Class 10 Mathematics — Chapter 3
Pair of Linear Equations in Two Variables: Question Bank
A. Multiple Choice Questions — MCQs
1.
The graph of a linear equation in two variables is a:
A. Circle
B. Straight line
C. Parabola
D. Triangle
Answer: B
2.
If two lines intersect at exactly one point, the pair of equations has:
A. No solution
B. One solution
C. Two solutions
D. Infinitely many solutions
Answer: B
3.
A pair of equations whose graphs are parallel lines is:
A. Consistent
B. Dependent
C. Inconsistent
D. Identical
Answer: C
4.
If two equations represent coincident lines, they have:
A. No solution
B. Exactly one solution
C. Exactly two solutions
D. Infinitely many solutions
Answer: D
5.
For a1x+b1y+c1=0
and a2x+b2y+c2=0,
the condition for a unique solution is:
A. a2a1=b2b1
B. a2a1=c2c1
C. a2a1=b2b1
D. b2b1=c2c1
Answer: C
6.
The condition a2a1=b2b1=c2c1
represents:
A. Intersecting lines
B. Parallel lines
C. Coincident lines
D. Perpendicular lines
Answer: B
7.
The condition a2a1=b2b1=c2c1
represents:
A. Parallel lines
B. Intersecting lines
C. Coincident lines
D. Perpendicular lines
Answer: C
8.
Which method involves expressing one variable in terms of the other?
A. Graphical method
B. Substitution method
C. Elimination method
D. Factorisation method
Answer: B
9.
In the elimination method, the main objective is to:
A. Draw a graph
B. Eliminate one variable
C. Find the slope
D. Factorise both equations
Answer: B
10.
If elimination produces the statement 0=0,
the pair of equations has:
A. No solution
B. One solution
C. Infinitely many solutions
D. Two solutions
Answer: C
11.
If elimination produces 0=5,
the pair of equations has:
A. One solution
B. No solution
C. Infinitely many solutions
D. Two solutions
Answer: B
12.
The equation 2x+3y−7=0
represents:
A. A straight line
B. A circle
C. A point
D. A parabola
Answer: A
13.
Which of the following represents a pair of coincident lines?
A. x+y=4,2x+2y=8
B. x+y=4,2x+2y=10
C. x+y=4,2x+3y=8
D. x+y=4,x−y=8
Answer: A
14.
Which pair represents parallel lines?
A. x+y=5,2x+2y=10
B. x+y=5,2x+2y=12
C. x+y=5,2x+3y=10
D. x−y=2,2x−y=5
Answer: B
15.
The solution of a pair of equations graphically is represented by:
A. The midpoint of the two lines
B. The common point of the two lines
C. The x-intercept only
D. The y-intercept only
Answer: B
16.
The substitution method is particularly convenient when:
A. One variable can be easily isolated
B. Both equations have no variables
C. The graph is unavailable
D. Both equations are quadratic
Answer: A
17.
If two lines have no common point, the corresponding pair is:
A. Consistent
B. Inconsistent
C. Dependent
D. Coincident
Answer: B
18.
If two equations have infinitely many common solutions, they represent:
A. Parallel lines
B. Intersecting lines
C. The same line
D. Perpendicular lines
Answer: C
19.
A two-digit number whose tens and units digits are x and y is written as:
A. x+y
B. xy
C. 10x+y
D. 10y+x
Answer: C
20.
When the digits x and y of a two-digit number are reversed, the number becomes:
A. x+y
B. 10x+y
C. 10y+x
D. xy
Answer: C
B. Fill in the Blanks
1.
The graph of a linear equation in two variables is a ________.
Answer: straight line
2.
Two intersecting lines have ________ common point.
Answer: one
3.
Parallel lines have ________ common solution.
Answer: no
4.
Coincident lines have ________ solutions.
Answer: infinitely many
5.
A pair having at least one solution is called a ________ pair.
Answer: consistent
6.
A pair having no solution is called an ________ pair.
Answer: inconsistent
7.
The substitution method involves expressing one ________ in terms of the other.
Answer: variable
8.
The elimination method aims to ________ one variable.
Answer: eliminate
9.
If elimination gives a true statement without variables, the pair has ________ solutions.
Answer: infinitely many
10.
If elimination gives a false statement without variables, the pair has ________ solution.
Answer: no
11.
The solution obtained from the graphical method is given by the ________ point of the two lines.
