THE GOAL • CBSE CLASS 10

PAIR OF LINEAR EQUATIONS
IN TWO VARIABLES

Chapter 3 • Basic Mathematics • 2025 PYQs

CBSE 2025 Basic Mathematics Previous Year Questions Chapter-wise Collection

Chapter 3: Pair of Linear Equations in Two Variables

Unique Previous Year Questions extracted from the CBSE Class 10 Basic Mathematics 2025 question paper sets, with major conceptual variations retained.

I

Multiple Choice Questions

1. Solution Identification
1 Mark
The number of solutions of the system of equations \[ x=3,\qquad y=-1 \] is:
(A) 0
(B) 1
(C) 2
(D) Infinite
Set 430/2/2
2. Parallel Lines – Number of Solutions
1 Mark
The number of solutions of the system of equations \[ x=a,\qquad x=b \] where \(a\neq b\), is:
(A) 0
(B) 1
(C) 2
(D) Infinite
Set 430/2/3
3. Finding the Constant
1 Mark
If \((0,0)\) is the solution of the equation \[ x+y=c-1, \] then the value of \(c\) is:
(A) 0
(B) 1
(C) \(-1\)
(D) Any real number
Set 430/2/1
4. Unique Solution Condition
1 Mark
For what value(s) of \(k\) does the system of equations \[ kx+2y=3 \] and \[ 2x+y=5 \] have a unique solution?
(A) \(k=\) a real number
(B) \(k\neq8\)
(C) \(k\neq4\)
(D) \(k=4\)
Set 430/3/1
5. Parallel Lines Condition
1 Mark
The value of \(m\) for which the lines \[ 14x+my=20 \] and \[ -3x+2y=16 \] are parallel, is:
(A) \(\frac{3}{14}\)
(B) \(-\frac{7}{3}\)
(C) \(-\frac{28}{3}\)
(D) \(-\frac{3}{28}\)
Set 430/4/2
6. Consistency Nature
1 Mark
The system of linear equations given by \[ x=a,\qquad y=b \] is:
(A) Consistent with a unique solution
(B) Consistent with infinitely many solutions
(C) Consistent with two solutions
(D) Inconsistent
Set 430/3/3
II

Assertion – Reason Question

7. Consistency and Infinitely Many Solutions
1 Mark
Assertion (A): The value of \(p\) for which the system of equations \[ 4x+py+8=0 \] and \[ 2x+2y+2=0 \] is consistent is \(4\).
Reason (R): The system of equations \[ a_1x+b_1y=c_1 \] and \[ a_2x+b_2y=c_2 \] is consistent with infinitely many solutions if \[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}. \]
Set 430/1/1
III

Very Short Answer Type Questions

8. Algebraic Solution – Fractions
2 Marks
Solve the following system of equations for \(x\) and \(y\):
\[ \frac{x}{2}+\frac{2y}{3}=-1 \] \[ x-\frac{y}{3}=3 \]
Set 430/1/1
9. Algebraic Solution – Decimals
2 Marks
Solve for \(x\) and \(y\):
\[ 0.1x+0.3y=1 \] \[ 0.2x-0.1y=-0.1 \]
Set 430/1/2
10. Infinitely Many Solutions
2 Marks
Find the value of \(c\) for which the following pair of linear equations has infinitely many solutions:
\[ cx+3y=c-3 \] \[ 12x+cy=c \]
Set 430/2/1
11. Arithmetic Solution
2 Marks
Solve for \(x\) and \(y\):
\[ 3x+2y=65 \] \[ 2x+3y=60 \]
Set 430/2/1
12. Graphical Solution
2 Marks
Solve the following system of equations graphically:
\[ x+2y=10 \] \[ y=3 \]
Set 430/3/2
13. Graphical Solution
2 Marks
Solve the following system of linear equations graphically:
\[ x+y=5 \] \[ x-y=3 \]
Set 430/3/3
IV

Short Answer Type Questions

14. Graphical Solution – Intersecting Lines
3 Marks
Solve the following system of equations graphically:
\[ x+3y=6 \] \[ 2x-3y=12 \]
Set 430/1/2
15. Word Problem – Complementary Angles
3 Marks
\(x\) and \(y\) are complementary angles such that \[ x:y=1:2. \] Express the given information as a system of linear equations in two variables and hence solve it.
Set 430/1/2
16. Graphical Intercepts
3 Marks
Solve graphically the following pair of linear equations:
\[ 2x-y=2 \] \[ 4x-y=4 \]
Also, write the coordinates of the points where the lines represented by these equations cut the \(y\)-axis.
Set 430/2/1
17. Word Problem – Cricket Coaching
3 Marks
An academy provided 10 bats and 5 balls for ₹32,500. Later, it bought 2 bats and 8 balls for ₹10,000.
If the price remained the same, find the cost of 1 bat and 1 ball.
Set 430/2/2
18. Word Problem – Fraction
3 Marks
A fraction becomes \[ \frac{1}{3}, \] when 1 is subtracted from the numerator and it becomes \[ \frac{1}{4}, \] when 8 is added to its denominator.
Find the fraction.
Set 430/3/1
19. Unknown \(k\) and Solutions
3 Marks
Find the value of \(k\) for which the following pair of linear equations will have infinitely many solutions:
\[ kx+3y-(k-3)=0 \] \[ 12x+ky-k=0 \]
Hence, find any two solutions of the given pair of equations.
Set 430/3/1
V

Long Answer Type Questions

20. Graphical Analysis
5 Marks
Determine graphically whether the following pair of linear equations has a unique solution or infinitely many solutions:
\[ 2x+3y=12 \] \[ x-y=1 \]
Set 430/5/1
21. Word Problem – Number Digits
5 Marks
The sum of a 2-digit number and the number obtained by reversing the order of its digits is 121. The two digits differ by 3.
(i) Represent the information in the form of a pair of linear equations.
(ii) Show that the equations have a unique solution.
(iii) Solve the equations and find the number.
Set 430/5/2
22. Graphical Solution – General
5 Marks
Solve the following pair of equations using graphical method:
\[ 3x-4y+3=0 \] \[ -2x+5y=9 \]
Set 430/5/3
23. Word Problem – Investment and Interest
5 Marks
Nidhi received simple interest of ₹1,200 when she invested ₹\(x\) at 6% p.a. and ₹\(y\) at 5% p.a. for 1 year.

Had she invested ₹\(x\) at 3% p.a. and ₹\(y\) at 8% p.a. for that year, she would have received simple interest of ₹1,260.

Find the values of \(x\) and \(y\).
Set 430/6/2
24. Graphical Solution – Extended
5 Marks
Solve the following pair of linear equations by graphical method:
\[ 2x+y=9 \] \[ x-2y=2 \]
Set 430/6/3
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