CBSE 9th Standard Mathematics Linear Equations in Two Variables Sample Question Papers Study Material - QB365 Set B | QB365

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CBSE 9th Standard Mathematics Linear Equations in Two Variables Sample Question Papers Study Material - QB365 Set B

Published on: 29/10/2025

Question Set B

CBSE 9th Standard Mathematics Linear Equations in Two Variables Sample Question Papers Study Material - QB365

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Std: 9th Standard CBSE - Mathematics

chapter: Linear Equations in Two Variables - Sample Question Papers

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Questions:

Generic 4

4 x 4 = 16

Ram is sitting on a chair in a corner of a huge park. His son Ravi is a student of class X. Mr. Ram asks him, "Draw a straight line passing through my feet him, "Draw a straight line passing through my feet and the centre of the park in 2 minutes. Ravi do so.
(i) Find the co-ordinate of the feet of Mr. Ram if the y-coordinate of the feet is -19.5 and the equation of the line x+y=0
(ii) Find the co-ordinate of the feet of mR. Ram if the y-coordinate of the feet is 20.5 and the equation of the line is x=y.
(iii) Which mathematical concept is used in the above problem?
(iv) Which values do you learn from Ravi?

Three vertices of a square PQRS are P(-4,0), Q(1,0), R(1,-5).Plot the points.Also, find the coordinates of the missing vertex S.

(i) If 3x + y + z =0, show that \(\\ 27{ x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }=9xyz.\)
(ii) Can we say that each of x, y, and z is a factor of \(\\ 27{ x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }?\)
(iii) Meenu finds that a perfect square number is a factor of \(\\ 27{ x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }\) . Is she correct? If so, which value of Meenu is depicted by her finding?
(iv) Which mathematical concept has been covered in this problem?
(v) Write the formulae used in the solution.

The polynomial \(p(x)=2{ x }^{ 3 }-3{ x }^{ 2 }+ax-3a+9\)when divided by x+1, leaves the remainder 16. Find the value of a. Also, find the remainder when p(x) is divided by x+2.

Generic 3

5 x 3 = 15

Given the equation 2x+y=7
(i) What is the value of x, when the value of y is 3?
(ii) What is the value of y, when the value of x is 4?
(iii) Find one more solution for the above equation.

The Auto fare in a city is as follows:
For the first kilometer the fare is Rs.5 and for the successive distance it is Rs.1 per kilometer.Taking a distance covered as x kilometer and total fare as Rs.y, write a linear equation for the above said data and draw its graph

Check which of the following are solutions of equation x-2y = 4 and which are not:
\((\sqrt{2}, 4\sqrt{2})\)

(Street Plan):A city has two main roads cross each other at the centre of the city. These two roads are along the North-South direction and East-West direction.All the other streets of the city run parallel to these roads and are 200 m apart.There are 5 streets in each direction.Using 1 cm = 200 m, draw a model of the city on your notebook.Represent the roads/streets by single line.

There are many cross-streets in your model.A particular cross-street is made by two streets, one running in the North-South direction and another in the East- West direction.Each cross-street is referred to in the following manner: If the 2nd street running in the North-South direction and 5th in the East-West direction meet at some crossing, then we will call this cross-street (2, 5).Using this convention, find:
(i) how many cross-streets can be referred to as (4,3)?
(ii) how many cross-streets can be referred to as (3,4)?

Determine which of the following polynomials has (x+1) a factor: \({ x }^{ 3 }-{ x }^{ 2 }-(2+\sqrt { 2 } )x+\sqrt { 2 } .\)

Generic 2

4 x 2 = 8

10.

What are the coordinates of a point that is:
(i) the mirror image of point (0,4) in x-axis.
(ii) the mirror image of point (-3,-5) in y-axis.

11.

In figure, \(\triangle ABC\) is an equilateral triangle with coordinates of B and C as (-4,0) and (4,0) respectively.Find the coordinates of the vertex.

12.

In which quadrant do the given point lie? (-4,-5)

13.

Factorise: \(4x^ 3+20x^2+33x+18\)

Multiple Choice Question

9 x 1 = 9

14.

In the figure, l is the graph of the equation:

(a)

y=x

(b)

x+y=0

(c)

x=2y

(d)

y=2x

15.

The equation of y-axis is

(a)

x=0

(b)

y=0

(c)

x=1

(d)

y=1

16.

The salary of Dr.Harikisham is thrice the salary of Manish Goyal.Write a linear equation in two variables to represent the statement.

(a)

x=3

(b)

x+3y=0

(c)

x=3y+3

(d)

x=y+3

17.

Write the coordinates of P

(a)

(2,2)

(b)

(-1,-2)

(c)

(1,-2)

(d)

(-1,2)

18.

The co-ordinates of point Q are:

(a)

(3,3.5)

(b)

(3.5,3)

(c)

(-3,3.5)

(d)

(-3,-3.5)

19.

The line of intersection of I and II quadrants is

(a)

x - axis

(b)

y - axis

(c)

vertical axis

(d)

None of these

20.

