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Mastersheet: Pair of Linear Equations in Two Variables (Ch. 3)
Student Name: Class: 10 CBSE Subject: Mathematics
✅ Rationalised Syllabus Note: This mastersheet strictly follows the CBSE Rationalised Syllabus (2024–25 onwards). The following topics have been deleted and are NOT included: Cross-Multiplication Method • Equations Reducible to a Pair of Linear Equations. Focus areas: Consistency/Inconsistency • Graphical Method • Substitution Method • Elimination Method • Word Problems.
Topic 1: MCQ / 1-Mark Questions
1.
The pair $2x + 3y = 5$ and $4x + 6y = 10$ has: [1M]
  1. A unique solution
  2. No solution
  3. Infinitely many solutions
  4. Two solutions
2.
For what value of $k$ does $3x + ky = 9$ and $6x + 4y = 18$ have infinitely many solutions? [1M]
  1. $k = 2$
  2. $k = 4$
  3. $k = 6$
  4. $k = -2$
3.
The pair of equations $x = 0$ and $x = 5$ has: [1M]
  1. No solution
  2. A unique solution
  3. Two solutions
  4. Infinitely many solutions
4.
If $\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$, the pair of linear equations has: [1M]
  1. No solution
  2. Infinitely many solutions
  3. A unique solution
  4. Cannot be determined
5.
The solution of $2x + y = 6$ and $x - y = 3$ is: [1M]
  1. $(3, 0)$
  2. $(0, 6)$
  3. $(2, 2)$
  4. $(1, 4)$
6.
The value of $k$ for which $kx - y = 2$ and $6x - 2y = 3$ has no solution is: [1M]
  1. $k = 2$
  2. $k = 3$
  3. $k = -3$
  4. $k = -2$
7.
Graphically, the pair of equations $6x - 3y + 10 = 0$ and $2x - y + 9 = 0$ represents two lines which are: [1M]
  1. Intersecting at exactly one point
  2. Coincident
  3. Parallel
  4. Intersecting at two points
8.
Asha has only ₹1 and ₹2 coins with her. If the total number of coins she has is 50 and the total amount of money with her is ₹75, then the number of ₹1 and ₹2 coins are, respectively: [1M]
  1. 25 and 25
  2. 15 and 35
  3. 35 and 15
  4. 35 and 25
Topic 2: Consistency, Inconsistency & Value of k
9.
On comparing the ratios $\dfrac{a_1}{a_2}$, $\dfrac{b_1}{b_2}$, $\dfrac{c_1}{c_2}$, find whether the pair $5x - 3y = 11$ and $-10x + 6y = -22$ is consistent or inconsistent. [2M]
10.
On comparing the ratios, find whether the following pairs of equations are consistent or inconsistent: [2M]
  1. $3x + 2y = 5$; $2x - 3y = 7$
  2. $2x - 3y = 8$; $4x - 6y = 9$
  3. $\dfrac{3}{2}x + \dfrac{5}{3}y = 7$; $9x - 10y = 14$
11.
Find the value of $k$ for which the pair $2x + 3y = 5$ and $kx - 6y = 8$ has a unique solution. [2M]
12.
Find the value of $k$ for which the pair $kx + 2y = 5$ and $3x + y = 1$ has: (i) a unique solution, (ii) no solution. [2M]
13.
Find the value of $k$ for which $2x + 3y = 7$ and $(k-1)x + (k+2)y = 3k$ has infinitely many solutions. [3M]
14.
Find the values of $a$ and $b$ for which the system $2x + 3y = 7$ and $(a-b)x + (a+b)y = 3a + b - 2$ has infinitely many solutions. [3M]
15.
For what value of $p$ does the pair $3x + py = 10$ and $6x + 2y = 20$ have: (i) no solution, (ii) infinitely many solutions, (iii) a unique solution? [3M]
Topic 3: Substitution Method
16.
Solve by substitution: $x + y = 14$ and $x - y = 4$. [2M]
17.
Solve by substitution: $s - t = 3$ and $\dfrac{s}{3} + \dfrac{t}{2} = 6$. [2M]
18.
Solve by substitution: $0.2x + 0.3y = 1.3$ and $0.4x + 0.5y = 2.3$. [2M]
19.
Solve by substitution: $\dfrac{3x}{2} - \dfrac{5y}{3} = -2$ and $\dfrac{x}{3} + \dfrac{y}{2} = \dfrac{13}{6}$. [3M]
20.
Solve by substitution: $\sqrt{2}\,x + \sqrt{3}\,y = 0$ and $\sqrt{3}\,x - \sqrt{8}\,y = 0$. [2M]
21.
Solve by substitution: $ax + by = a - b$ and $bx - ay = a + b$. [3M]
Topic 4: Elimination Method
22.
Solve by elimination: $3x + 4y = 10$ and $2x - 2y = 2$. [2M]
23.
Solve by elimination: $x + y = 5$ and $2x - 3y = 4$. [2M]
24.
Solve by elimination: $3x - 5y - 4 = 0$ and $9x = 2y + 7$. [2M]
25.
Solve by elimination: $\dfrac{x}{2} + \dfrac{2y}{3} = -1$ and $x - \dfrac{y}{3} = 3$. [2M]
26.
Solve by elimination: $3x - 4y = 1$ and $-4x + 5y = 2$. [2M]
27.
Solve the equations: $99x + 101y = 499$ and $101x + 99y = 501$. [3M]
28.
