Pair of Linear Equations in Two Variable Test

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Pair of Linear Equations in Two Variable Test - 1

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Question 1
1 / -0

Two intersecting lines are drawn in the graph given below. Find the point of intersection of these lines.

A
(2, -2)

B
(-2, 2)

C
(1, -1)

D
(-1, -5)

Solution

The lines are intersecting at 2 on the x-axis and at -2 on the y-axis. So, the point of intersection is (2, -2). [1]
Question 2
1 / -0

The equations x = -a and y = -b graphically represent lines which are

A
intersecting at (a, b)

B
parallel

C
coincident

D
intersecting at (-a, -b)

Solution

x = -a
y = -b
Question 3
1 / -0

Use the given graph to find the solution of y = 10 – x and x = 2 + y.

A
x = 6, y = 8

B
x = 6, y = 4

C
x = 5, y = 7

D
x = 4, y = 6

Solution

y = 10 - x
or x + y = 10 ---(1)
Now, x = 2 + y
or x - y = 2 ---(2)
Adding (1) and (2), we get:
2x = 12
x =
x = 6 [½]
Putting x = 6 in equation (1), we get:
6 + y = 10
y = 10 - 6
y = 4 [½]
Question 4
1 / -0

For what value(s) of k does the given system of equations have no solution?

kx + 2y = 3
3x + 6y = 10

A
k = 3/10

B
k = 1

C
Either k = 3/10 or k = 1

D
None of these

Solution

kx + 2y = 3
3x + 6y = 10
Comparing above equations with
a₁x + b₁y = c₁
a₂x + b₂y = c₂We get
, ,

For no solution:
(a₁/a₂₎ = (b₁/b₂)≠^₍c₁/c₂)
(k/3) = (2/6) ≠ (3/10)
k = 1
and k≠ (9/10)
The given system of equations will have no solution if k = 1
Question 5
1 / -0

For what value of k does the pair of linear equations given below will not have unique solution?

2x - ky = k - 1
3x - 15y = 2k - 7

A
10

B
^-10

C
Inappropriate information

D
Other than 10

Solution
Question 6
1 / -0

For what value of p does the pair of linear equations given below have not unique solution?

2x - 3y = 5
(2p - 1)x + py = 7

A
3

B
1

C

D
^_-2

Solution

2x - 3y = 5
(2p - 1)x + py = 7
Comparing the above equations with a₁x + b₁y = c₁and a₂x + b₂y = c₂, we get

For a unique solution,
∴
Or 2p ≠ -3(2p - 1)
2p ≠ -6p + 3
Or 8p ≠ 3
p ≠
Question 7
1 / -0

The cost of 3 pencils boxes and 3 erasers boxes is Rs. 150. Four times the cost of pencil box is 20 more than the cost of eraser box. Find the respective costs of a pencil box and an eraser box.

A
Rs. 14, Rs. 36

B
Rs. 15, Rs. 30

C
Rs. 10, Rs. 15

D
Rs. 21, Rs. 13

Solution

Let x and y be the costs of a pencil box and an eraser box, respectively.
According to the question:
3x + 3y = 150
x + y = 50 ... (i)
Also, 4x - y = 20 ... (ii)
Putting the value of y from (i) in (ii), we get 4x - (50 - x) = 20
4x - 50 + x = 20
5x = 70
x = 14
So, x + y = 50
y = 50 - 14 = 36
Question 8
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Find the values of x and y that solve the given set of equations.

2x + 3y = 2
4x² + y² = 4

A
x = 1, y = 0 and x = , y =

B
x = 1, y = 2 and x =, y =

C
x = 0, y = 1 and x = , y =

D
x = 1, y = 0 and x = , y =

Solution

2x + 3y = 2 ..(i)
4x² + y² = 4 ..(ii)
Rearrange (i) with subject as y
y =
Substitute the value of y in (ii)
4x² + = 4
4x² + = 4
36x² + 4x² 8x + 4 36 = 0
40x² 8x 32 = 0
Or, 5x² x 4 = 0
(5x + 4) (x - 1) = 0
x = and x = 1
When x = , y =
= = or
When x = 1, y = = 0
x = 1, y = 0
or x = , y =
Question 9
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Which of the following values of p is not correct for the system of equations pm – 4n = 8 and 24m – 8n = 12 to have a unique solution?

