Pair of Linear Equations in Two Variables Class 10 Important Questions | NCERT Maths Chapter-3

Pair of Linear Equations in Two Variables Class 10 Important Questions

The start of class 10 marks the beginning of the foundation for class 11 and class 12. It is very important study the basics in order to understand each and every chapter properly. In this page, we have provided all the important question for cbse class 9 that could be asked in the examination. Students also need to study the ncert solutions for class 10 in order to gain more knowledge and understanding the lessons. Questions and Answers are way to learn the new things in a proper way. NCERT textbooks downloads for class 9 in pdf are also available for the students if they need more help. By downloading these books, they can study from it. Our experts also prepared revision notes for class 9 so that students should see the details of each and every chapters. Class 9 important questions are the best to revise all the chapters in the best way.

Question 1.
The age of the father is twice the sum of the ages of his 2 children. After 20 years, his age will be equal to the sum of the ages of his children. Find the age of the father. (2012)
Solution:
Let the present ages of his children be x years and y years.
Then the present age of the father = 2(x + y) …(i)
After 20 years, his children’s ages will be
(x + 20) and (y + 20) years
After 20 years, father’s age will be 2(x + y) + 20
According to the Question,
⇒ 2(x + y) + 20 = x + 20 + y + 20
⇒ 2x + 2y + 20 = x + y + 40
⇒ 2x + 2y – x – y = 40 – 20
⇒ x + y = 20 …[From (i)
∴ Present age of father = 2(20) = 40 years

Question 2.
A two digit number is seven times the sum of its digits. The number formed by reversing the digits is 18 less than the given number. Find the given number. (2013)
Solution:
Let unit’s place digit be x and ten’s place digit bey.
Then original number = x + 10y
and reversed number = 10x + y
According to the Question,
x + 10y = 7(x + y)
x + 10y = 7x + 7y
⇒ 10y – 7y = 7x – x
⇒ 3y = 6x ⇒ y = 2x …(i)
(x + 10y) – (10x + y) = 18
x + 10y – 10x – y = 18
⇒ 9y – 9x = 180
⇒ y – x = 2 …[Dividing by 9
⇒ 2x – x = 2 …[From (i)
∴ x = 2
Putting the value of ‘x’ in (i), we get y = 2(2) = 4
∴ Required number = x + 10y
= 2 + 10(4) = 42

Question 3.
Sita Devi wants to make a rectangular pond on the road side for the purpose of providing drinking water for street animals. The area of the pond will be decreased by 3 square feet if its length is decreased by 2 ft. and breadth is increased by 1 ft. Its area will be increased by 4 square feet if the length is increased by 1 ft. and breadth remains same. Find the dimensions of the pond. (2014)
Solution:
Let length of rectangular pond = x
and breadth of rectangular pond = y
Area of rectangular pond = xy
According to Question,

∴Length of rectangular pond = 7 ft.
Breadth of rectangular pond = 4 ft.

Question 4.
On reversing the digits of a two digit number, number obtained is 9 less than three times the original number. If difference of these two numbers is 45, find the original number. (2014)
Solution:
Let unit’s place digit be x and ten’s place digit bey.
∴ Original number = x + 10y Reversed number = 10x + y
According to the Question,
10x + y = 3(x + 10y) – 9
⇒ 10x + y = 3x + 30y – 9
⇒ 10x + y – 3x – 30y = -9
⇒ 7x – 29y = -9 …(i)
10x + y – (x + 10y) = 45
⇒ 9x – 9y = 45
⇒ x – y = 5 …[Dividing both sides by 9
⇒ x – 5 + y …(ii)
Solving (i),
7x – 29y = -9
7(5 + y) – 29y = -9 …[From (ii)
35+ 7y – 29y = -9
-22y = -9 – 35
-22y = -44 ⇒ y = 4422 = 2
Putting the value of y in (ii),
x = 5 + 2 = 7
∴ Original number = x + 10y
= 7 + 10(2) = 27

Question 5.
Speed of a boat in still water is 15 km/h. It goes 30 km upstream and returns back at the same point in 4 hours 30 minutes. Find the speed of the stream. 2017D
Solution:
Let the speed of the stream = x km/hr
Speed of the boat in still water = 15 km/hr
then, the speed of the boat upstream = (15 – x) km/hr
and the speed of the boat downstream = (15 + x) km/hr

∴ Speed of stream = 5 km/hr

Question 6.
The owner of a taxi company decides to run all the taxis on CNG fuel instead of petrol/diesel. The taxi charges in city comprises of fixed charges together with the charge for the distance covered. For a journey of 12 km, the charge paid is 789 and for journey of 20 km, the charge paid is ₹145.
What will a person have to pay for travelling a distance of 30 km? (2014)
Solution:
Let the fixed charges = 7x
and the charge per km = ₹y
According to the Question,

