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## Maths Chapter 4 Linear Equations in Two VariablesClass IX Chapter 4– Linear Equations in Two Variables Maths

Page 1 of 19
Exercise 4.1
Question 1:
The cost of a notebook is twice the cost of a pen. Write a linear equation in two
variables to represent this statement.
(Take the cost of a notebook to be Rs x and that of a pen to be Rs y.)
Let the cost of a notebook and a pen be x and y respectively.
Cost of notebook = 2 × Cost of pen
x = 2y
x – 2y = 0
Question 2:
Express the following linear equations in the form ax + by + c = 0 and indicate the
values of a, b, c in each case:
(i) (ii) (iii) – 2x + 3 y = 6
(iv) x = 3y (v) 2x = – 5y (vi) 3x + 2 = 0
(vii) y – 2 = 0 (viii) 5 = 2x
(i)
= 0
Comparing this equation with ax + by + c = 0,
a = 2, b = 3,
(ii)
Comparing this equation with ax + by + c = 0,
a = 1, b = , c = -10
(iii) – 2x + 3 y = 6

Class IX Chapter 4– Linear Equations in Two Variables Maths
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– 2x + 3 y – 6 = 0
Comparing this equation with ax + by + c = 0,
a = -2, b = 3, c = -6
(iv) x = 3y
1x – 3y + 0 = 0
Comparing this equation with ax + by + c = 0,
a = 1, b = -3, c = 0
(v) 2x = -5y
2x + 5y + 0 = 0
Comparing this equation with ax + by + c = 0,
a = 2, b = 5, c = 0
(vi) 3x + 2 = 0
3x + 0.y + 2 = 0
Comparing this equation with ax + by + c = 0,
a = 3, b = 0, c = 2
(vii) y – 2 = 0
0.x + 1.y – 2 = 0
Comparing this equation with ax + by + c = 0,
a = 0, b = 1, c = -2
(vii) 5 = 2x
– 2x + 0.y + 5 = 0
Comparing this equation with ax + by + c = 0,
a = -2, b = 0, c = 5

Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 3 of 19
Exercise 4.2
Question 1:
Which one of the following options is true, and why?
y = 3x + 5 has
(i) a unique solution, (ii) only two solutions, (iii) infinitely many solutions
y = 3x + 5 is a linear equation in two variables and it has infinite possible solutions.
As for every value of x, there will be a value of y satisfying the above equation and
vice-versa.
Hence, the correct answer is (iii).
Question 2:
Write four solutions for each of the following equations:
(i) 2x + y = 7 (ii) πx + y = 9 (iii) x = 4y
(i) 2x + y = 7
For x = 0,
2(0) + y = 7
y = 7
Therefore, (0, 7) is a solution of this equation.
For x = 1,
2(1) + y = 7
y = 5
Therefore, (1, 5) is a solution of this equation.
For x = -1,
2(-1) + y = 7
y = 9
Therefore, (-1, 9) is a solution of this equation.
For x = 2,
2(2) + y = 7
y = 3
Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 4 of 19
Therefore, (2, 3) is a solution of this equation.
(ii) πx + y = 9
For x = 0,
π(0) + y = 9
y = 9
Therefore, (0, 9) is a solution of this equation.
For x = 1,
π(1) + y = 9
y = 9 – π
Therefore, (1, 9 – π) is a solution of this equation.
For x = 2,
π(2) + y = 9
y = 9 – 2π
Therefore, (2, 9 -2π) is a solution of this equation.
For x = -1,
π(-1) + y = 9
y = 9 + π
(-1, 9 + π) is a solution of this equation.
(iii) x = 4y
For x = 0,
0 = 4y
y = 0
Therefore, (0, 0) is a solution of this equation.
For y = 1,
x = 4(1) = 4
Therefore, (4, 1) is a solution of this equation.
For y = -1,
x = 4(-1)
x = -4
Therefore, (-4, -1) is a solution of this equation.

Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 5 of 19
For x = 2,
2 = 4y

 Therefore, is a solution of this equation. Question 3: Check which of the following are solutions of the equation x – 2y = 4 and which are not: (i) (0, 2 (ii) (2, 0) (iii) (4, 0)

(iv) (v) (1, 1)
(i) (0, 2)
Putting x = 0 and y = 2 in the L.H.S of the given equation,
x – 2y = 0 – 2×2 = – 4 ≠ 4
L.H.S ≠ R.H.S
Therefore, (0, 2) is not a solution of this equation.
(ii) (2, 0)
Putting x = 2 and y = 0 in the L.H.S of the given equation,
x – 2y = 2 – 2 × 0 = 2 ≠ 4
L.H.S ≠ R.H.S
Therefore, (2, 0) is not a solution of this equation.
(iii) (4, 0)
Putting x = 4 and y = 0 in the L.H.S of the given equation,
x – 2y = 4 – 2(0)
= 4 = R.H.S
Therefore, (4, 0) is a solution of this equation.
(iv)

Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 6 of 19
Putting and in the L.H.S of the given equation,
L.H.S ≠ R.H.S
Therefore, is not a solution of this equation.
(v) (1, 1)
Putting x = 1 and y = 1 in the L.H.S of the given equation,
x – 2y = 1 – 2(1) = 1 – 2 = – 1 ≠ 4
L.H.S ≠ R.H.S
Therefore, (1, 1) is not a solution of this equation.
Question 4:
Find the value of k, if x = 2, y = 1 is a solution of the equation 2x + 3y = k.
Putting x = 2 and y = 1 in the given equation,
2x + 3y = k
2(2) + 3(1) = k
4 + 3 = k
k = 7
Therefore, the value of k is 7.

Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 7 of 19
Exercise 4.3
Question 1:
Draw the graph of each of the following linear equations in two variables:
(i) (ii) (iii) y = 3x (iv) 3 = 2x + y
(i)
It can be observed that x = 0, y = 4 and x = 4, y = 0 are solutions of the above
equation. Therefore, the solution table is as follows.

 x 0 4 y 4 0

The graph of this equation is constructed as follows.
(ii)
It can be observed that x = 4, y = 2 and x = 2, y = 0 are solutions of the above
equation. Therefore, the solution table is as follows.

 x 4 2 y 2 0

Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 8 of 19
The graph of the above equation is constructed as follows.
(iii) y = 3x
It can be observed that x = -1, y = -3 and x = 1, y = 3 are solutions of the above
equation. Therefore, the solution table is as follows.

 x – 1 1 y – 3 3

The graph of the above equation is constructed as follows.
(iv) 3 = 2x + y

Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 9 of 19
It can be observed that x = 0, y = 3 and x = 1, y = 1 are solutions of the above
equation. Therefore, the solution table is as follows.

 x 0 1 y 3 1

The graph of this equation is constructed as follows.
Question 2:
Give the equations of two lines passing through (2, 14). How many more such lines
are there, and why?
It can be observed that point (2, 14) satisfies the equation 7x – y = 0 and
x – y + 12 = 0.
Therefore, 7x – y = 0 and x – y + 12 = 0 are two lines passing through point (2,
14).
As it is known that through one point, infinite number of lines can pass through,
therefore, there are infinite lines of such type passing through the given point.
Question 3:
If the point (3, 4) lies on the graph of the equation 3y = ax + 7, find the value of a.
Putting x = 3 and y = 4 in the given equation,

Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 10 of 19
3y = ax + 7
3 (4) = a (3) + 7
5 = 3a
Question 4:
The taxi fare in a city is as follows: For the first kilometre, the fares is Rs 8 and for
the subsequent distance it is Rs 5 per km. Taking the distance covered as x km and
total fare as Rs y, write a linear equation for this information, and draw its graph.
Total distance covered = x km
Fare for 1
st kilometre = Rs 8
Fare for the rest of the distance = Rs (x – 1) 5
Total fare = Rs [8 + (x – 1) 5]
y = 8 + 5x – 5
y = 5x + 3
5x – y + 3 = 0
It can be observed that point (0, 3) and satisfies the above equation.
Therefore, these are the solutions of this equation.