Rene Descartes belonged to

(a)

15^th Century

(b)

16^th Century

(c)

17^th Century

(d)

18^th Century

21.

Product of \((x-\frac{1}{x})(x+\frac{1}{x})(x^2+\frac{1}{x^2})\) is:

(a)

\(x^4+\frac{1}{x^4}\)

(b)

\(x^3+\frac{1}{x^3}-2\)

(c)

\(x^4-\frac{1}{x^4}\)

(d)

\(x^2+\frac{1}{x^2}+2\)

22.

The value of the polynomial \(x^2-x-1\) at x=-1 is:

(a)

-3

(b)

(c)

-1

(d)

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Answers Key:

Generic 4

4 x 4 = 16

(i) According to the questions, the eqiuation of line is
x+y=0..(i)
The line passes through a point whose y-coordinate is -19.5
y=-19.5
Put the value of y in equation (i), we get
x-19.5=0
x=19.5
So the required coordinates are (19.5-19.5)
(ii) According to the question, the equation of line is
x=y...(ii)
The line passes through a point whose x-coordinate is 20.5
x=20.5
Put this value of x in (i), we get
y=20.5
So the required coordinates are(20.5,20.5)
(iii) The coordinates of all the points lying on a line satisfy the equation of the line.
(iv) Obedience and respect for elders or parents.

The coordinates of the missing vertex S are (-4,-5)

(i) We know that if x + y + z = 0, then
\({ x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }=3xyz\)
\(\therefore 27{ x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }\)
\(={ (3x) }^{ 2 }+{ (y) }^{ 3 }+{ (z) }^{ 3 }\)
= 3(3x)(y)(z) | \(\therefore\) 3x + y + z = 0
= 9xyz
(ii) Clearly each of x, y, and z is a factor of \(27{ x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }\)
(ii) We have seen that 9 is the factor of \(27{ x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }\) and 9 is a perfect square. hence the result. So, Meenu is correct. So, the value 'Expertness' is depicted by her finding.
(iv) The mathematical concept ' Polynomials' has been covered in this problem.
(v) The formulae used in the solution are as follow:
1. If x + y + z = 0, then, \(27{ x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }\)= 3xyz
2. Concept of factors
3. Concept of perfect square numbers.

\(p(x)=2{ x }^{ 3 }-3{ x }^{ 2 }+ax-3a+9\)
By remainder theorem,
\(p(-1)=16\ x+1=0\quad \Rightarrow x=-1\)
\(\Rightarrow 2{ (-1) }^{ 3 }-3{ (-1) }^{ 2 }+a(-1)-3a+9=16\)
\(\Rightarrow -2-3-a-3a+9=16\)
\(\Rightarrow 4a=-12\)
\(\Rightarrow a=-3\)
\(\therefore p(x)=2{ x }^{ 3 }-3{ x }^{ 2 }-3x-3 \times(-3)+9\)
\(={ 2x }^{ 3 }-3{ x }^{ 2 }-3x+18\)
\(\therefore \) Remainder when p(x) is divided by x+2 =p(-2)
By remainder theorem: \(x+2=0\Rightarrow x=-2\)
\(={ 2(-2) }^{ 3 }-3{ (-2) }^{ 2 }-3(-2)+18\)
\(=-16-12+6+18=-4\)

Generic 3

5 x 3 = 15

(i) when y=3, then
2x+3=7
\(\Rightarrow\)2x=4
\(\Rightarrow\)x-=2
(ii)when x=4, then
2(4)+y=7
\(\Rightarrow\) y=7-8=-1
(iii) when x=1, then
2+y=7\(\Rightarrow\) y=5
\(\therefore \)One more solution is (1,5)

y=5+2(x-1) ⇒ y=3+2x

The given equation is x - 2y = 4
Put \(x=\sqrt{2}\), y=\(4\sqrt{2}\) in (1), we get
x-2y =\(\sqrt{2}-2(4\sqrt{2})\)
\(=\sqrt{2}-8\sqrt{2}=-7\sqrt{2}\) which is not 4.
\((\sqrt{2},\ 4\sqrt{2})\) which is not 4.
\((\sqrt{2}, 4\sqrt{2})\) is not a soluton of (1)

Both the cross-streets are marked in the figure given.They are uniquely found because of the two reference lines we have used for locating them.

\({ x }^{ 3 }-{ x }^{ 2 }-(2+\sqrt { 2 } )x+\sqrt { 2 } .\)
Let \(p(x)=\)\({ x }^{ 3 }-{ x }^{ 2 }-(2+\sqrt { 2 } )x+\sqrt { 2 } .\)
The zero of x+1 is -1
\(p(-1)=(-1)^{ 3 }+3(-1)^{ 2 }+(2-\sqrt { 2 } )(-1)+\sqrt { 2 }\)
\( \\ =-1-1+2+\sqrt { 2 } +\sqrt { 2 } =2\sqrt { 2 } \neq 0\)
\(\therefore\) By factors theorem, x+1 is not a factor of \({ x }^{ 3 }-{ x }^{ 2 }-(2+\sqrt { 2 } )x+\sqrt { 2 } .\)