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Practice Test: Pair of Linear Equations in Two Variables - Class 10 MCQ


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25 Questions MCQ Test Mathematics (Maths) Class 10 - Practice Test: Pair of Linear Equations in Two Variables

Practice Test: Pair of Linear Equations in Two Variables for Class 10 2024 is part of Mathematics (Maths) Class 10 preparation. The Practice Test: Pair of Linear Equations in Two Variables questions and answers have been prepared according to the Class 10 exam syllabus.The Practice Test: Pair of Linear Equations in Two Variables MCQs are made for Class 10 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Practice Test: Pair of Linear Equations in Two Variables below.
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Practice Test: Pair of Linear Equations in Two Variables - Question 1

If A : Homogeneous system of linear equations is always consistent. R : x = 0, y = 0 is always a solution of the homogeneous system of equations with unknowns x and y, then which of the following statement is true?

Detailed Solution for Practice Test: Pair of Linear Equations in Two Variables - Question 1

Homogeneous system of  linear equations are of the form ax+by=0 and cx+dy=0. These lines pass through the origin. Thus, there is always at least one solution, the point (0,0).So, Homogeneous system of linear is always consistent, meaning that it has at least one solution which is (0,0). So B is the exact explanation.

Practice Test: Pair of Linear Equations in Two Variables - Question 2

Rozly can row downstream 20km in 2 hours, and the upstream 4km in 2 hours. What will be the speed of rowing in still water?       

Practice Test: Pair of Linear Equations in Two Variables - Question 3

The pair of linear equations 2x + ky – 3 = 0, 6x + 2/3y + 7 = 0 has a unique solution if –

Detailed Solution for Practice Test: Pair of Linear Equations in Two Variables - Question 3

Practice Test: Pair of Linear Equations in Two Variables - Question 4

The pair of linear equations 2kx + 5y = 7, 6x – 5y = 11 has a unique solution if –

Detailed Solution for Practice Test: Pair of Linear Equations in Two Variables - Question 4

Given :

2 k x + 5 y – 7 = 0  ...( i )
6 x – 5 y – 1 = 0   ... ( ii )
Pair of linear equations has a unique solution.
We know for unique solution.

Comparing from ( i ) and ( ii ) we have

Put these values in formula.

Thus we get answer many values of k but leaving k ≠ -3.

Practice Test: Pair of Linear Equations in Two Variables - Question 5

The pair of equations 3x + 4y = k, 9x + 12y = 6 has infinitely many solutions if –

Detailed Solution for Practice Test: Pair of Linear Equations in Two Variables - Question 5

An equation has infinitely many solutions when the lines are coincident.
The lines are coincident when 
So 3x + 4y = k, 9x + 12y = 6 are coincident when

Practice Test: Pair of Linear Equations in Two Variables - Question 6

The pair of linear equations 2x + 5y = k, kx + 15y = 18 has infinitely many solutions if –

Detailed Solution for Practice Test: Pair of Linear Equations in Two Variables - Question 6

An equation has infinitely many solutions when the lines are coincident.
The lines are coincident when 
So 2x + 5y = k, kx + 15y = 18 are coincident when

Practice Test: Pair of Linear Equations in Two Variables - Question 7

The pair of linear equations 3x + 5y = 3, 6x + ky = 8 do not have any solution if –

Detailed Solution for Practice Test: Pair of Linear Equations in Two Variables - Question 7

Practice Test: Pair of Linear Equations in Two Variables - Question 8

The pair of linear equations 3x + 7y = k, 12x + 2ky = 4k + 1 do not have any solution if

Detailed Solution for Practice Test: Pair of Linear Equations in Two Variables - Question 8

3/12 = 7/2k  [ applying a1/a2=b1/b2 ]
3 x 2k = 7 x 12
k=14

Practice Test: Pair of Linear Equations in Two Variables - Question 9

8 girls and 12 boys can finish work in 10 days while 6 girls and 8 boys can finish it in 14 days. Find the time taken by the one girl alone that by one boy alone to finish the work.       

Practice Test: Pair of Linear Equations in Two Variables - Question 10

The pair of linear equations kx + 4y = 5, 3x + 2y = 5 is consistent only when –

Detailed Solution for Practice Test: Pair of Linear Equations in Two Variables - Question 10

kx + 4y = 5, 3x + 2y = 5
Here, a1=k,b1=4,c1​=−5
and a2=3,b2=2,c2=-5

So , The equation is consistent when 

k ≠ 6

Practice Test: Pair of Linear Equations in Two Variables - Question 11

The pair of linear equations 7x + ky = k, 14x + 2y = k + 1 has infinitely many solutions if –

Practice Test: Pair of Linear Equations in Two Variables - Question 12

The pair of linear equations 13x + ky = k, 39x + 6y = k + 4 has infinitely many solutions if –

Practice Test: Pair of Linear Equations in Two Variables - Question 13

The pair of linear equations x + y = 3, 2x + 5y = 12 has a unique solution x = x1, y = y1 then value of x1 is –

