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Page No 74:

Question 1:

Mark the correct alternative in each of the following:

If (4, 19) is a solution of the equation y = ax + 3, then a=

(a) 3

(b) 4

(c) 5

(d) 6

Answer:

We are given (4, 19)as the solution of equation

Substituting x = 4 and y = 19, we get

Therefore, the correct answer is (b).

Page No 74:

Question 2:

If (a, 4) lies on the graph of 3x + y = 10, then the value of a is

(a) 3

(b) 1

(c) 3

(d) 4

Answer:

We are given (a, 4) lies on the graph of linear equation 3x + y = 10.

So, the given co-ordinates are the solution of the equation 3x + y = 10.

Therefore, we can calculate the value of a by substituting the value of given co-ordinates in equation 3x + y = 10.

Substituting x = a and y = 4 in equation 3x + y = 10, we get

No option is correct.

Page No 74:

Question 3:

The graph of the linear equation 2xy = 4 cuts x- axis at

(a) (2, 0)

(b) (−2, 0)

(c) (0, −4)

(d) (0, 4)

Answer:

We are given,

we get,

We will substitute in to get the co-ordinates for the graph of at x axis

Co-ordinates for the graph of are .

Therefore, the correct answer is (a).

Page No 74:

Question 4:

How many linear equations are satisfied by x = 2  and y = −3?

(a) Only one

(b) Two

(c) Three

(d) Infinitely many

Answer:

There are infinite numbers of linear equations that are satisfied by as

(i) Every solution of the linear equation represent a point on the line.

(ii) Every point on the line is the solution of the linear equation.

Therefore, the correct answer is (d).

Page No 74:

Question 5:

The equation x − 2 = 0 on number line is represented by

(a) a line

(b) a point

(c) infinitely many lines

(d) two lines

Answer:

The equation is represented by a point on the number line.

Therefore, the correct answer is (b).

Page No 74:

Question 6:

x = 2, y = −1 is a solution of the linear equation

(a) x + 2y = 0

(b) x + 2y = 4

(c) 2x + y = 0

(d) 2x + y = 5

Answer:

We are given as the solution of linear equation, which we have to find?

The equation is which can be proved by

Substituting in the equation, we get

Therefore, the correct answer is (a).



Page No 75:

Question 7:

If (2k − 1, k) is a solution of the equation 10x − 9y = 12, then k =

(a) 1

(b) 2

(c) 3

(d) 4

Answer:

We are given as the solution of equation

Substituting, we get

Therefore, the correct answer is (b).

Page No 75:

Question 8:

The distance between the graph of the equations x = −3 and x = 2 is

(a) 1

(b) 2

(c) 3

(d) 5

Answer:

Distance between the graph of equations, say D

D = Distance of co-ordinate on negative side of x axis + Distance of co-ordinate on positive side of x axis

Distance of co-ordinate on negative side of x axis = x = 3 units

Distance of co-ordinate on positive side of x axis = x = 2 units

Therefore, the correct answer is (d).

Page No 75:

Question 9:

The distance between the graphs of the equations y = −1 and y = 3 is

(a) 2

(b) 4

(c) 3

(d) 1

Answer:

Distance between the graph of equations, say D

D = Distance of co-ordinate on negative side of y axis + Distance of co-ordinate on positive side of y axis

Distance of co-ordinate on negative side of y axis = y = 1 units

Distance of co-ordinate on positive side of y axis = y = 3 units

Therefore, the correct answer is (b).

Page No 75:

Question 10:

The point of the form (a, a) always lies on
(a) x-axis
(b) y-axis
(c) on the line y = x
(d) on the line x + y = 0

Answer:

Since, the given point (aa) has the same value of x and y-coordinates. Therefore, the point (aa) must lie on the line = x.

Hence, the correct answer is option (c).

Page No 75:

Question 11:

Any point on the x-axis is of the form
(a) (x, y)
(b) (0, y)
(c) (x, 0)
(d) (x, x)

Answer:

Any point on the x-axis has an x-coordinates but the y-coordinates value will be always zero.
Therefore any point on x-axis is in the form (x, 0).

Hence, the correct answer is option (c).

Page No 75:

Question 12:

The equation of the x-axis is of the form
(a) x = 0
(b) y = 0
(c) x + y = 0
(d) x = y

Answer:

Any point on the x-axis has an x-coordinates but the y-coordinates value will be always zero.
Therefore any point on x-axis is in the form (x, 0).
Thus, the equation of y-axis is: y = 0

Hence, the correct answer is option (b).

Page No 75:

Question 13:

The linear equation 2x – 5y = 7 has
(a) a unique solution
(b) two solutions
(c) infinitely many solutions
(d) no solutions

Answer:

2x – 5y = 7 is a linear equation in 2 variables. 