Answer: common/intersection
12.
A pair of coincident lines represents a ________ pair.
Answer: dependent
13.
A two-digit number with tens digit x and units digit y is ________.
Answer: 10x+y
14.
The reversed form of 10x+y is ________.
Answer: 10y+x
15.
In the elimination method, suitable non-zero constants are used to make coefficients of one variable ________.
Answer: equal
C. True or False
1.
Every pair of linear equations has exactly one solution.
Answer: False
2.
Two parallel lines have no common point.
Answer: True
3.
Coincident lines have infinitely many common points.
Answer: True
4.
A consistent pair must always have infinitely many solutions.
Answer: False
5.
An inconsistent pair has no solution.
Answer: True
6.
The graphical representation of a linear equation in two variables is a straight line.
Answer: True
7.
The elimination method removes one variable from the equations.
Answer: True
8.
The substitution method requires one variable to be expressed in terms of another.
Answer: True
9.
The statement 0=7 indicates infinitely many solutions.
Answer: False
10.
The statement 0=0 can indicate infinitely many solutions.
Answer: True
11.
Coincident lines are inconsistent.
Answer: False
12.
Parallel lines form an inconsistent pair.
Answer: True
D. Assertion–Reason Questions
Choose:
A. Both Assertion and Reason are true, and Reason correctly explains Assertion.
B. Both are true, but Reason does not correctly explain Assertion.
C. Assertion is true, but Reason is false.
D. Assertion is false, but Reason is true.
1.
Assertion: Two intersecting lines have a unique common solution.
Reason: Intersecting lines meet at exactly one point.
Answer: A
2.
Assertion: Parallel lines form an inconsistent pair.
Reason: Parallel lines have no common point.
Answer: A
3.
Assertion: Coincident lines have infinitely many solutions.
Reason: Every point on one line also lies on the other.
Answer: A
4.
Assertion: The substitution method eliminates a variable by adding two equations.
Reason: In substitution, one variable is first expressed in terms of another.
Answer: D
5.
Assertion: If elimination gives 0=5, the equations have no solution.
Reason: 0=5 is a false statement.
Answer: A
6.
Assertion: Graphical methods are always the most accurate method for fractional solutions.
Reason: Fractional coordinates can be difficult to read accurately from a graph.
Answer: D
E. Match the Following
| Column A | Column B |
|---|---|
| 1. Intersecting lines | A. No solution |
| 2. Parallel lines | B. Infinitely many solutions |
| 3. Coincident lines | C. Unique solution |
| 4. Consistent pair | D. At least one solution |
| 5. Inconsistent pair | E. No common solution |
Answers:
1–C, 2–A, 3–B, 4–D, 5–E
F. Identify the Nature of the Solution
For each pair, state whether it has one solution, no solution, or infinitely many solutions.
1.
x+y=6 2x+2y=12
Answer: Infinitely many solutions
2.
x+y=6 2x+2y=15
Answer: No solution
3.
x+y=6 2x+3y=10
Answer: One solution
4.
3x−2y+5=0 6x−4y+10=0
Answer: Infinitely many solutions
5.
2x+3y−4=0 4x+6y−9=0
Answer: No solution
G. Very Short Answer Questions
1.
What is a consistent pair of linear equations?
Answer: A pair having at least one solution.
2.
What is an inconsistent pair?
Answer: A pair having no solution.
3.
What does a point of intersection of two lines represent?
Answer: The common solution of the two equations.
4.
What do coincident lines represent?
Answer: Infinitely many common solutions.
5.
Name the two algebraic methods discussed in the chapter.
Answer: Substitution and elimination.
6.
What is the main purpose of elimination?
Answer: To remove one variable.
7.
What is the first major step in substitution?
Answer: Express one variable in terms of the other.
8.
What does 0=0 indicate after elimination?
Answer: Infinitely many solutions.
9.
What does 0=9 indicate after elimination?
Answer: No solution.
10.
What is the graphical representation of a pair of linear equations?
Answer: Two straight lines.
H. Concept-Based Questions
1.
Why can a pair of linear equations have infinitely many solutions?
Answer: When both equations represent the same straight line, every point on that line satisfies both equations.
2.
Why do parallel lines have no solution?
Answer: They never meet, so they have no common point.
3.
Why is the graphical method sometimes inconvenient?
Answer: When the solution has fractional or decimal coordinates, accurately reading the point from a graph can be difficult.