 x 0 y 3 0

The graph of this equation is constructed as follows.
Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 11 of 19
Here, it can be seen that variable x and y are representing the distance covered and
the fare paid for that distance respectively and these quantities may not be negative.
Hence, only those values of x and y which are lying in the 1
considered.
Question 5:
From the choices given below, choose the equation whose graphs are given in the
given figures.
For the first figure For the second figure

 (i) y = x (i) y = x +2 (ii) x + y = 0 (ii) y = x – 2 (iii) y = 2x (iii) y = – x + 2 (iv) 2 + 3y = 7x (iv) x + 2y = 6

Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 12 of 19
Points on the given line are (-1, 1), (0, 0), and (1, -1).
It can be observed that the coordinates of the points of the graph satisfy the
equation x + y = 0. Therefore, x + y = 0 is the equation corresponding to the graph
as shown in the first figure.
Hence, (ii) is the correct answer.

Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 13 of 19
Points on the given line are (-1, 3), (0, 2), and (2, 0). It can be observed that the
coordinates of the points of the graph satisfy the equation y = – x + 2.
Therefore, y = – x + 2 is the equation corresponding to the graph shown in the
second figure.
Hence, (iii) is the correct answer.
Question 6:
If the work done by a body on application of a constant force is directly proportional
to the distance travelled by the body, express this in the form of an equation in two
variables and draw the graph of the same by taking the constant force as 5 units.
Also read from the graph the work done when the distance travelled by the body is
(i) 2 units (ii) 0 units
Let the distance travelled and the work done by the body be x and y respectively.
Work done
distance travelled
y
x
y = kx
Where, k is a constant
If constant force is 5 units, then work done y = 5x
It can be observed that point (1, 5) and (-1, -5) satisfy the above equation.
Therefore, these are the solutions of this equation. The graph of this equation is
constructed as follows.

Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 14 of 19
(i)From the graphs, it can be observed that the value of y corresponding to x = 2 is
10. This implies that the work done by the body is 10 units when the distance
travelled by it is 2 units.
(ii) From the graphs, it can be observed that the value of y corresponding to x = 0 is
0. This implies that the work done by the body is 0 units when the distance travelled
by it is 0 unit.
Question 7:
Yamini and Fatima, two students of Class IX of a school, together contributed Rs 100
towards the Prime Minister’s Relief Fund to help the earthquake victims. Write a
linear equation which satisfies this data. (You may take their contributions as Rs x
and Rs y.) Draw the graph of the same.
Let the amount that Yamini and Fatima contributed be x and y respectively towards
the Prime Minister’s Relief fund.
Amount contributed by Yamini + Amount contributed by Fatima = 100
x + y = 100
It can be observed that (100, 0) and (0, 100) satisfy the above equation. Therefore,
these are the solutions of the above equation. The graph is constructed as follows.

Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 15 of 19
Here, it can be seen that variable x and y are representing the amount contributed
by Yamini and Fatima respectively and these quantities cannot be negative. Hence,
only those values of x and y which are lying in the 1
Question 8:
In countries like USA and Canada, temperature is measured in Fahrenheit, whereas
in countries like India, it is measured in Celsius. Here is a linear equation that
converts Fahrenheit to Celsius:
(i) Draw the graph of the linear equation above using Celsius for x-axis and
Fahrenheit for y-axis.
(ii) If the temperature is 30°C, what is the temperature in Fahrenheit?
(iii) If the temperature is 95°F, what is the temperature in Celsius?
(iv) If the temperature is 0°C, what is the temperature in Fahrenheit and if the
temperature is 0°F, what is the temperature in Celsius?
(v) Is there a temperature which is numerically the same in both Fahrenheit and
Celsius? If yes, find it.
(i)

Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 16 of 19
It can be observed that points (0, 32) and (-40, -40) satisfy the given equation.
Therefore, these points are the solutions of this equation.
The graph of the above equation is constructed as follows.
(ii) Temperature = 30°C
Therefore, the temperature in Fahrenheit is 86°F.
(iii) Temperature = 95°F
Therefore, the temperature in Celsius is 35°C.

Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 17 of 19
(iv)
If C = 0°C, then
Therefore, if C = 0°C, then F = 32°F
If F = 0°F, then
Therefore, if F = 0°F, then C = -17.8°C
(v)
Here, F = C
Yes, there is a temperature, -40°, which is numerically the same in both Fahrenheit
and Celsius.

Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 18 of 19
Exercise 4.4
Question 1:
Give the geometric representation of y = 3 as an equation
(I) in one variable
(II) in two variables
In one variable, y = 3 represents a point as shown in following figure.
In two variables, y = 3 represents a straight line passing through point (0, 3) and
parallel to x-axis. It is a collection of all points of the plane, having their y-coordinate
as 3.
Question 2:
Class IX Chapter 4– Linear Equations in Two Variables Maths
Page 19 of 19
Give the geometric representations of 2x + 9 = 0 as an equation
(1) in one variable
(2) in two variables
(1) In one variable, 2x + 9 = 0 represents a point as shown in the
following figure.
(2) In two variables, 2x + 9 = 0 represents a straight line passing through point
(-4.5, 0) and parallel to y-axis. It is a collection of all points of the plane, having
their x-coordinate as 4.5.

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## CBSE Class 10 Mathematics Solved Sample Paper 2019

In this article you will get CBSE Class 10 Mathematics: Important 1 Mark Questions. All the questions have been provided with proper solution to help you make an easy and quick preparation.

In order to master a subject, on must have a set of questions based on various topic and concepts discussed in that subject. Solving different questions helps to assess your preparedness and familiarise with the different ways in which a concept can be tested. This ultimately helps you to be more prepared and confidents for the exams.

Here, we are providing a set of most important 1 mark questions to prepare for CBSE class 10 Mathematics board exam 2019. All these questions have been provided with detailed solutions explained by the subject experts.

Students may download the whole question bank for CBSE Class 10 Maths Exam 2019 and practice the same while revising the Maths syllabus. Solving various questions will also help to increase your speed and accuracy.

Given below are some sample questions from the set of CBSE Class 10 Mathematics: Important 1 Mark Questions:

Q.  If the HCF of 65 and 117 is expressible in the form of 65 m – 117, then find the value of m.

Sol.

We have, 65 = 13 × 5

And 117 = 13 × 9

Hence, HCF = 13

According to question 65m – 117 = 13

⇒ 65m = 13 + 117 = 130

⇒ m = 130/65 = 2

Q. If the common difference of an A.P. is 3, then find a20 – a15.

Sol.

Let the first term of the AP be a.

an = a(n − 1)d

a20 – a15 = [a + (20 – 1)d] – [a + (15 – 1)d]

= 19d – 14d

= 5d

= 5 × 3

Q. Find the value of a so that the point (3, a) lies on the line represented by 2x – 3y = 5.

Sol.

Since point (3, a) lies on line 2x – 3y = 5.
Then, 2 x 3 – 3 x a = 5
⟹       6 – 5 = 3a
⟹       a = 1/3

Q. For what value of k will k + 9, 2 k ‒ 1 and 2k + 7 are the consecutive terms of an A.P.?

Sol.

If three terms x, y and z are in A.P. then, 2 y = x + z

Since k + 9, 2 k ‒ 1 and 2 k + 7 are in A.P.

&there4;          2(2k − 1) = (k + 9) + (2k + 7)
⟹       4k – 2 = 3k + 16

⟹       k = 18

Q. Find the median using an empirical relation, when it is given that mode and mean are 8 and 9 respectively.

Sol.

The relation between Mean, Median and Mode of the given data is:

Mode = 3Median − 2Mean

⟹       8 = 3Median − 2 × 9

⟹       3Median = 8 + 18

⟹       Median = 26/3

⟹       Median = 8.67

In CBSE Class 10 Mathematics Board Exam 2019, Section – A will comprise 6 questions of 1 mark each.

