists.b)There are no solutions.c)A unique non-trivial solution exists.d)Multiple non-trivial solutions existCorrect answer is option 'D'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about For the set of equationsx1 + 2x+ x3+ 4x4 = 03x1 + 6x2 + 3x3 + 12x4 = 0a)Only the trivial solution x1 = x2 = x3 = x4 = 0 exists.b)There are no solutions.c)A unique non-trivial solution exists.d)Multiple non-trivial solutions existCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For the set of equationsx1 + 2x+ x3+ 4x4 = 03x1 + 6x2 + 3x3 + 12x4 = 0a)Only the trivial solution x1 = x2 = x3 = x4 = 0 exists.b)There are no solutions.c)A unique non-trivial solution exists.d)Multiple non-trivial solutions existCorrect answer is option 'D'. Can you explain this answer?.
For the set of equationsx1 + 2x+ x3+ 4x4 = 03x1 + 6x2 + 3x3 + 12x4 = 0a)Only the trivial solution x1 = x2 = x3 = x4 = 0 exists.b)There are no solutions.c)A unique non-trivial solution exists.d)Multiple non-trivial solutions existCorrect answer is option 'D'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about For the set of equationsx1 + 2x+ x3+ 4x4 = 03x1 + 6x2 + 3x3 + 12x4 = 0a)Only the trivial solution x1 = x2 = x3 = x4 = 0 exists.b)There are no solutions.c)A unique non-trivial solution exists.d)Multiple non-trivial solutions existCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For the set of equationsx1 + 2x+ x3+ 4x4 = 03x1 + 6x2 + 3x3 + 12x4 = 0a)Only the trivial solution x1 = x2 = x3 = x4 = 0 exists.b)There are no solutions.c)A unique non-trivial solution exists.d)Multiple non-trivial solutions existCorrect answer is option 'D'. Can you explain this answer?.
Solutions for For the set of equationsx1 + 2x+ x3+ 4x4 = 03x1 + 6x2 + 3x3 + 12x4 = 0a)Only the trivial solution x1 = x2 = x3 = x4 = 0 exists.b)There are no solutions.c)A unique non-trivial solution exists.d)Multiple non-trivial solutions existCorrect answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mechanical Engineering.
Download more important topics, notes, lectures and mock test series for Mechanical Engineering Exam by signing up for free.
Here you can find the meaning of For the set of equationsx1 + 2x+ x3+ 4x4 = 03x1 + 6x2 + 3x3 + 12x4 = 0a)Only the trivial solution x1 = x2 = x3 = x4 = 0 exists.b)There are no solutions.c)A unique non-trivial solution exists.d)Multiple non-trivial solutions existCorrect answer is option 'D'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
For the set of equationsx1 + 2x+ x3+ 4x4 = 03x1 + 6x2 + 3x3 + 12x4 = 0a)Only the trivial solution x1 = x2 = x3 = x4 = 0 exists.b)There are no solutions.c)A unique non-trivial solution exists.d)Multiple non-trivial solutions existCorrect answer is option 'D'. Can you explain this answer?, a detailed solution for For the set of equationsx1 + 2x+ x3+ 4x4 = 03x1 + 6x2 + 3x3 + 12x4 = 0a)Only the trivial solution x1 = x2 = x3 = x4 = 0 exists.b)There are no solutions.c)A unique non-trivial solution exists.d)Multiple non-trivial solutions existCorrect answer is option 'D'. Can you explain this answer? has been provided alongside types of For the set of equationsx1 + 2x+ x3+ 4x4 = 03x1 + 6x2 + 3x3 + 12x4 = 0a)Only the trivial solution x1 = x2 = x3 = x4 = 0 exists.b)There are no solutions.c)A unique non-trivial solution exists.d)Multiple non-trivial solutions existCorrect answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice For the set of equationsx1 + 2x+ x3+ 4x4 = 03x1 + 6x2 + 3x3 + 12x4 = 0a)Only the trivial solution x1 = x2 = x3 = x4 = 0 exists.b)There are no solutions.c)A unique non-trivial solution exists.d)Multiple non-trivial solutions existCorrect answer is option 'D'. Can you explain this answer? tests, examples and also practice Mechanical Engineering tests.
|
|
Explore Courses for Mechanical Engineering 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
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup