Consistency and Inconsistency MCQs Quiz | Class 10
Welcome to the Class X Mathematics (Code 041) quiz on Unit II: Algebra. This quiz focuses on ‘Consistency and Inconsistency MCQs Quiz | Class 10’, covering key concepts such as consistent vs inconsistent systems of linear equations and their interpretation. Attempt all 10 multiple-choice questions, then submit to view your results. You can also download a detailed answer PDF for review.
Understanding Consistency and Inconsistency in Linear Equations
Linear equations are fundamental in mathematics, and understanding the nature of their solutions is crucial. When we have a system of two linear equations in two variables (say, x and y), these equations geometrically represent straight lines. The ‘consistency’ or ‘inconsistency’ of the system refers to whether these lines intersect, are parallel, or coincide, which in turn determines if there is a solution, no solution, or infinitely many solutions.
What is a System of Linear Equations?
A pair of linear equations in two variables can be written in the general form:
- Equation 1: a1x + b1y + c1 = 0
- Equation 2: a2x + b2y + c2 = 0
Here, a1, b1, c1, a2, b2, c2 are real numbers, and a12 + b12 ≠ 0 and a22 + b22 ≠ 0.
Consistency and its Interpretation
A system of linear equations is said to be consistent if it has at least one solution. This means the lines represented by the equations either intersect at a single point or are coincident (overlap completely).
- Unique Solution (Consistent and Independent):
- Algebraic Condition: a1/a2 ≠ b1/b2
- Graphical Interpretation: The lines intersect at exactly one point. This point is the unique solution to the system.
- Infinitely Many Solutions (Consistent and Dependent):
- Algebraic Condition: a1/a2 = b1/b2 = c1/c2
- Graphical Interpretation: The lines are coincident; one line lies exactly on top of the other. Every point on the line is a solution.
Inconsistency and its Interpretation
A system of linear equations is said to be inconsistent if it has no solution. This occurs when the lines represented by the equations are parallel and distinct, meaning they never intersect.
- No Solution (Inconsistent):
- Algebraic Condition: a1/a2 = b1/b2 ≠ c1/c2
- Graphical Interpretation: The lines are parallel and never intersect. Hence, there is no common point, and thus no solution.
Summary Table
| Ratio of Coefficients | Graphical Representation | Consistency | Number of Solutions |
|---|---|---|---|
| a1/a2 ≠ b1/b2 | Intersecting Lines | Consistent | Unique Solution |
| a1/a2 = b1/b2 = c1/c2 | Coincident Lines | Consistent (Dependent) | Infinitely Many Solutions |
| a1/a2 = b1/b2 ≠ c1/c2 | Parallel Lines | Inconsistent | No Solution |
Quick Revision Points
- Consistent: At least one solution.
- Intersecting lines: Unique solution.
- Coincident lines: Infinitely many solutions.
- Inconsistent: No solution.
- Parallel lines: No common point.
- Memorize the three ratio conditions for quick analysis!
Practice Questions
- If two linear equations have a unique solution, what kind of lines do they represent graphically?
- Write a pair of linear equations that are consistent and dependent.
- Can a system of linear equations be consistent but have no solution? Explain.
- For what value of ‘k’ will the system of equations kx + 3y = k-3 and 12x + ky = k have no solution?
- If a system of equations is given by x + y = 5 and 2x + 2y = 10, how many solutions does it have and what is its consistency?
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