Here you get the CBSE Class 10 Mathematics chapter 3, Pair of Linear Equations in Two Variables: NCERT Exemplar Problems and Solutions (PartIIIA). This part of the chapter includes solutions of Question Number 1 to 11 from Exercise 3.1 of NCERT Exemplar Problems for Class 10 Mathematics Chapter: Pair of Linear Equations in Two Variables. This exercise comprises of only the Short Answer Type Questions framed from various important topics in the chapter. Each question is provided with a detailed solution.
NCERT Exemplar Solution for CBSE Class 10 Mathematics: Pair of Linear Equations (PartI)
NCERT Exemplar problems are a very good resource for preparing the critical questions like Higher Order Thinking Skill (HOTS) questions. All these questions are very important to prepare for CBSE Class 10 Mathematics Board Examination 20172018 as well as other competitive exams.
NCERT Exemplar Solution for CBSE Class 10 Mathematics: Pair of Linear Equations (PartII)
Find below the NCERT Exemplar problems and their solutions for Class 10 Mathematics Chapter, Pair of Linear Equations in Two Variables:
Exercise 3.3
Short Answer Type Questions (Q. No. 1 to 11)
Quesntion1. For which value(s) of l, do the pair of linear equations lx + y = l^{2} and
x+ly = 1 have
(i) no solution?
(ii) infinitely many solutions?
(iii) a unique solution?
Solution:
Given, lx + y = l^{2} and x + ly 1 ….(i)
Here, a_{1} = l, b_{1}, = 1, c, = l^{2} and a_{2} = 1, b_{2} = l, c_{2} = 1
(iii) Condition for a unique solution is:
Therefore, all real values of l except ±1 equation will have unique solution.
Quesntion2. For which value (s) of k will the pair of equations
kx + 3y = k 3,
12x+ ky = k
have no solution?
Solution:
Given,
kx + 3y – (k 3) = 0 and 12x + ky – k = 0
On comparing with a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 we have
Here, a_{1} = k, b_{1}=3 and c_{1} = (k 3)
And a_{2} = 12, b_{2} = k and c_{2} = k …..(i)
Condition for a pair of linear equations to have no solution is:
Thus, from (iii) and (iv), it’s clear that at k =  6 the given pair of linear equations will have no solution.
Quesntion3. For which values of a and b will the following pair of linear equations has infinitely many solutions?
x + 2y = 1
(a  b) x + (a + b) y = a + b  2
Solution:
Given equation are:
x + 2y −1 = 0 and (a  b) x + (a + b) y – (a + b  2) = 0 ….(i)
On comparing with a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 we have:
Quesntion4. Find the values of p in (i) to (iv) and p and q in (v) for the following pair of equations
(i) 3x y  5 = 0 and 6x  2y  p = 0, if the Lines represented by these
equations are parallel.
(ii) x + py = 1 and px y = 1, if the pair of equations has no solution.
(iii) 3x + 5y = 7 and 2px  3y =1
if the lines represented by these equations are intersecting at a unique point.
(iv) 2x + 3y 5 = 0 and px  6y  8 = 0,
if the pair of equations has a unique solution.
(v) 2x + 3y = 7 and 2px + py = 28 qy,
if the pair of equations has infinitely many solutions.
Solution:
(i) Given, 3x  y 5 = 0 and 6x2y  p = 0
On comparing with a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 we have:
c_{1} = 5 and a_{2} = 6, b_{2} = 2, c_{2} = p
As, the lines represented by these equations are parallel, therefore
Therefore, the given pair of linear equations are parallel for all real values of p except 10.
(ii) Given, x + py 1 = 0 and pxy1 = 0 …..(i)
We have, a_{1} = 1, b_{1} = p, c_{1}= 1and a_{2} = p, b_{2} = 1 and c_{2} = 1
As, the pair of linear equations has no solution i.e., both lines are parallel to each other.
Therefore, the given pair of linear equations has no solution for p = 1.
(iii) Given, 3x + 5y  7 = 0 and 2px3y1 = 0
Here, a_{1} = 3, b_{1} = 5, c_{1} = 7 and a_{2} = 2p, b_{2} = 3, c_{2} = 1
As, the lines are intersecting at a unique point i.e., it has a unique solution.
Therefore, the lines represented by these equations are intersecting at a unique point for all great values of p except 9/10.
(iv) Given, 2x + 3y5 = 0 and px  6y 8 = 0
Here,c_{1} = 5 and a_{2} = p, b_{2} = 6, c_{2} = −8
As, the pair of linear equations has a unique solution.
Given, the pair of linear equations has a unique solution for all values of p except 4.
(v) Given, 2x + 3y = 7 and 2px + py = 28 qy
Here, a_{1} = 2, b_{2}, = 3, c_{1} _{= }7 and a_{2} = 2p, b_{2} = (p + q), c_{2} = 28
Since, the pair of equations has infinitely many solutions i.e., both lines are coincident.
⟹ 4 + q =12
Therefore, q = 8
Therefore, the pair of equations has infinitely many solutions for the values of p = 4and q = 8
Quesntion5. Two straight paths are represented by the equations x  3y = 2 and 2x + 6y = 5. Check whether the paths cross each other or not.
Solution:
Hence, two straight paths represented by the given equations never cross each other because they are parallel to each other.
Quesntion6. Write a pair of linear equations which has the unique solution
x =  1 and y 3. How many such pairs can you write?
Solution:
Hence, infinitely many pairs of linear equations are possible.
Quesntion7. If 2x + y = 23 and 4x  y = 19, then find the values of 5y 2x and
Solution:
Quesntion8. Find the values of x and y in the following rectangle
Quesntion9. Solve the following pairs of equations
Solution:
Hence, the required values of x and y are 1.2 and 2.1, respectively
Hence, the required values of x and y are 6 and 8, respectively
Hence, the required values of x and y are 3 and 2, respectively.
(v) Given pair of linear equations is
43x + 67y = 24 ...(i)
And 67x + 43y = 24 ...(ii)
Now taking 43 × (i) + 67 ×(ii), we get
Hence, the required values of x and y are 1 and 1, respectively.
(vi) Given pair of linear equations is
Hence, the required values of x and y are a^{2} and b^{2}, respectively.
Now, multiplying both sides of equation (i) by LCM (10, 5) = 10, we get
x + 2y – 10 = 0
Þ x + 2y = 10 ... (iii)
Again, multiplying both sides of (iv) by LCM (8, 6) = 24, we get
3x + 4y = 360 ... (iv)
Now, (iv) – 2 × (iii) gives:
Hence, the solution of the pair of equations is x = 340, y =  165 and the required value of l is
Quesntion11. By the graphical method, find whether the following pair of equations are consistent or not. If consistent, solve them.
(i) 3x + y + 4 = 0, 6x  2y + 4 = 0
(ii) x  2y = 6, 3x  6y = 0
(iii) x + y = 3, 3x + 3y = 9
Solution:
(i) Given pair of equations is:
3x+ y + 4= 0 and 6x  2y + 4 = 0
On comparing with a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 we have
and c_{1} = 4 and a_{2} = 6, b_{2} =  2 and c_{2} = 4
So, the given pair of linear equations has a unique solution and thereby it is consistent.
We have, 3x + y + 4 = 0
Þ y =  4  3x ….(i)
If x = 0, y =  4
x =  1, y =  1
x =  2, y = 2
x 
1 
2 

