Number of Solutions (Conditions) MCQs Quiz | Class 10
Welcome to the Class X Mathematics quiz on Number of Solutions (Conditions) from Unit II: Algebra. This quiz focuses on understanding the conditions for unique solutions, infinitely many solutions, or no solution for a pair of linear equations. Test your knowledge, then submit your answers to see your score and download a detailed answer PDF for review.
Understanding Solutions of Linear Equations
Linear equations in two variables, such as ax + by + c = 0, represent a straight line in a two-dimensional coordinate system. When we consider a pair of such equations, we are essentially looking for points (x, y) that satisfy both equations simultaneously. Graphically, this means finding the intersection points of the two lines.
Conditions for Number of Solutions
Based on the relationship between their coefficients, a pair of linear equations can have a unique solution, no solution, or infinitely many solutions.
1. Unique Solution (Consistent and Independent)
- Algebraic Condition: For equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, a unique solution exists if:
a1 / a2 is NOT equal to b1 / b2
- Graphical Interpretation: The two lines intersect at exactly one point. This point is the unique solution (x, y) that satisfies both equations.
2. No Solution (Inconsistent)
- Algebraic Condition: A pair of linear equations has no solution if:
a1 / a2 is equal to b1 / b2 is NOT equal to c1 / c2
- Graphical Interpretation: The two lines are parallel and distinct. Since parallel lines never intersect, there is no common point (x, y) that satisfies both equations.
3. Infinitely Many Solutions (Consistent and Dependent)
- Algebraic Condition: A pair of linear equations has infinitely many solutions if:
a1 / a2 is equal to b1 / b2 is equal to c1 / c2
- Graphical Interpretation: The two lines are coincident, meaning they lie exactly on top of each other. Every point on one line is also a point on the other line, hence there are infinitely many common points (solutions).
Summary Table of Conditions
| Type of Solution | Algebraic Condition | Graphical Representation |
|---|---|---|
| Unique Solution | a1/a2 ≠ b1/b2 | Intersecting Lines |
| No Solution | a1/a2 = b1/b2 ≠ c1/c2 | Parallel Lines |
| Infinitely Many Solutions | a1/a2 = b1/b2 = c1/c2 | Coincident Lines |
Quick Revision Points
- A system of linear equations describes the relationship between two lines.
- Comparing the ratios of coefficients (a1/a2, b1/b2, c1/c2) helps determine the nature of solutions.
- Intersecting lines imply a unique solution.
- Parallel lines imply no solution.
- Coincident lines imply infinitely many solutions.
- “Consistent” systems have at least one solution (unique or infinitely many).
- “Inconsistent” systems have no solution.
Practice Questions
Test your understanding with these additional questions:
- For what value of ‘p’ does the pair of equations 4x + py + 8 = 0 and 2x + 2y + 2 = 0 have a unique solution?
- Find the value of ‘k’ for which the equations kx + 3y = k-3 and 12x + ky = k have infinitely many solutions.
- Check whether the pair of equations 5x – 8y + 1 = 0 and 3x – (24/5)y + (3/5) = 0 is consistent or inconsistent.
- What type of lines are represented by the equations x + 2y – 4 = 0 and 2x + 4y – 12 = 0?
- If the system of equations 2x – 3y = 7 and (a+b)x – (a+b-3)y = 4a+b has infinitely many solutions, find the values of ‘a’ and ‘b’.
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