4.
Why is the elimination method so named?
Answer: Because one variable is eliminated first, leaving an equation in the other variable.
5.
Why is the substitution method so named?
Answer: Because the value or expression for one variable is substituted into the other equation.
I. Application-Based Questions
1. Age Situation
The present ages of two people are represented by x and y. A condition involving their ages produces two linear equations.
Question: Which algebraic methods can be used to find their ages?
Answer: Substitution or elimination; the pair can also be represented graphically.
2. Cost Situation
The cost of one item is x and another is y. Two different combinations of the items give two total costs.
Question: What mathematical model is formed?
Answer: A pair of linear equations in x and y.
3. Taxi Charges
A taxi fare consists of a fixed charge and a charge based on distance.
Let the fixed charge be x and the charge per kilometre be y.
For two different distances, two fare equations can be formed.
Question: Which method can be used to determine x and y?
Answer: Substitution or elimination.
4. Number Problem
The tens and units digits of a two-digit number are x and y.
Question: Write the number and its reversed form.
Answer:
Original number: 10x+y
Reversed number: 10y+x
5. Geometry Interpretation
Two rails are represented by two linear equations.
Question: What does it mean geometrically if the equations have no common solution?
Answer: The rails are represented by parallel lines and therefore do not cross.
J. Competency-Based Questions
1.
A student obtains a2a1=b2b1
while comparing two equations.
Can the student immediately conclude that the pair is inconsistent?
Answer: No. The ratio c2c1 must also be compared. If a2a1=b2b1=c2c1,
the pair is inconsistent; if all three ratios are equal, it has infinitely many solutions.
2.
While solving a pair by elimination, a student obtains 12=12.
What should the student conclude?
Answer: The equations have infinitely many solutions.
3.
A student obtains 7=2
after eliminating a variable.
What does this tell the student?
Answer: The equations have no solution and are inconsistent.
4.
A graph shows two lines meeting at (−3,4).
What is the solution of the pair?
Answer: (−3,4)
5.
A graph shows two identical lines lying exactly on each other.
What is the nature of the solution?
Answer: Infinitely many solutions; the pair is dependent and consistent.
K. Error-Spotting Questions
1.
A student says:
“Parallel lines have infinitely many solutions because they never meet.”
Identify the error.
Answer: Parallel lines have no common point, so the pair has no solution.
2.
A student obtains 0=0 and concludes that there is no solution.
Is the conclusion correct?
Answer: No. A true identity such as 0=0 indicates infinitely many solutions.
3.
A student obtains 0=4 and concludes that the equations have infinitely many solutions.
Is the conclusion correct?
Answer: No. 0=4 is false, so the pair has no solution.
4.
A student says:
“If two lines intersect, the equations are inconsistent.”
Correct the statement.
Answer: If two lines intersect at one point, the equations are consistent and have a unique solution.
L. Numerical Practice — Direct Answer Type
1.
Find the nature of the solution: 2x+4y=8 x+2y=4.
Answer: Infinitely many solutions.
2.
Find the nature of the solution: 3x+6y=9 x+2y=5.
Answer: No solution.
3.
Find the solution: x+y=7 x−y=1.
Answer: x=4, y=3
4.
Find the solution: 2x+y=9 x+y=6.
Answer: x=3, y=3
5.
Solve: 3x+y=11 x+y=7.
Answer: x=2, y=5
M. Method-Identification Questions
1.
A student writes x=5−2y and puts this expression into the second equation.
Which method is being used?
Answer: Substitution method.
2.
A student multiplies two equations so that the coefficients of y become equal and then subtracts them.
Which method is being used?
Answer: Elimination method.
3.
A student plots two equations and reads their common point.
Which method is being used?
Answer: Graphical method.
Quick Revision Answer Map
| Situation | Result |
|---|---|
| Lines intersect | Unique solution |
| Lines are parallel | No solution |
| Lines coincide | Infinitely many solutions |
| Pair has at least one solution | Consistent |
| Pair has no solution | Inconsistent |
| Coincident pair | Dependent and consistent |
| 0=0 after elimination | Infinitely many solutions |
| False statement after elimination | No solution |
| Express one variable in terms of another | Substitution |
| Remove one variable | Elimination |
| Common point of two graphs | Solution |
| Two-digit number with digits x,y | 10x+y |
| Reversed number | 10y+x |