Class 10th Maths class 10 maths pair of linear equations in two variables Hots Questions

Q 1 :

The pair of linear equations 2x + 3y = 5 and 4x + 6y = 10 is

inconsistent
consistent
dependent consistent
none of these

1

2

3

4

(3)

Dependent Consistent

Q 2 :

If the system of equations $3 x + y = 1 and (2 k - 1) x + (k - 1) y = 2 k + 1$ is inconsistent, then k =

-1
0
1
2

1

2

3

4

(4)

$3 x + y = 1$ ...(i)

and $(2 k - 1) x + (k - 1) y = 2 k + 1$ ...(ii)

Comparing eq. (i) with $a_{1} x + b_{1} y + c_{1} = 0$ and eq. (ii)

with $a_{2} x + b_{2} y + c_{2} = 0,$ we get

$a_{1} = 3, a_{2} = 2 k - 1, b_{1} = 1, b_{2} = k - 1, c_{1} = - 1$ and $c_{2} = - (2 k + 1)$

Since, system is inconsistent, then $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$

$\Rightarrow \frac{3}{2 k - 1} = \frac{1}{k - 1} \neq \frac{- 1}{- (2 k + 1)} \Rightarrow \frac{3}{2 k - 1} = \frac{1}{k - 1} or \frac{1}{k - 1} \neq \frac{1}{2 k + 1}$

$\Rightarrow 3 k - 3 = 2 k - 1 or 2 k + 1 \neq k - 1 \Rightarrow k = 2 or k \neq - 2$

Hence, the value of $k$ is 2.

Q 3 :

In the given figure, graphs of two linear equations are shown. The pair of these linear equations is:

consistent with unique solution.
consistent with infinitely many solutions.
inconsistent.
inconsistent but can be made consistent by extending these lines.

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2

3

4

(4) inconsistent but can be made consistent by extending these lines.

Q 4 :

The pair of equations x = 2a and y = 3b $(a, b \neq 0)$ graphically represents straight lines which are :

coincident
parallel
intersecting at (2a, 3b)
intersecting at (3b, 2a)

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(3) intersecting at (2a, 3b)

Q 5 :

The value of k for which the system of equations 3x – y + 8 = 0 and 6x – ky + 10 = 0 has infinitely many solutions, is

– 2
2
1/2
− 1/2

1

2

3

4

(2)

Given equation are $3 x - y + 8 = 0$ and $6 x - k y + 16 = 0$

Here, $a_{1} = 3, b_{1} = - 1, c_{1} = 8$

$a_{2} = 6, b_{2} = - k, c_{2} = 16$

For Infinite many solution, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \Rightarrow \frac{3}{6} = \frac{- 1}{- k} = \frac{8}{16} \Rightarrow \frac{1}{k} = \frac{1}{2} \Rightarrow k = 2$

Q 6 :

Consider the equation changes from y = 3x + 7 to y = 3x − 4. What is true about the changes to the graph?
(i) The coefficient of x remains same.
(ii) The value of y at x = 0 decreases.
(iii) Both lines are on the same side of the origin.
(iv) The line shifts downward.

(i), (iii) and (iv)
(ii) and (iv) only
(i), (ii) and (iv)
(i), (ii), (iii) and (iv)

1

2

3

4

(3)

When the equation changes from y = 3x + 7 to y = 3x − 4, the graph changes as shown in the figure.

(i) Yes, the coefficient of x i.e., 3 remains same.
(ii) The value of y at x = 0 decreases from 7 to −4.
(iii) Clearly, both lines are on the opposite side of the origin.
(iv) Yes, the line shifts downward.

Q 7 :

Consider the lines given by the equations y = (1/3)x + 5 and y = (1/3)x − 2. Which of the following statements are true regarding these lines?

(i) They intersect in 1st quadrant.
(ii) They are parallel to each other.
(iii) They intersect at the point where x = −1.
(iv) They have different values of y at x = 0.

(i), (iii) and (iv)
(i), (ii), (iii) and (iv)
(ii) and (iv)
(i) and (iv)

1

2

3

4

(3)

Here, a?/a? = (1/3)/(1/3) = 1, b?/b? = (−1)/(−1) = 1, c?/c? = 5/(−2).
The two lines are parallel (as a?/a? = b?/b? ≠ c?/c?). Therefore, they do not intersect.

At x = 0, y = (1/3)×0 + 5 = 5.
At x = 0, y = (1/3)×0 − 2 = −2.
Hence, statements (ii) and (iv) are correct

Q 8 :

For the equations 2x + y = 8 and 4x + 2y = 10, select the correct statements:

(i) The lines are identical.
(ii) The lines are parallel.
(iii) The second equation can be reduced to 2x + y = 5.
(iv) There is no solution to this system.