Number of variables is two and number of equation is one and moreover, it is the equation of line which will pass through infinite points and all points will satisfy the given equation.

Hence, the correct answer is option (c).

Page No 75:

Question 14:

The graph of y = 6 is a line
(a) parallel to x-axis at a distance 6 units from the origin. 
(b) parallel to y-axis at a distance 6 units from the origin.
(c) making an intercept 6 on the x-axis.
(d) making an intercept 6 on both the axes.

Answer:

The given equation is = 6.
Compare the above equation with the generalized equation of a line y = mx + c
m = 0, which means the line is parallel to the x-axis.
c = 6, which means it is at 6 units from the origin.

Hence, the correct answer is option (a).

Page No 75:

Question 15:

The graph of the linear equation 2x + 3y = 6 cuts the y-axis at the point
(a) (2, 0)
(b) (0, 3)
(c) (3, 0)
(d) (0, 2)

Answer:

The given equation is 2x + 3y = 6.
Since the graph cuts the y-axis, at that point value of x=0.
Put x = 0 into the equation 2x + 3y = 6,
20+3y=63y=6y=2

Thus, the required point is (0, 2).

Hence, the correct answer is option (d).

Page No 75:

Question 16:

If (2, 0) is a solution of the linear equation 2x + 3y = k, then the value of k is
(a) 4
(b) 6
(c) 5 
(d) 2

Answer:

Since (2, 0) is a solution of the linear equation 2x + 3y = k,
Put x = 2 and y = 0 in 2x + 3y = k, we get
22+30=k
k=4

Hence, the correct answer is option (a).

Page No 75:

Question 17:

The equation 2x + 5y = 7 has a unique solution, if x, y are
(a) natural numbers
(b) positive real numbers
(c) real numbers
(d) rational numbers

Answer:

In the case of natural numbers, only one pair is there which satisfies the equation. The pair is (1, 1). 
Therefore, if x and y are natural numbers, the given equation has a unique solution.

Hence, the correct answer is option (a).

Page No 75:

Question 18:

The graph of the equation 2x + 3y = 6 is a line which meets the x and y axes respectively at the points
(a) (3, 0) and (0, 2)
(b) (2, 0) and (0, 3)
(c) (0, 3) and (2, 0)
(d) (3, 2) and (2, 3)

Answer:

The given equation is 2x+3y=6.
Since the graph cuts the y-axis , at that point value of x=0.
Put x = 0 into the equation 2x+3y=6,
20+3y=63y=6y=2
The required point is (0, 2).
Also,
2x + 3y = 6 to meet the x-axis, put y = 0.
∴ 2x + 3 × 0 = 6
⇒ 2x = 6
⇒ x = 3
Thus, the coordinates of x-axis are (3, 0).

Hence, the correct answer is option (a).

Page No 75:

Question 19:

Any point on the line y = x is of the form
(a) (a, –a)
(b) (0, a)
(c) (a, a)
(d) (a, 0)

Answer:


Any point on the line y = x will have x and y coordinates the same.
So, any point on the line y = x is of the form (aa).

Hence, the correct answer is option (c).

Page No 75:

Question 20:

Any point on the line y = –x is of the form
(a) (a, –a)
(b) (–a, –a)
(c) (a, a)
(d) (a, 0)

Answer:

Any point on the line y = –x will have x and y coordinate with opposite signs.
So, any point on the line y = –x is of the form (a, –a).

Hence, the correct answer is option (a).

Page No 75:

Question 21:

How many linear equations in x and y can be satisfied by x = 1 and y = 2?
(a) only one
(b) two
(c) three
(d) infinitely many

Answer:

Let the linear equation be ax+by+c=0    .....(1)
On putting x = 1 and y = 2, in above equation we get;
a+2b+c=0, where a,b and c, are real numbers.
So, different values of a, b and c satisfy a + 2b + c = 0.
Thus, infinitely many linear equations in x and y can be satisfied by x = 1 and y = 2.

Hence, the correct answer is option (d).



Page No 76:

Question 22:

Any solution of the linear equation 2x + 0y + 9 = 0 in two variables is of the form
(a) -92, a
(b)  b,-92
(c)  0,-92
(d) (–9, 0)

Answer:

The given linear equation = 2x + 0y + 9 = 0

2x+9=02x=-9x=-92

As the coefficient is 0 it can take any value and will not affect the answer.

Thus, the linear equation in two variables is of the form (-92,a).

Hence, the correct answer is option (a).