Practice Test: Pair of Linear Equations in Two Variables - Question 14

The pair of linear equations 3x – 5y + 1 = 0, 2x – y + 3 = 0 has a unique solution x = x1, y = y1 then y1 =

Practice Test: Pair of Linear Equations in Two Variables - Question 15

The pair of linear equations x + 2y = 5, 7x + 3y = 13 has a unique solution,

Detailed Solution for Practice Test: Pair of Linear Equations in Two Variables - Question 15

Given equations are:
x+2y=5 ....(1)
7x+3y=13 .....(2)
Multiplying equation (1) by 7
7x+14y=35 .....(3)
Subtracting equation (2) from equation (3), we have
11y=22
y=2
Now putting y value in equation (1)
⇒x+2(2)=5
⇒x=5−4
⇒x=1
x=1,y=2

Practice Test: Pair of Linear Equations in Two Variables - Question 16

The pair of linear equations x + 2y = 5, 3x + 12y = 10 has –

Practice Test: Pair of Linear Equations in Two Variables - Question 17

If the sum of the ages of a father and his son in years is 65 and twice the difference of their ages in years is 50, then the age of the father is –

Practice Test: Pair of Linear Equations in Two Variables - Question 18

A fraction becomes  4/5 when 1 is added to each of the numerator and denominator. However, if we subtract 5 from each then it becomes 1/2. The fraction is –

Practice Test: Pair of Linear Equations in Two Variables - Question 19

Three chairs and two tables cost Rs. 1850. Five chairs and three tables cost Rs. 2850. Then the total cost of one chair and one table is –

Practice Test: Pair of Linear Equations in Two Variables - Question 20

Six years hence a man's age will be three times the age of his son and three years ago he was nine times as old as his son. The present age of the man is –

Detailed Solution for Practice Test: Pair of Linear Equations in Two Variables - Question 20

Let the present age of man is x and of son is y.
Six years hence,
Man’s age =x+6
Son’s age=y+6
Man’s age is 3 times son’s age
x+6=3(y+6)
x+6=3y+18
x=3y+12    …...1
Three years ago,
Man’s age =x-3
Son’s age=y-3
Man’s age was 9 times as of son
x-3=9(y-3)
x-3=9y-27
x=9y-24   ….2
From 1 and 2
3y+12=9y-24
6y=36
y=6
x=3*6+12=18+12=30 years

Practice Test: Pair of Linear Equations in Two Variables - Question 21

The solution of the equations x - y = 2 and x + y = 4 is:

Detailed Solution for Practice Test: Pair of Linear Equations in Two Variables - Question 21

x - y  = 2
x = 2 + y
Substituting the value of x in the second equation we get;
2 + y + y = 4
2 + 2y = 4
2y = 2
Y = 1
Now putting the value of y, we get;
x = 2 + 1 = 3
Hence, the solutions are x = 3 and y = 1.

Practice Test: Pair of Linear Equations in Two Variables - Question 22

The pairs of equations 9x + 3y + 12 = 0 and 18x + 6y + 26 = 0 have

Detailed Solution for Practice Test: Pair of Linear Equations in Two Variables - Question 22

Given, 9x + 3y + 12 = 0 and 18x + 6y + 26 = 0 
a1/a2 = 9/18 = 1/2 
b1/b2 = 3/6 = 1/2 
c1/c2 = 12/26 = 6/13 
Since, a1/a2 = b1/b2 ≠ c1/c2 
So, the pairs of equations are parallel and the lines never intersect each other at any point, therefore there is no possible solution.

Practice Test: Pair of Linear Equations in Two Variables - Question 23

The pairs of equations x + 2y - 5 = 0 and -4x - 8y + 20 = 0 have:

Detailed Solution for Practice Test: Pair of Linear Equations in Two Variables - Question 23

a1/a2 = 1/-4
b1/b2 = 2/-8 = 1/-4
c1/c2 = -5/20 = -¼
This shows:
a1/a2 = b1/b2 = c1/c2
Therefore, the pair of equations has infinitely many solutions.

Practice Test: Pair of Linear Equations in Two Variables - Question 24

If the lines 3x + 2ky – 2 = 0 and 2x + 5y + 1 = 0 are parallel, then what is the value of k?

Detailed Solution for Practice Test: Pair of Linear Equations in Two Variables - Question 24

The condition for parallel lines is:
a1/a2 = b1/b2 ≠ c1/c2
Hence, 3/2 = 2k/5
k = 15/4

Practice Test: Pair of Linear Equations in Two Variables - Question 25

If one equation of a pair of dependent linear equations is -3x + 5y - 2 = 0. The second equation will be:

Detailed Solution for Practice Test: Pair of Linear Equations in Two Variables - Question 25

The condition for dependent linear equations is:
a1/a2 = b1/b2 = c1/c2
For option a,
a1/a2 = b1/b2 ≠ c1/c2 =  ½

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