(i), (iii) and (iv)
(i), (ii), (iii) and (iv)
(ii), (iii) and (iv)
(i) and (iv)

1

2

3

4

(3)

We have, 2x + y = 8 …(i)
And, 4x + 2y = 10 ⇒ 2(x + y) = 2×5 ⇒ 2x + y = 5.
Here, a?/a? = 2/2 = 1, b?/b? = 1/1 = 1, c?/c? = 8/5.
Since a?/a? = b?/b? ≠ c?/c?, the lines are parallel. So, there is no point of intersection.
Hence, no solution.

Q 9 :

Given the equations y = −2x + 3 and y = 2x − 1, identify the correct descriptions about their graph and intersection:

(i) The lines intersect in the fourth quadrant.
(ii) The lines intersect in the first quadrant.
(iii) They are parallel lines.
(iv) Area of triangle formed by two lines with y-axis is 2 sq. units.

(i), (iii) and (iv)
(i), (ii), (iii) and (iv)
(ii) and (iii)
(ii) and (iv)

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2

3

4

(4)

Solving the system graphically, they intersect at (1, 1) in the first quadrant.
The lines are not parallel.
Area of ΔABC = (1/2) × AB × PC = (1/2) × 4 × 1 = 2 sq. units.

Q 10 :

The pair of equations ax + 2y = 9 and 3x + by = 18 represents parallel lines, where a, b are integers, if:

a = b
3a = 2b
2a = 3b
ab = 6

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2

3

4

(4)

Given, ax + 2y = 9 and 3x + by = 18 represent parallel lines.
∴ a?/a? = b?/b? ≠ c?/c? ⇒ a/3 = 2/b ≠ -9/-18
⇒ a/3 = 2/b ⇒ ab = 6

Q 11 :

The pair of equations, x = 0 and x = -4 has

a unique solution
no solution
infinitely many solutions
only solution (0, 0)

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2

3

4

(2)

Since the lines x = 0 and x = -4 are parallel to each other, therefore there is no solution for the given pair of equations

Q 12 :

If 2x + 3y = 15 and 3x + 2y = 25, then the value of x – y is

–10
8
10
–8

1

2

3

4

(3)

Given linear equations, we have:

2x + 3y = 15 ...(i)
3x + 2y = 25 ...(ii)

Subtracting (i) from (ii), we get:
x – y = 25 – 15
⇒ x – y = 10

Q 13 :

3 chairs and 1 table cost Rs 900; whereas 5 chairs and 3 tables cost Rs 2,100. If the cost of 1 chair is Rs x and the cost of 1 table is Rs y, then the situation can be represented algebraically as

3x + y = 900, 3x + 5y = 2100
x + 3y = 900, 3x + 5y = 2100
3x + y = 900, 5x + 3y = 2100
x + 3y = 900, 5x + 3y = 2100

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2

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(3)

Given situations can be represented algebraically as:

3x + y = 900 and 5x + 3y = 2100

Q 14 :

Tanisha and Aditya have some chocolates with them such that

• If Tanisha were to give 6 chocolates to Aditya, the new quantity of chocolates with each of them would be equal.

• Instead, if Aditya were to give 3 chocolates to Tanisha, then Tanisha would have four times as many chocolates as Aditya initially had.

Which of these pairs of equations would help us find the number of chocolates that they have?

(Note: Assume the initial number of chocolates with Tanisha as 'x' and that with Aditya as 'y'.)

x – 6 = y + 6; x + 3 = 4(y – 3)
x – 6 = y + 6; x + 3 = 4y
x + 6 = y – 6; x – 3 = 4y
x – y = 6; x = y

1

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4

(2)

Initial number of chocolates with Tanisha = x

Initial number of chocolates with Aditya = y

Case I: x – 6 = y + 6

Case II: x + 3 = 4y

Q 15 :

The sum of the digits of a two-digit number is 9. If 27 is subtracted from the number, its digits are interchanged. Which of these is the product of the digits of the number?

8
14
18
20

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3

4

(3)

Let x be the unit place digit and y be the ten’s place digit.

Two-digit number = 10y + x

According to the question,

x + y = 9 ...........(i)

10y + x – 27 = 10x + y ⇒ y – x = 3 ...........(ii)

Adding (i) and (ii), we get 2y = 9 + 3 ⇒ y = 6

Putting y = 6 in (i), we get x + 6 = 9 ⇒ x = 3

So, product of the digits of the number = 3 × 6 = 18

Topic Question Set

Q 1 :

The pair of linear equations 2x + 3y = 5 and 4x + 6y = 10 is

Q 2 :

If the system of equations $3 x + y = 1 and (2 k - 1) x + (k - 1) y = 2 k + 1$ is inconsistent, then k =

Q 3 :

In the given figure, graphs of two linear equations are shown. The pair of these linear equations is:

Q 4 :

The pair of equations x = 2a and y = 3b $(a, b \neq 0)$ graphically represents straight lines which are :

Q 5 :

The value of k for which the system of equations 3x – y + 8 = 0 and 6x – ky + 10 = 0 has infinitely many solutions, is

Q 6 :

Consider the equation changes from y = 3x + 7 to y = 3x − 4. What is true about the changes to the graph?
(i) The coefficient of x remains same.
(ii) The value of y at x = 0 decreases.
(iii) Both lines are on the same side of the origin.
(iv) The line shifts downward.