y 
4 
1 
2 
Points 
B 
C 
A 
And 6  2 + 4 = 0 …..(ii)
Þ 2y = 6x + 4
Þ y = 3x + 2
If x = 0, y = 2
x = 1, y =  1
x = 1, y = 5
x 
1 
1 

y 
1 
2 
5 
Points 
C 
Q 
P 
Plotting (i) and (ii) as per the respective values of x and y, we get two lines AB and PQ respectively that intersect at C (1.1).
Thus, the given pair of linear equations has no solution, i.e., inconsistent.
(iii) Given pair of equations is:
x + y = 3 and 3x + 3y = 9
On comparing with a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 we have
a_{1} = 1, b_{1} = 1 and c_{1}, =  3 and a_{2} = 3, b_{2} = 3 andc_{2} =  9
So, the given pair of lines is coincident. Therefore, these lines have. Hence, the given pair of linear equations has infinitely many solutions, i.e., consistent.
Now, x + y = 3 ….(i)
Or y = 3  x
If x = 0, y = 3
x = 3, y = 0
x

3 

y 
3 

Points 
A 
B 
Also 3x + 3y = 9 ….(ii)
Or 3y = 9  3x
Plotting (i) and (ii) for respective set of values for x and y, we get two lines AB and CD respectively, that are coincident.
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