Page No 76:

Question 23:

If a linear equation has solutions (–2, 2), (0, 0) and (2, –2), then it is of the form
(a) y = –x
(b) y = x
(c) y = 2x
(d) x = 2y

Answer:

Let the linear equation be ax + by + c = 0      .....(1)
Thus as, (–2, 2), (0, 0) and (2, –2) are the solution of the linear equation. Therefore, it satisfies the equation (1).
At (–2, 2), the equation is –2a + 2b + c = 0    .....(2)
At (0, 0), the equation is c = 0                         .....(3)
At (2, –2), the equation is 2a – 2b + c = 0       .....(4)

From (2) and (3), we get
c = 0 and a = b
On putting a = b and c = 0 in (1), we have
ax + ay = 0
⇒ x + y = 0

Thus, y = –x is the required form of linear equation.

Hence, the correct answer is option (a).

Page No 76:

Question 24:

A linear equation in two variables is of the form ax + by + c = 0, where
(a) a ≠ 0, b ≠ 0
(b) a = 0, b ≠ 0
(c) a ≠ 0, b = 0
(d) a = 0, c = 0

Answer:

A linear equation in two variables is of the form ax by c = 0, where a ≠ 0, b ≠ 0.

Hence, the correct answer is option (a).

Page No 76:

Question 25:

If we multiply or divide both sides of a linear equation with non-zero number, then the solution of the linear equation
(a) changes
(b) remains some
(c) changes in case of division only
(d) change in case of multiplication only.

Answer:

We know, that if we multiply or divide both sides of a linear equation with a non-zero number, then the solution of the linear equation remains the same.

Hence, the correct answer is option (b).

Page No 76:

Question 26:

The graph of the linear equation y = x passes through the point 
(a) 32, -32
(b) 0, 32
(c) (1, 1)
(d) -12, 12

Answer:

The linear equation y = x has the same values for x-coordinates and y-coordinates.

Thus, out of all options only (1, 1) satisfies the equation y = x.

Hence, the correct answer is option C.

Page No 76:

Question 27:

If the graph of the equation 4x + 3y = 12 cuts the coordinate axes at A and B, then hypotenuse of right triangle AOB is of length

(a) 4 units

(b) 3 units

(c) 5 units

(d) none of these

Answer:

We are given,

We get,

Now, substituting in, we get

Substituting in,we get

Thus, we have the following table exhibiting the abscissa and ordinates of points on the line represented by the given equation

x

0

3

y

4

0

We are given that the graph of equation cuts the co-ordinate axes and and forms the right angle triangle AOB.

Length of AB in right angled triangle AOB

The length of hypotenuse AB of triangle AOB is 5 units.

Hence, the correct answer is option (c).

Page No 76:

Question 28:

Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has the following four choices (a), (b), (c), and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) ​Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): The graph of the linear equation 2x – 9y + 18 = 0 meets x-axis at (–9, 0).
Statement-2 (Reason): Coordinates of points on the y-axis are of the form (0, a), where a is a variable.

Answer:

Statement - 1: True
Given that, 2x – 9y + 18 = 0 
At x-aixs, y = 0

2x+18=02x=-18x=-9x,y=-9,0

Statement - 2: Coordinates of points on the y-axis are of the form (0, a), where a is a variable.

Thus, Statement-2 is true but Statement-2 is not a correct explanation for Statement-1.

Hence, the correct answer is option (b).


 

Page No 76:

Question 29:

Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has the following four choices (a), (b), (c), and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) ​Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): The graph of equation x = –5 is a line parallel to the y-axis at a distance of 5 units to the left of the y-axis.
Statement-2 (Reason): The line parallel to the y-axis at a distance of units to the left of the y-axis is given by the equation x = –a.

Answer:

Statement-1: True
The equation of a line parallel to the y-axis is of the form x = a. Since the line is 5 units to the left of the y-axis
 x=-5

Statement-2: The line parallel to the y-axis at a distance of units to the left of the y-axis is given by the equation x = –a.

Thus, Statement-2 is true and Statement-2 is a correct explanation for Statement-1.

Hence, the correct answer is option (a).

Page No 76:

Question 30:

Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) ​Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): The graph of the linear equation y = mx + c passes through the origin.
Statement-2 (Reason): The linear equation ax + by = 0 represents a straight line passing through the origin.

Answer:

Statement-1:  False,
The graph of the linear equation y = mx may pass through the origin depending c=0 or c0.

Statement -2: True
Comparing ax by = 0 with  y = mx c, we have
y=-abx+0
Since c=0, the above equation passes through the origin.

Thus, Statement-1 is false and Statement-2 is true.

Hence, the correct answer is option (d).

 

Page No 76:

Question 31:

Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) ​Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): The graph of the linear equation y = 4 is a line parallel to x-axis at a distance of 4 units above it.
Statement-2 (Reason): The line parallel to x-axis at a distance a units above the x-axis is given by the equation y = –a.

Answer:

Statement-1: True
The equation of a line parallel to the x-axis is of the form y = a. Since the line is 4 units to the above of the x-axis
 y=4

Statement-2: False, the line parallel to x-axis at a distance a units above the x-axis is given by the equation y = a

Hence, the correct answer is option (c).



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