Putting the value of y in (i), we get
x + 12(7) = 89
x + 84 = 89 ⇒ x = 89 – 84 = 5
Total fare for 30 km = x + 30y = 5 + 30(7)
= 5 + 210 = ₹215

Question 7.
A boat takes 4 hours to go 44 km downstream and it can go 20 km upstream in the same time. Find the speed of the stream and that of the boat in still water. (2015)
Solution:
Let the speed of the stream = y km/hr
Let the speed of boat in still water = x km/hr
then, the speed of the boat in downstream = (x + y) km/hr
and, the speed of the boat in upstream = (x – y) km/hr

From (i), x = 11 – 3 = 8
∴ Speed of the stream, y =3 km/hr
Speed of the boat in still water, x = 8 km/hr

Question 8.
A man travels 300 km partly by train and partly by car. He takes 4 hours if the travels 60 km by train and the rest by car. If he travels 100 km by train and the remaining by car, he takes 10 minutes longer. Find the speeds of the train and the car separately. (2017D)
Solution:
Let the speed of the train = x km/hr
Let the speed of the car = y km/ hr
According to the Question,

∴ Speed of the train = 60 km/hr
and Speed of the car = 80 kn/hr

Question 9.
The owner of a taxi company decides to run all the taxis on CNG fuel instead of petrol/diesel. The taxi charges in city comprises of fixed charges together with the charge for the distance covered. For a journey of 13 km, the charge paid is ₹129 and for a journey of 22 km, the charge paid is ₹210.
What will a person have to pay for travelling a distance of 32 km? (2014 )
Solution:
Let fixed charge be ₹x and the charge for the distance = ₹y per km
According to the Question,
For a journey of 13 km,
x + 13y = 129 ⇒ x = 129 – 13y …(/)
For a journey of 22 km, x + 22y = 210 …(ii)
⇒ 129 – 13y + 22y = 210 …[From (i)
⇒ 9y = 210 – 129 = 81
⇒ 9y = 81 ⇒ y = 9
From (i), x = 129 – 13(9)
= 129 – 117 = 12
∴ Fixed charge, x = ₹12
∴ The charge for the distance, y = ₹9 per km
To pay for travelling a distance of 32 km
= x + 32y = 12 + 32(9) = 12 + 288 = ₹300

Question 10.
Solve the following pair of linear equations graphically:
x + 3y = 6 ; 2x – 3y = 12
Also find the area of the triangle formed by the lines representing the given equations with y-axis. (2012, 2015)
Solution:

By plotting points and joining them, the lines intersesct at A(6, 0)
∴ x = 6, y = 0
Line x + 3y = 6 intersects Y-axis at B(0, 2) and Line 2x – 3y = 12 intersects Y-axis at C(0, -4). Therefore, Area of triangle formed by the lines with y-axis.
Area of triangle
= 12 × base × corresponding altitude
= 12 × BC × AO = 12 × 6 × 6 = 18 sq. units

Question 11.
Draw the graphs of following equations:
2x – y = 1; x + 2y = 13
Find the solution of the equations from the graph and shade the triangular region formed by the lines and the y-axis. (2013)
Solution:

By plotting the points and joining them, the lines intersect at A(3,5).
∴ x = 3, y = 5
Here ∆ABC is the required triangle.

Question 12.
Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and x-axis. (2012, 2017D)
Solution:

Lines intersect at (2, 3)
∴ x = 2, y = 3
Vertices of ∆ABC are A(2, 3), B(-1, 0) and C(4, 0)

Question 13.
Amit bought two pencils and three chocolates for ₹11 and Sumeet bought one pencil and two chocolates for ₹7. Represent this situation in the form of a pair of linear equations. Find the price of one pencil and that of one chocolate graphically. (2017OD)
Solution:
Let the price of one pencil = ₹x and the price of one chocolate = ₹y.
As per the Question,

Lines intersect at (1, 3).
∴ x = 1, y = 3
Therefore the price of one pencil = ₹1 and price of one chocolate = ₹3

Question 14.
7x – 5y – 4 = 0 is given. Write another linear equation, so that the lines represented by the pair are:
(i) intersecting
(ii) coincident
(iii) parallel (2015 OD)
Solution:
7x – 5y – 4 = 0

Important Links

NCERT Quick Revision Notes- Pair of Linear Equations in Two Variables

NCERT Solution-Pair of Linear Equations in Two Variables

Important MCQs- Pair of Linear Equations in Two Variables

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