Q 8 :

For the equations 2x + y = 8 and 4x + 2y = 10, select the correct statements:

(i) The lines are identical.
(ii) The lines are parallel.
(iii) The second equation can be reduced to 2x + y = 5.
(iv) There is no solution to this system.

Q 10 :

The pair of equations ax + 2y = 9 and 3x + by = 18 represents parallel lines, where a, b are integers, if:

Q 11 :

The pair of equations, x = 0 and x = -4 has

Q 12 :

If 2x + 3y = 15 and 3x + 2y = 25, then the value of x – y is

Q 13 :

3 chairs and 1 table cost Rs 900; whereas 5 chairs and 3 tables cost Rs 2,100. If the cost of 1 chair is Rs x and the cost of 1 table is Rs y, then the situation can be represented algebraically as

Q 15 :

The sum of the digits of a two-digit number is 9. If 27 is subtracted from the number, its digits are interchanged. Which of these is the product of the digits of the number?

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Topic Question Set

Q 1 : The pair of linear equations 2x + 3y = 5 and 4x + 6y = 10 is

Q 2 : If the system of equations 3x+y=1 and (2k-1)x+(k-1)y=2k+1 is inconsistent, then k =

Q 3 : In the given figure, graphs of two linear equations are shown. The pair of these linear equations is:

Q 4 : The pair of equations x = 2a and y = 3b (a,b≠0) graphically represents straight lines which are :

Q 5 : The value of k for which the system of equations 3x – y + 8 = 0 and 6x – ky + 10 = 0 has infinitely many solutions, is

Q 6 : Consider the equation changes from y = 3x + 7 to y = 3x − 4. What is true about the changes to the graph? (i) The coefficient of x remains same. (ii) The value of y at x = 0 decreases. (iii) Both lines are on the same side of the origin. (iv) The line shifts downward.

Q 8 : For the equations 2x + y = 8 and 4x + 2y = 10, select the correct statements: (i) The lines are identical. (ii) The lines are parallel. (iii) The second equation can be reduced to 2x + y = 5. (iv) There is no solution to this system.

Q 10 : The pair of equations ax + 2y = 9 and 3x + by = 18 represents parallel lines, where a, b are integers, if:

Q 11 : The pair of equations, x = 0 and x = -4 has

Q 12 : If 2x + 3y = 15 and 3x + 2y = 25, then the value of x – y is

Q 13 : 3 chairs and 1 table cost Rs 900; whereas 5 chairs and 3 tables cost Rs 2,100. If the cost of 1 chair is Rs x and the cost of 1 table is Rs y, then the situation can be represented algebraically as

Q 15 : The sum of the digits of a two-digit number is 9. If 27 is subtracted from the number, its digits are interchanged. Which of these is the product of the digits of the number?

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Q 1 :

The pair of linear equations 2x + 3y = 5 and 4x + 6y = 10 is

Q 2 :

If the system of equations $3 x + y = 1 and (2 k - 1) x + (k - 1) y = 2 k + 1$ is inconsistent, then k =

Q 3 :

In the given figure, graphs of two linear equations are shown. The pair of these linear equations is:

Q 4 :

The pair of equations x = 2a and y = 3b $(a, b \neq 0)$ graphically represents straight lines which are :

Q 5 :

The value of k for which the system of equations 3x – y + 8 = 0 and 6x – ky + 10 = 0 has infinitely many solutions, is

Q 6 :

Consider the equation changes from y = 3x + 7 to y = 3x − 4. What is true about the changes to the graph?
(i) The coefficient of x remains same.
(ii) The value of y at x = 0 decreases.
(iii) Both lines are on the same side of the origin.
(iv) The line shifts downward.

Q 8 :

For the equations 2x + y = 8 and 4x + 2y = 10, select the correct statements:

(i) The lines are identical.
(ii) The lines are parallel.
(iii) The second equation can be reduced to 2x + y = 5.
(iv) There is no solution to this system.

Q 10 :

The pair of equations ax + 2y = 9 and 3x + by = 18 represents parallel lines, where a, b are integers, if:

Q 11 :

The pair of equations, x = 0 and x = -4 has

Q 12 :

If 2x + 3y = 15 and 3x + 2y = 25, then the value of x – y is

Q 13 :

3 chairs and 1 table cost Rs 900; whereas 5 chairs and 3 tables cost Rs 2,100. If the cost of 1 chair is Rs x and the cost of 1 table is Rs y, then the situation can be represented algebraically as

Q 15 :

The sum of the digits of a two-digit number is 9. If 27 is subtracted from the number, its digits are interchanged. Which of these is the product of the digits of the number?