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Practice Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables MCQ Questions with Answers based on the latest NCERT syllabus. Learn how to determine whether a pair of linear equations has one solution, no solution, or infinitely many solutions using coefficient relationships, with detailed solutions and board exam tips.
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Chapter Information
Subject: Mathematics
Class: 10
Chapter: Pair of Linear Equations in Two Variables
Question Type: Multiple Choice Questions (MCQs)
Difficulty Level: Board Exam Level
Based On: NCERT Latest Syllabus
Introduction:
One of the most important concepts in this chapter is determining the number of solutions of a pair of linear equations. Instead of drawing graphs every time, mathematicians use the relationship between the coefficients of the two equations.
By comparing the ratios of the coefficients of $$x$$, $$y$$, and the constant terms, we can quickly determine whether the equations have one unique solution, no solution, or infinitely many solutions.
This concept is frequently tested in CBSE Board examinations because it combines algebraic understanding with logical reasoning. Students should memorize the conditions carefully and practice applying them to different types of equations.
What You Will Learn?
✔ Conditions for One Solution
✔ Conditions for No Solution
✔ Conditions for Infinitely Many Solutions
✔ Comparing Coefficient Ratios
✔ Algebraic Interpretation
✔ Board Exam Preparation
Why This Topic Is Important?
These conditions allow students to determine the nature of solutions without solving the equations completely, saving valuable time in examinations.
Exam Relevance
These questions are useful for:
✔ CBSE Board Exams
✔ State Board Exams
✔ School Unit Tests
✔ Half-Yearly Exams
✔ Annual Exams
✔ Scholarship Examinations
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Q1. If
$$\frac{a_1}{a_2}\neq\frac{b_1}{b_2}$$
then the pair of linear equations has:
Question Difficulty:
🟢 Easy
A) No Solution
B) One Unique Solution
C) Infinitely Many Solutions
D) Cannot be Determined
Answer:
$$\text{One Unique Solution}$$
Useful Formula for this Question:
For one unique solution:
$$\frac{a_1}{a_2}\neq\frac{b_1}{b_2}$$
Concept Behind This Question:
Different slopes mean the two lines intersect at exactly one point.
Solution:
Since
$$\frac{a_1}{a_2}\neq\frac{b_1}{b_2}$$
the two lines intersect at exactly one point.
Therefore, the pair of equations has:
$$\text{One Unique Solution}$$
Hence, the correct answer is:
$$\text{One Unique Solution}$$
Exam Tip:
This is one of the most frequently asked direct formula questions in board exams.
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Q2. If
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$$
then the pair of linear equations has:
Question Difficulty:
🟡 Medium
A) One Unique Solution
B) No Solution
C) Infinitely Many Solutions
D) Exactly Two Solutions
Answer:
$$\text{No Solution}$$
Useful Formula for this Question:
For no solution:
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$$
Concept Behind This Question:
The lines are parallel and never intersect.
Solution:
Equal ratios of coefficients of
$$x$$
and
$$y$$
but unequal constant ratios indicate parallel lines.
Parallel lines never meet.
Therefore, there is:
$$\text{No Solution}$$
Hence, the correct answer is:
$$\text{No Solution}$$
Exam Tip:
Parallel lines always mean no common solution.
————————————————–
Q3. If
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$$
then the pair of linear equations has:
Question Difficulty:
🟡 Medium
A) No Solution
B) One Unique Solution
C) Infinitely Many Solutions
D) Exactly Three Solutions
Answer:
$$\text{Infinitely Many Solutions}$$
Useful Formula for this Question:
For infinitely many solutions:
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$$
Concept Behind This Question:
The equations represent the same straight line.
Solution:
All three ratios are equal.
Hence, both equations represent coincident lines.
Therefore, there are:
$$\text{Infinitely Many Solutions}$$
Hence, the correct answer is:
$$\text{Infinitely Many Solutions}$$
Exam Tip:
Equal ratios everywhere mean the same line.