Solve: $\dfrac{x+y}{xy} = 2$ and $\dfrac{x-y}{xy} = 6$, where $x, y \neq 0$. (Hint: divide numerators.) [3M]
Topic 5: Graphical Method – Solving & Area
29.
Solve graphically: $x - y = 1$ and $2x + y = 8$. Also find the area of the triangle formed by the two lines and the $y$-axis. [4M]
30.
Solve graphically: $2x + y = 6$ and $2x - y + 2 = 0$. Find the area of the triangle formed by these lines and the $x$-axis. [4M]
31.
Solve graphically: $x - y + 1 = 0$ and $3x + 2y - 12 = 0$. Also find the vertices and area of the triangle formed by the lines and the $x$-axis. [4M]
32.
The path of two cars on a road is represented by $y = 3x + 4$ and $6x - 2y + 8 = 0$. Will the cars ever meet? Justify your answer. [2M]
33.
Determine graphically whether the system $x - 2y = 2$ and $4x - 2y = 5$ is consistent. If consistent, find the solution. [3M]
Topic 6: Word Problems – Numbers, Fractions & Digits
34.
The sum of a two-digit number and the number obtained by reversing its digits is 66. If the digits differ by 2, find the number. How many such numbers are there? [3M]
35.
A two-digit number is 3 more than 4 times the sum of its digits. If 18 is added to the number, its digits are reversed. Find the number. [3M]
36.
The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes $\dfrac{1}{2}$. Find the fraction. [3M]
37.
A fraction becomes $\dfrac{1}{3}$ when 1 is subtracted from the numerator and it becomes $\dfrac{1}{4}$ when 8 is added to its denominator. Find the fraction. [3M]
38.
The sum of two numbers is 1000 and 500 is less than their difference. Find the two numbers. [2M]
39.
If $2x + y = 35$ and $3x + 4y = 65$, find the value of $\dfrac{x}{y}$. [2M]
Topic 7: Word Problems – Ages
40.
Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu? [3M]
41.
The ages of two friends Ani and Biju differ by 3 years. Ani's father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju. [3M]
42.
Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob's age was seven times that of his son. What are their present ages? [3M]
43.
A person starts his job with a certain monthly salary and earns a fixed increment every year. If his salary after 4 years is ₹15,000 and his salary after 10 years is ₹18,000, find his starting salary and the annual increment. [3M]
Topic 8: Word Problems – Speed, Distance & Stream
44.
Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If they travel in the same direction, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. Find the speeds of the two cars. [3M]
45.
Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current. [3M]
46.
A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water. [4M]
47.
A train covered a certain distance at a uniform speed. If the speed had been 10 km/h more, it would have taken 2 hours less. If the speed were 10 km/h less, it would have taken 3 hours more. Find the distance covered. [4M]
Topic 9: Word Problems – Cost, Work & Geometry
48.
The cost of 5 oranges and 3 apples is ₹35 and the cost of 2 oranges and 4 apples is ₹28. Find the cost of each. [3M]
49.
2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone and 1 man alone. [4M]
50.
Susan invested money in two schemes A and B at 8% and 9% p.a. She received ₹1860 as annual interest. Had she interchanged the amounts, she would have received ₹20 more. Find the amounts invested. [3M]
51.
The area of a rectangle reduces by 80 sq. units if its length is reduced by 5 units and breadth increased by 2 units. If the length is increased by 10 units and breadth decreased by 5 units, the area increases by 50 sq. units. Find the dimensions. [4M]
52.
Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden. [2M]
53.
The larger of two supplementary angles exceeds the smaller by 18°. Find both angles. [2M]
54.
The angles of a triangle are $x$, $y$ and $40°$. The difference between the two unknown angles $x$ and $y$ is $30°$. Find $x$ and $y$. [2M]
55.
The population of a village is 5000. If in a year, the number of males increases by 5% and that of females by 3%, the population grows to 5202. Find the number of males and females. [3M]
Topic 10: Assertion-Reason & Case Study Questions
56.
Assertion (A): The pair of equations $x + 2y = 5$ and $3x + 6y = 15$ has infinitely many solutions.
Reason (R): If $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$, the pair has infinitely many solutions. [1M]
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is NOT the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
57.
Assertion (A): The pair $2x + 3y = 7$ and $4x + 6y = 11$ is inconsistent.
Reason (R): If $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$, the pair has no solution. [1M]
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is NOT the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
58.
Assertion (A): The solution of $x + y = 10$ and $x - y = 4$ is $x = 7, y = 3$.
Reason (R): The point of intersection of two lines gives the unique solution of the pair. [1M]
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is NOT the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
59.
Case Study: The School Canteen
The school canteen charges ₹15 for each plate of noodles and ₹10 for each plate of rice. One day, total orders were 100 plates and the total revenue was ₹1200. [4M]
  1. Form a pair of linear equations representing the above situation.
  2. Find the number of plates of noodles ordered.
  3. Find the number of plates of rice ordered.
  4. If next day the revenue was ₹1400 with 100 plates sold, find the new order breakup.
60.
Case Study: The Journey
A journey of 600 km is partly by train (speed $x$ km/h) and partly by bus (speed $y$ km/h). The journey takes 8 hours in all. If the train portion is 120 km and the bus covers the rest, the journey takes 20 minutes longer. [4M]
  1. Form a pair of linear equations in $x$ and $y$.
  2. Find the speed of the train.
  3. Find the speed of the bus.
  4. Which method (substitution or elimination) did you use and why?
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