Find below the NCERT Exemplar problems and their solutions for Class 10 Mathematics Chapter, Pair of Linear Equations in Two Variables:

Exercise 3.1

Multiple Choice Questions (MCQs)

Quesntion1. Graphically, the pair of equations

6x - 3y + 10 = 0

2x - y + 9 = 0

represents two lines which are

(a)    intersecting at exactly one point

(b)   intersecting exactly two points

(c)    coincident

(d)   parallel

Solution: (d)  

Explanation:

On comparing with a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 we have a₁ = 6, b₁ = ‒3, c₁ = 10 and a₂ = 2, b₂y = ‒1, c₂ = 9.

So, the given system of equations have no solution, this will be possible if two lines never intersect each other.

In other words both the lines are parallel to each other.

Rough figure is shown below for reference,

Quesntion2.The pair of equations x + 2y + 5 = 0 and - 3x- 6y + 1 = 0 has

(a) a unique solution

(b) exactly two solutions

(c) infinitely many solutions

(d) no solution

Solution: (d)

Given, equations are x + 2y + 5 = 0 and - 3x - 6y + 1 = 0

On comparing with a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, we have

Therefore, the pair of equations has no solution.

Quesntion3. If a pair of linear equations is consistent, then the lines will be

(a) parallel

(b) always coincident

(c) intersecting or coincident

(d) always intersecting

Solution: (c)   

If two lines are parallel then, they have no solution pair of linear equations is inconsistent;

If two lines are coincident then, they have infinite solution and pair of linear equations is consistent;

If two lines are intersecting then, they have unique solution and pair of linear equations is consistent.

Quesntion4.The pair of equations y = 0 and y = ‒7 has

(a) one solution

(b) two solutions

(c) infinitely many solutions  

(d) no solution

Solution: (d)  

If we draw a rough graph of both the lines (as shown below) we will find that both lines are parallel. So, they have no solution.

Quesntion5. The pair of equations x = a and y = b graphically represents lines which are

(a) parallel

(b) intersecting at (b, a)

(c) coincident 

(d) intersecting at (a, b)

Solution: (d)

x = a is the equation of a straight line which is parallel to the y–axis lying at a distance ‘a’ from it. Also, y = b is the equation of a straight line parallel to the x–axis lying at a distance ‘b’ from it. Thus the two lines come to intersect at a point (a, b).

Quesntion6. For what value of k, do the equations 3x -y + 8 = 0 and 6x -k y = -16 represent coincident lines?

Quesntion7. If the lines given by 3x+ 2ky = 2 and 2x + 5y = 1 are parallel, then the value of k is

Quesntion8. The value of c for which the pair of equations cx - y = 2 and 6x - 2y = 3 will have infinitely many solutions is

(a) 3

(b) -3

(c) -12

(d) no value

Solution: (d)  

Condition for two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have infinitely many solutions is:

As, c has different values, therefore, there will be no value of c for which the pair of equations will have infinitely many solutions.

Quesntion9. One equation of a pair of dependent linear equations is - 5x + 7y - 2 = 0. The second equation can be

(a) 10x + 14y + 4 = 0 

(b) 10x -14y + 4 = 0

(c) 10x + 14y + 4=0

(d) 10x -14y + 4=0

Solution: (d)  

Condition for two linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to be dependent is:

Quesntion10. A pair of linear equations which has a unique solution x = 2 and y = - 3 is

(a) x + y = 1 and 2x + 3y = - 5

(b) 2x + y = - 11 and 4x + 10y = - 22

(c) 2x - y = 1 and 3x - 2y = 0

(d) x - 4y - 14 = 0 and 5x - y - 13 = 0

Solution: (b)    

If x = 2, y = ‒3 is a unique solution of a pair of equations, then these values will satisfy both pair of equations.

So, we will check by putting x = 2, y = ‒3 in all options and check which one of the four options is correct.

In option (b),

Putting x = 2, y = ‒3 in LHS of equation, 2x + y = - 11

⇒ 2(2) + 5(-3) = 4 - 15 = 11 = RHS

Putting x = 2, y = ‒3 in LHS of equation 4x+ 10y = ‒22

⇒4 (2) + 10(-3) = 8 - 30 = -22 = RHS

Quesntion11. If x = a and y = b is the solution of the equations x - y = 2 and x + y = 4, then the values of a and b are, respectively

(a) 3 and 5

(b) 5 and 3

(c) 3 and 1

(d) 1 and -3

Solution: (c)   

As, x = a and y = b being the solution of the equations x - y = 2 and x + y = 4, must satisfy both the equations.

Thus, on putting x = a and y = b in both the equations, we get:

            a - b = 2 …(i)

And      a + b = 4 ...(ii)

On solving equations (i) and (ii), we get 2a = 6 or a = 3.

⟹ a = 3 and b = 1.

Quesntion12. Aruna has only Rs. 1 and Rs. 2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is Rs. 75, then the numbers of Rs. 1 and Rs.2 coins are, respectively

(a) 35 and 15

(b) 35 and 20

(c) 15 and 35

(d) 25 and 25

Solution: (d)

Let total number of Rs. 1 coins = x

And total number of Rs. 2 coins = y

Now, according to the conditions given in question x + y = 50 …(i)

Also,    x × 1+ y×2=75

Þ        x + 2y = 75      ...(ii)

Solving (i) and (ii)

(x + 2y) - (x + y) = 75 - 50

Þ        y = 25

As, x + y =50

Therefore, for y = 25, we get x = 25.

Quesntion13. The father's age is six times his son's age. Four years hence, the age of the father will be four times his son's age. The present ages (in year) of the son and the father are, respectively

(a) 4 and 24

(b) 5 and 30

(c) 6 and 36    

(d) 3 and 24

Solution: (c)

Let present age of father = x

And present age of son = y  

It is given in question that, x = 6y      …(i)

According to the question,

(x + 4) = 4 (y + 4)

⇒ x - 4y = 12  ...(ii)

Using equations (i) and (ii), we get:

        6y - 4y = 12

Þ    2y = 12

Þ    y = 6

If y = 6, then x = 36

Therefore, present age of father is 36 year and present age of son is 6 year.