————————————————–
Q4. Which graphical representation corresponds to
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$$
Question Difficulty:
🟡 Medium
A) Intersecting Lines
B) Coincident Lines
C) Parallel Lines
D) Perpendicular Lines
Answer:
$$\text{Parallel Lines}$$
Useful Formula for this Question:
Parallel lines satisfy:
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$$
Concept Behind This Question:
Parallel lines have the same slope but different intercepts.
Solution:
Since the coefficient ratios are equal but the constant ratios differ, the lines never intersect.
Therefore, they are:
$$\text{Parallel Lines}$$
Hence, the correct answer is:
$$\text{Parallel Lines}$$
Exam Tip:
Parallel lines → No intersection → No solution.
————————————————–
Q5. Which graphical representation corresponds to
$$\frac{a_1}{a_2}\neq\frac{b_1}{b_2}$$
Question Difficulty:
🟢 Easy
A) Coincident Lines
B) Parallel Lines
C) Intersecting Lines
D) Horizontal Lines Only
Answer:
$$\text{Intersecting Lines}$$
Useful Formula for this Question:
Different coefficient ratios indicate intersecting lines.
Concept Behind This Question:
Intersecting lines meet at exactly one point.
Solution:
Since
$$\frac{a_1}{a_2}\neq\frac{b_1}{b_2}$$
the slopes of the lines are different.
Hence, the lines intersect at one point.
Therefore, the correct answer is:
$$\text{Intersecting Lines}$$
Exam Tip:
Different slopes always intersect exactly once.
————————————————–
Important Formulas and Concepts
General Form:
$$a_1x+b_1y+c_1=0$$
$$a_2x+b_2y+c_2=0$$
One Unique Solution:
$$\frac{a_1}{a_2}\neq\frac{b_1}{b_2}$$
No Solution:
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$$
Infinitely Many Solutions:
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$$
————————————————–
FAQs
Q. Why do intersecting lines have one solution?
Answer:
Because they meet at exactly one common point.
Q. Why do parallel lines have no solution?
Answer:
Because they never intersect.
Q. Why do coincident lines have infinitely many solutions?
Answer:
Because every point on the common line satisfies both equations.
————————————————–
Common Mistakes Students Make
❌ Comparing only one ratio instead of all required ratios.
❌ Confusing parallel and coincident lines.
❌ Forgetting the condition for infinitely many solutions.
❌ Solving the equations unnecessarily instead of using ratio conditions.
————————————————–
Quick Revision Notes
✔ One Solution:
$$\frac{a_1}{a_2}\neq\frac{b_1}{b_2}$$
✔ No Solution:
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$$
✔ Infinite Solutions:
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$$
✔ Intersecting Lines → One Solution
✔ Parallel Lines → No Solution
✔ Coincident Lines → Infinitely Many Solutions
————————————————–
Conclusion:
These Class 10 Maths Chapter 3 MCQs help students master the relationship between coefficient ratios and the number of solutions of a pair of linear equations. Understanding these conditions improves speed, accuracy, and confidence in solving board examination questions.
Related links
- Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables part-2
- Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables – Part 1
- Class 10 Maths Chapter 2 Polynomials (Graphical Zeroes) – Part 13
- Class 10 Maths Chapter 2 Polynomials graphical representation of polynomials – Part 12
- Class 10 Maths Chapter 2 Polynomials (Cubic Polynomials ) – Part 11
- Class 10 Maths Chapter 2 Polynomials zeroes of polynomials – Part 10
- Class 10 Maths Chapter 2 Polynomials formation – Part 9
- Class 10 Maths Chapter 2 Forming Quadratic Polynomials – Part 8
- Class10 Maths Chapter2 Polynomials MCQ – Part 7
- Class10 Maths Chapter2 Polynomials, missing coefficients – Part 6
- Class 10 Maths Chapter 2 Polynomials MCQ – Part 5
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