If D = 0 and at least one auxiliary determinant is non-zero, what is the solution status of the pair?
Answer and explanation
Correct answer: No solution
For a pair of linear equations, let D be the determinant of the coefficients of x and y, while Dx and Dy are the auxiliary determinants. Cramer's rule gives a unique solution only when D is non-zero. If D = 0, the coefficient rows are dependent, so the represented lines are either coincident or parallel. When at least one auxiliary determinant is non-zero, the corresponding ratios are not all equal; hence the equations are inconsistent. Geometrically, the lines are distinct and parallel. They have no common point, so the pair has no solution. Therefore option C is correct, whereas infinitely many solutions require D = Dx = Dy = 0.
Frequently asked questions
What is the correct answer to this question?
No solution
Why is this the correct answer?
For a pair of linear equations, let D be the determinant of the coefficients of x and y, while Dx and Dy are the auxiliary determinants. Cramer's rule gives a unique solution only when D is non-zero. If D = 0, the coefficient rows are dependent, so the represented lines are either coincident or parallel. When at least one auxiliary determinant is non-zero, the corresponding ratios are not all equal; hence the equations are inconsistent. Geometrically, the lines are distinct and parallel. They have no common point, so the pair has no solution. Therefore option C is correct, whereas infinitely many solutions require D = Dx = Dy = 0.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Pair of Linear Equations in Two Variables. Topic: Conditions for solvability.
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