A
12

B
24

C
6

D
8

Solution

We will compare the given equations with a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.

This is the condition for the system of equations to have a unique solution.

≠

p ≠ 12
Question 10
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If 4y – 7 = and – 13 = 6y, find the value of (5x – 2y).

A
– 25.67

B
– 42.02

C
– 73.9

D
– 79.61

Solution

4y – 7 = Or 4y – = 0
Or 4y – = 7
Or = 7 …. (1)
– 13 = 6y
Or – 6y = 13 …. (2)
Multiplying equations (1) by 6 and (2) by 4, we get
+ 24y = 42 …. (3)
- 24y = 52 … (4)
Adding equations (3) and (4),
= 42 + 52
= 94
= 94
Or 2 = 94x
Or x =
x =
Putting x = in equation (1), we get
+ 4y = 7
– 3 47 + 4y = 7
– 141 + 4y = 7
4y = 7 + 141 (Add 141 on both sides)
4y = 148
y =
y = 37
Now, 5x – 2y = – 2 37
=
= 0.1 – 74
= – 73.9 (approximately)
Question 11
1 / -0

The length and the breadth of a rectangular plot are and b, respectively. The perimeter of the plot is 24 m. If the length is 4 m more than the breadth, what is the area of the plot?

A
32 m²

B
96 m²

C
64 m²

D
24 m²

Solution

We know,
Perimeter = 2(l + b)
2( + b) = 24
+ b = 12 … (1)
= 4 + b … (2)
From (1) and (2), we get
= 8
b = 12 - 8 = 4
Area = b
Area = 8 4 = 32 m²
Question 12
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In a box, there are 10 pens in total, out of which some are blue and some are red. When twice the number of blue pens is added to three times the number of red pens, the result obtained is 23. How many red pens are there in the box?

A
7

B
3

C
4

D
2

Solution

Let the number of blue pens be x and the number of red pens be y.
According to question:
x + y = 10 … (i)
and 2x + 3y = 23 … (ii)
Multiply (i) by 2 and subtract (ii) from (i),

y = 3
Number of red pens = 3
Question 13
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I have only 1-dollar and 50-cent coins amounting to $216. How many 50-cent coins do I have if the number of 1-dollar coins is 18 more than that of 50-cent coins?

A
150

B
140

C
132

D
66

Solution

Let the number of 50 cent coins be n.
Thus, number of 1 dollar coins = n + 18
Now, 50n + (n + 18) × 100 = 21,600
Or, n = 132
Hence, answer option 3 is correct.
Question 14
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If 2 boys and 5 girls complete a piece of work in 3 days, while 6 boys and 6 girls complete the same work in 2 days, then how many days will 1 girl alone take to finish the work?

A
36 days

B
18 days

C
12 days

D
10 days

Solution

Let the work done by 1 boy in 1 day be B.
Let the work done by 1 girl in 1 day be G.
So, 3(2B + 5G) = W
Or 6B = W - 15G
Or 12B = 2(W - 15G) ...(i)
Also,
2(6B + 6G) = W
Or 12B + 12G = W
Or 12B = W - 12G ...(ii)
Equating (i) and (ii), we get
2(W - 15G) = W - 12G
Or W = 18G
Thus, a girl will take 18 days to finish the work.
Question 15
1 / -0

A fraction reduces to if we subtract 2 from the numerator and 1 from the denominator. If we subtract 1 from the numerator and 2 from the denominator, the fraction becomes Which of the following is the correct representation of the equations, where n is numerator and d is denominator?

A

B

C

D

Solution

3n – 6 = d – 1
3n – d = 5
Also,

5n – 5 = 3d – 6
5n – 3d = –1
Equations are:
3n – d = 5
5n – 3d = –1