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If S is the set of distinct values of 'b' for which the following system of linear equations x + y + z = 1 x + ay + z = 1 ax + by + z = 0 has no solution, then S is :
- a)a singleton
- b)an empty set
- c)an infinite set
- d)a finite set containing two or more elements
Correct answer is option 'A'. Can you explain this answer?
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If S is the set of distinct values of 'b' for which the follow...
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If S is the set of distinct values of 'b' for which the follow...
Explanation:
Given System of Equations:
The given system of linear equations is:
x + y + z = 1
x + ay + z = 1
ax + by + z = 0
Inconsistent System:
For the system to have no solution, it must be inconsistent, i.e., the determinant of the coefficients of x, y, and z must be zero.
Determinant Calculation:
The determinant of the coefficients of x, y, and z is given by:
| 1 1 1 |
| 1 a 1 |
| a b 1 |
Determinant Condition:
For the system to have no solution, the determinant must be zero.
Therefore, solving for the determinant gives:
1(1-b) - a(1-a) = 0
1 - b - a + a^2 = 0
a^2 - a - b + 1 = 0
Discriminant Calculation:
The discriminant of the above quadratic equation is Δ = (-1)^2 - 4(1)(-b+1) = 1 + 4b - 4 = 4b - 3.
No Real Roots Condition:
For the system to have no real roots, the discriminant must be negative.
Therefore, 4b - 3 < />
4b < />
b < />
Conclusion:
The set of distinct values of b for which the system has no solution is a singleton set {b} where b < 3/4.="" hence,="" the="" correct="" answer="" is="" option="" 'a'.="" 3/4.="" hence,="" the="" correct="" answer="" is="" option="" />
Given System of Equations:
The given system of linear equations is:
x + y + z = 1
x + ay + z = 1
ax + by + z = 0
Inconsistent System:
For the system to have no solution, it must be inconsistent, i.e., the determinant of the coefficients of x, y, and z must be zero.
Determinant Calculation:
The determinant of the coefficients of x, y, and z is given by:
| 1 1 1 |
| 1 a 1 |
| a b 1 |
Determinant Condition:
For the system to have no solution, the determinant must be zero.
Therefore, solving for the determinant gives:
1(1-b) - a(1-a) = 0
1 - b - a + a^2 = 0
a^2 - a - b + 1 = 0
Discriminant Calculation:
The discriminant of the above quadratic equation is Δ = (-1)^2 - 4(1)(-b+1) = 1 + 4b - 4 = 4b - 3.
No Real Roots Condition:
For the system to have no real roots, the discriminant must be negative.
Therefore, 4b - 3 < />
4b < />
b < />
Conclusion:
The set of distinct values of b for which the system has no solution is a singleton set {b} where b < 3/4.="" hence,="" the="" correct="" answer="" is="" option="" 'a'.="" 3/4.="" hence,="" the="" correct="" answer="" is="" option="" />
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If S is the set of distinct values of 'b' for which the follow...
ANS is wrond b condain infinit values on the basis of the matrix
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If S is the set of distinct values of 'b' for which the following system of linear equations x + y + z = 1 x + ay + z = 1 ax + by + z = 0 has no solution, then S is :a)a singletonb)an empty setc)an infinite setd)a finite set containing two or more elementsCorrect answer is option 'A'. Can you explain this answer?
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If S is the set of distinct values of 'b' for which the following system of linear equations x + y + z = 1 x + ay + z = 1 ax + by + z = 0 has no solution, then S is :a)a singletonb)an empty setc)an infinite setd)a finite set containing two or more elementsCorrect answer is option 'A'. 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 If S is the set of distinct values of 'b' for which the following system of linear equations x + y + z = 1 x + ay + z = 1 ax + by + z = 0 has no solution, then S is :a)a singletonb)an empty setc)an infinite setd)a finite set containing two or more elementsCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If S is the set of distinct values of 'b' for which the following system of linear equations x + y + z = 1 x + ay + z = 1 ax + by + z = 0 has no solution, then S is :a)a singletonb)an empty setc)an infinite setd)a finite set containing two or more elementsCorrect answer is option 'A'. Can you explain this answer?.
If S is the set of distinct values of 'b' for which the following system of linear equations x + y + z = 1 x + ay + z = 1 ax + by + z = 0 has no solution, then S is :a)a singletonb)an empty setc)an infinite setd)a finite set containing two or more elementsCorrect answer is option 'A'. 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 If S is the set of distinct values of 'b' for which the following system of linear equations x + y + z = 1 x + ay + z = 1 ax + by + z = 0 has no solution, then S is :a)a singletonb)an empty setc)an infinite setd)a finite set containing two or more elementsCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If S is the set of distinct values of 'b' for which the following system of linear equations x + y + z = 1 x + ay + z = 1 ax + by + z = 0 has no solution, then S is :a)a singletonb)an empty setc)an infinite setd)a finite set containing two or more elementsCorrect answer is option 'A'. Can you explain this answer?.
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