CHAPTER 3 : PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
CBSE Class 10 Mathematics — Basic + Standard | PYQs 2026
BASIC QUESTION PAPER 2026
SECTION A : MULTIPLE CHOICE QUESTIONS (1 Mark Each)
1.
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]
2.
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 ≠ 8
(C) k ≠ 4
(D) k = 4
[Set 430/3/1]
3.
The system of linear equations given by
x = a and
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]
4.
The number of solutions of the system of equations
x = 3, y = −1 is:
(A) 0
(B) 1
(C) 2
(D) Infinite
[Set 430/2/2]
5.
The number of solutions of the system of equations
x = a and
x = b, where
a ≠ b, is:
(A) 0
(B) 1
(C) 2
(D) Infinite
[Set 430/2/3]
6.
For what value(s) of
k is the system of equations
kx + 2y = 3 and
2x + y = 5
inconsistent?
(A) k = Any real number
(B) k ≠ 2
(C) k ≠ 4
(D) k = 4
[Set 430/3/2]
SECTION B : ASSERTION–REASON QUESTIONS (1 Mark Each)
7.
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
a₁x + b₁y = c₁ and
a₂x + b₂y = c₂
is consistent with infinitely many solutions if
a₁/a₂ = b₁/b₂ = c₁/c₂
.
[Sets: 430/1/1, 430/1/2, 430/1/3]
SECTION C : VERY SHORT ANSWER QUESTIONS (2 Marks Each)
Type A : Algebraic Solution
8.
Solve the system for
x and
y:
x/2 + 2y/3 = −1
x − y/3 = 3
[Set 430/1/1]
9.
Solve for
x and
y:
0.1x + 0.3y = 1
0.2x − 0.1y = −0.1
[Set 430/1/2]
10.
Solve for
x and
y:
3x + 2y = 65
2x + 3y = 60
[Sets: 430/2/1, 430/2/2, 430/2/3]
Type B : Infinite Solutions Condition
11.
Find the value of
c for which the following pair of
linear equations has infinitely many solutions:
cx + 3y = c − 3
12x + cy = c
[Sets: 430/2/1, 430/2/2, 430/2/3]
Type C : Graphical Solutions
12.
Solve graphically:
2x − 3y = −6
x = 3
[Set 430/3/1]
13.
Solve graphically:
x + 2y = 10
y = 3
[Set 430/3/2]
14.
Solve graphically:
x + y = 5
x − y = 3
[Set 430/3/3]
SECTION D : SHORT ANSWER QUESTIONS (3 Marks Each)
Type A : Graphical Solutions with Intercepts
15.
Solve the following system of equations graphically:
x + 3y = 6
2x − 3y = 12
[Sets: 430/1/1, 430/1/2, 430/1/3]
16.
Solve graphically:
2x − y = 2
4x − y = 4
Also, write the coordinates of the points where the lines represented by
these equations cut the y-axis.
[Sets: 430/2/1, 430/2/2, 430/2/3]
Type B : Word Problems
17.
Angles:
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.
[Sets: 430/1/1, 430/1/2, 430/1/3]
18.
Costing:
An academy bought 10 bats and 5 balls for ₹32,500.
Later, the academy bought 2 bats and 8 balls for ₹10,000.
Find the cost of 1 bat and 1 ball.
[Sets: 430/2/2, 430/2/3]
19.
Fractions:
A fraction becomes
1/3 when 1 is subtracted
from the numerator and it becomes
1/4 when
8 is added to its denominator. Find the fraction.
[Sets: 430/3/1, 430/3/2, 430/3/3]
Type C : Parameters for Infinite Solutions
20.
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.
[Sets: 430/3/1, 430/3/2, 430/3/3]
STANDARD QUESTION PAPER 2026
SECTION A : MULTIPLE CHOICE QUESTIONS (1 Mark Each)
Type A : Conditions for Solutions
1.
If a pair of linear equations in two variables is represented by two
coincident lines, then the pair of equations has:
(A) A unique solution
(B) Two solutions
(C) No solution
(D) An infinite number of solutions
[Set 30/1/1 Q5, Set 30/1/2 Q6, Set 30/1/3 Q14]
2.
If the pair of linear equations
a₁x + b₁y + c₁ = 0 and
a₂x + b₂y + c₂ = 0
is consistent and dependent, then:
(A)
a₁/a₂ ≠ b₁/b₂
(B)
a₁/a₂ ≠ b₁/b₂ = c₁/c₂
(C)
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
(D)
a₁/a₂ = b₁/b₂ = c₁/c₂
[Set 30/2/1 Q6, Set 30/2/3 Q12]
3.