Students must practice the important questions given here. This will help them to track their preparedness for the final exam and also get an idea about the important topics of class 10 Maths to be prepared for the exam. Moreover, the solutions provided here will give students an idea about how to write proper solution to each question in the board exam so as to score optimum marks

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## CBSE Class 10 Mathematics Exam 2019: Important 1 mark questions with solutions

They say that it is the little drops of water that make a mighty ocean. Thus, if you are somebody who is looking forward to scoring more than 90 and perhaps even top the upcoming board exams, you cannot ignore the one mark questions. Unless and until you score well in these you cannot dream of scoring well overall. In order to aid the preparation level of the students who will be appearing for the board exams in a matter of days now, we have carefully compiled a list of important one mark mathematics questions. Thus, if you are someone who is appearing for your class 10 board exams later this month, here is a list of questions that you must be thorough with. Range Of Prime Numbers
Range Of Prime Numbers
1. What is the range of the first ten prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29?
i. 28
ii. 26
iii. 29
iv. 27
Ans: iv
Mean And Standard Deviation
Mean And Standard Deviation
2. If the mean and the standard deviation of a data is 48 and 12 respectively, what is their coefficient of variation?
i. 42
ii. 25
iii. 28
iv. 48
Ans: ii
Equation Of A Straight Line With A Slope Equation Of A Straight Line With A Slope
3. The equation of a straight line having a slope of 3 and a y intercept of -4 is
i. 3x – y – 4 = 0
ii. 3x + y – 4 = 0
iii. 3x – y + 4 = 0
iv. 3x + y + 4 = 0
Ans: I
Slope Of The Line Slope Of The Line
4. Slope of the line joining the points (3, -2) and (-1, a) is -3, 2. Find out the value of a
i. 1
ii. 2
iii. 3
iv. 4
Ans: I
Height Of The Tower Height Of The Tower 5. If a vertical stick that is 12 m long casts a shadow 8 m long on the ground and at the same time a tower casts a shadow 40 m long on the ground, then what is the height of the tower?
i. 40 m
ii. 50 m
iii. 75 m
iv. 60 m
Ans: iv
Similar Triangles
6. Triangles ABC and DEF are similar. Their areas are 100 sq cm and 49 sq cm respectively. If one side of the triangle is 8cm, then what is the corresponding side of the other triangle?
i. 4 cm
ii. 3 cm
iii. 9 cm
iv. 6 cm
Ans:iv
Series And Sequences Series And Sequences
7. What is the 8th term in this series 1, 1,2, 3, 5, 8, ….
i. 25
ii. 24
iii. 23
iv. 21
Ans: iv
Volume Of Two Cylinders Volume Of Two Cylinders 8. The ratio of the respective heights and the respective radii of two cylinders are 1:2 and 2:1 respectively. What is the ratio of their respective volumes?
i. 4 : 1
ii. 1 : 4
iii. 2 : 1
iv. 1 : 2
Ans: iii
Curved Surface Area Of Cones
9. If the circumference at the base of a right circular cone and the slant height are 120 pi cm and 10 cm respectively, then what does the curved surface area of the cone equal to?
i. 1200 pi cm sq
ii. 600 pi cm sq
iii. 300 pi cm sq
iv. 600 cm sq
Ans: ii A
rithmetic Progression
10. If a, b, c, l, m and n are in arithmetic progression, then 3a + 7, 3b + 7, 3c + 7, 3l +7, 3m +7 and 3n+7 forms a what?
i. A geometric progression
ii. An arithmetic progression
iii. A constant sequence
iv. Neither an arithmetic progression nor a geometric progression
Ans: ii
Volume Of A Sphere Volume Of A Sphere 12. The radius of a sphere is half the radius of another sphere. What is the ratio of their respective volumes?
i. 1 : 8
ii. 2 : 1
iii. 1 : 2
iv. 8: 1
Ans: I
Variance
13. If the variance of 14, 18, 22, 26 and 30 is 32, then what is the variance of 28, 36, 44, 52 and 60?
i. 64
ii. 128
iii. 34
iv. 32
Ans: ii
Point Of Intersection
14. What is the point of intersection of the straight lines y = 0 and x= – 4
i. ( 0, -4 )
ii. ( – 4, 0 )
iii. ( 0, 4 )
iv. ( 4, 0 )
Ans: I
Series And Sequences
15. Which of the following statements is not true?
i. A sequence is a real valued function defined on N.
ii. Every function represents a sequence.
iii. A sequence may have infinitely many terms.
iv. A sequence may have a finite number of terms.
Ans: i