DELHI PUBLIC SCHOOL, MIHAN NAGPUR
Class X (2025-26)
Subject: Mathematics Worksheet Topic: - Pair of L.E in two variables
Q.1) For what value of k, do the equations 3(𝑘 − 1)𝑥 + 4𝑦 = 24 and 15𝑥 + 20𝑦 = 8(𝑘 + 13) have
infinite solutions?
a) ½ b) 6 c) ⅓ d) 2
Q.2) What is the nature of the graphs of a dependent system?
a) Parallel lines b) Perpendicular lines
c) Intersecting lines d) Coincident lines
Q.3) What is the number of solutions of the pair of linear equations 4𝑝 − 6𝑞 + 18 = 0 and
2𝑝 − 3𝑞 + 9 = 0?
a) 0 b) 1 c) 2 d) Infinitely many
( )
Q.4) If (p, p) is the solution of equations 𝑎𝑥 + 𝑏𝑦 + 𝑡 − 𝑠 = 0 and 𝑏𝑥 +𝑎𝑦 + (𝑠 − 𝑟) = (0 ≠ 𝑏)
then, which of the following is true?
a) 2𝑟 = 𝑠 + 𝑡 b) 2𝑡 = 𝑟 + 𝑠 c) 2𝑠 = 𝑟 + 𝑡 d) 𝑟 + 𝑠 + 𝑡 = 0
Q.5) If the pair of linear equations 5𝑥 + 𝑘𝑦 − 3 = 0 and 8𝑥 + 25𝑦 + 𝑘 = 0 has a unique solution,
which of the following is true?
a) k=83 b) k is not an integer c) k=67 d) k≠14
Q.6) If an ordered pair satisfying the equations 2𝑥 − 3𝑦 = 18 and 4𝑥 − 𝑦 = 16.also satisfies the
equations 5𝑥 − 𝑝𝑦−23 = 0, then find the value of p.
a) 2 b) 5 c) 4 d) 1
Q.7) If a pair of linear equations is consistent, then the lines will be
a) parallel b) always coincident
c) intersecting or coincident d) always intersecting
Q.8) The pair of equations y = 0 and y = –7 has
a) one solution b) two solutions
c) infinitely many solutions d) no solution
Q.9) The value of 'K' for which the system of equations x+2y−3=0 and 6x+Ky+7=0 has no solution, is
a) 12 b) 15 c) 8 d) 13
Q.10) The sum of the successors of two numbers is 40 and the difference of their predecessors is 8.
Find the numbers.
a) 23, 13 b) 23, 15 c) 40, 8 d) 21, 15
Q.11) For which value(s) of k will the pair of equations 𝑘𝑥 + 3𝑦 = 𝑘 – 3 and 12𝑥 + 𝑘𝑦 = 𝑘 have
no solution?
Q.12) Find p, – 3x + 5y = 7 and 2px – 3y = 1, if the lines represented by these equations are
intersecting at a unique point.
Q.13) 2x + 3y = 7 and 2px + py = 28 – qy, if the pair of equations have infinitely many solutions.
Q.14) The students of a class are made to stand in rows. If 3 students are extra in a row, there would
be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of
students in the class.
Q.15) Graphically, solve the following pair of equations: 2x + y = 6 ; 2x – y + 2 = 0
Find the ratio of the areas of the two triangles formed by the lines representing these equations
with the x-axis and the lines with the y-axis.
Q.16) Solve for x and y i) 𝑚𝑥 − 𝑛𝑦 = 𝑚2 + 𝑛2 ; 𝑥 + 𝑦 = 2𝑚
ii) 47x+ 31 y = 63 ; 31x+47 y=15
iii) 99𝑥 + 101𝑦 = 499 ; 101𝑥 + 99𝑦 = 501
iv) 3𝑥 + 2𝑦 = 2𝑦 + 𝑦 + 3 = 4𝑥 + 3𝑦 − 3
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Q.17) i) If 279𝑥 − 631𝑦 = 910 ∶ 631𝑥 − 279𝑦 = 910 then find 𝑥 + 𝑦.
ii) If 2x + y = 23 and 4x – y = 19, find the values of 5y – 2x and y/x – 2.
Q.18) Find whether the following equation has no solution, unique solution or infinitely many
solutions.
24 3
5𝑥 − 8𝑦 + 1 = 0 ; 3𝑥 − 𝑦 + = 0.
5 5
Q.19) A and B are two points 150 km apart on a highway. Two cars start from A and B at the same
time. If they move in the same direction, they meet in 15 hours. But if they move in the opposite
direction, they meet in 1 hour. Find their speeds.
Q.20) It is common that Governments revise travel fares from time to time based on various factors
such as inflation (a general increase in prices and fall in the purchasing value of money) on
different types of vehicles like auto, Rickshaws, taxis, Radio cab etc. The auto charges in a city
comprise a fixed charge together with the charge for the distance covered. Study the following
situations:
Name of the city Distance travelled (Km) Amount paid (Rs.)
City A 10 75
15 110
City B 8 91
14 145
Situation 1: In city A, for a journey
of 10 km, the charge paid is Rs 75 and for journey of 15 km, the charge paid is Rs 110.
Situation 2: In a city B, for a journey of 8km, the charge paid is Rs91 and for a journey of 14 km, the
charge paid is Rs 145.
a) Refer situation 1: If a man travels 50km then find out the amount he has to pay.
b) Refer situation 2: What will man have to pay for traveling 30 km?
Q.21) On selling a tea set at 5%loss and a lemon set at 15% gain, a shopkeeper gains ₹70. If he sells the
tea set at 5%gain and the lemon set at 10% gain, he gains ₹130. Find the cost price of lemon set
and tea set.
Q.22) The income of X and Y are in the ratio of 8: 7 and their expenditures are in the ratio 19: 16.
If each saves ₹ 1250, find their incomes.
Q.23) Find the four angles of a cyclic quadrilateral ABCD in which ∠A = (2x - 3) °, ∠B = (y + 7) °,
∠C = (2y + 17) ° and ∠D = (4x - 9) °.
Q.24) The angles of a triangle are x, y and 400 . The difference between the two angles x and y is 300 .
Find x and y.
Q.25) The present age of a father is three years more than three times the age of the son. Three years
hence, father's age will be 10 years more than twice the age of the son. Determine their present
ages.
Q.26) A book store shopkeeper gives books on rent for reading. He has variety of books in his store
related to fiction, stories and quizzes etc. He takes a fixed charge for the first two days and an
additional charge for subsequent day. Amruta paid Rs 22 for a book and kept for 6 days; while
Radhika paid Rs 16 for keeping the book for 4 days.
a) Write the algebraic representation for the situation
of amount paid by Radhika and Amruta.
b) Find the fixed charge for a book and the additional
charge for each subsequent day for a book.
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