The pair of linear equations
3x/2 + 5y/3 = 7
and
9x + 10y = 14
is:
(A) Consistent
(B) Inconsistent
(C) Consistent with one solution
(D) Consistent with many solutions
[Set 30/2/2 Q3]
Type B : Finding the Value of k
4.
The value of
k for which the system of linear
equations
(3/2)x + (5/3)y = 5
and
2x + ky = 7
is inconsistent, is:
(A) 4/3
(B) 3/4
(C) 3/1
(D) 3
[Set 30/4/1 Q8, Set 30/4/2 Q16, Set 30/4/3 Q11]
5.
The value of
k for which the system of linear
equations
kx − y − 2 = 0
and
6x − 2y − 3 = 0
has infinitely many solutions, is:
(A) 1/2
(B) 3
(C) 4
(D) Not exist
[Set 30/4/3 Q3]
Type C : Parallel and Coincident Lines
6.
Equation of another line parallel to the line represented by
2x − 6y = 7
is:
(A) y = 3x − 7
(B) 2x = 9 − 6y
(C) x − 3y = 7
(D) x = 7/2 − 3y
[Set 30/3/2 Q16]
7.
The equation of a line coincident with the line
2.5x − 2y = 3
is:
(A) 5x − 4y = 3
(B) 5x − 4y + 6 = 0
(C) 15x − 12y − 3 = 0
(D) 5x − 4y − 6 = 0
[Set 30/3/3 Q14]
SECTION B : ASSERTION–REASON QUESTIONS (1 Mark Each)
8.
Assertion (A):
The system of linear equations
3x − 5y + 7 = 0 and
−6x + 10y + 14 = 0
is inconsistent.
Reason (R):
When two linear equations don’t have unique solution,
they always represent parallel lines.
[Set 30/5/1 Q19, Set 30/5/2 Q19, Set 30/5/3 Q20]
SECTION C : SHORT ANSWER QUESTIONS (3 Marks Each)
Type A : Algebraic Word Problem
9.
Class Test Marks:
In a class test, Veer scored 6 more than twice as many marks as Kevin scored.
If one of them had scored 4 more marks, their total score would have been 40.
Find the marks obtained by Veer and Kevin.
[Set 30/4/1 Q26(a), Set 30/4/2 Q28(a), Set 30/4/3 Q29(a)]
Type B : Graphical Solutions
10.
Solve the linear equations graphically:
3x + y = 14
y = 2
[Set 30/4/1 Q26(b), Set 30/4/2 Q28(b), Set 30/4/3 Q29(b)]
11.
Solve graphically:
x = 4
3x − 2y = 6
[Set 30/5/1 Q30]
12.
Solve graphically:
y = −3
x + 2y = 4
[Set 30/5/2 Q28]
13.
Solve graphically:
x = −3
5x − 2y = −5
[Set 30/5/3 Q27]
SECTION D : LONG ANSWER QUESTIONS (5 Marks Each)
Type A : Graphical Representation and Consistency
14.
Represent the following pair of linear equations graphically and hence
comment on the condition of consistency of this pair:
x − 5y = 6
2x − 10y = 12
[Set 30/2/3 Q34]
15.
Solve graphically:
x − 2y = 3
3x − 8y = 7
[Set 30/3/1 Q34(a)]
Type B : Complex Word Problems
16.
Aarush and Tanish — Stationery:
Aarush bought 2 pencils and 3 chocolates for ₹11 and Tanish bought
1 pencil and 2 chocolates for ₹7 from the same shop.
Represent this situation in the form of a pair of linear equations.
Find the price of 1 pencil and 1 chocolate, graphically.
[Set 30/2/1 Q32, Set 30/2/2 Q32]
17.
Age Problem:
Five years ago, Adil was thrice as old as Bharat.
Ten years later Adil shall be twice as old as Bharat.
(i) Form the linear equations.
(ii) Show that the system is consistent with a unique solution.
(iii) Find the present ages of Adil and Bharat.
[Set 30/3/1 Q34(b)]
18.
Boat and Stream:
Venkat can row a boat in still water at the speed of 12 km/h.
He ferries tourists 15 km upstream and 18 km downstream in 3 hours.
Find the speed of the stream.
Note: This problem results in a quadratic equation but is typically
introduced in the linear equations chapter through reducible forms.
[Set 30/5/1 Q32, Set 30/5/2 Q32]
SECTION E : CASE STUDY QUESTION (4 Marks)
19.
School Contribution — Graphical Method:
Two sections, A and B, of class X contributed a total of ₹1500
for the Uttarakhand flood victims. The contribution from X-A was
₹100 less than that of X-B.
Graphically, find the amounts contributed by both sections.
[Set 30/1/2 Q35]