Important Questions for CBSE Class 10 Maths Chapter 3 - Pair of Linear Equations in Two Variables 2 Mark Question

CBSE Class 10 Maths Chapter-3 Pair of Linear Equations in Two Variables – Free PDF Download

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CBSE Class 10 Maths Chapter-3 Pair of Linear Equations in Two Variables Important Questions

CBSE Class 10 Maths Important Questions Chapter 3 – Pair of Linear Equations in Two Variables

2 Mark Questions

1. Half the perimeter of a rectangle garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
Ans. Let length of rectangular garden = x metres
Let width of rectangular garden = y metres
According to given conditions, perimeter = 36 m
 x + y = 36 ……(i)
And x = y+4
⇒ x– y=4 ……..(ii)
Adding eq. (i) and (ii),
2x = 40
 x = 20 m
Subtracting eq. (ii) from eq. (i),
2y = 32
 y = 16 m
Hence, length = 20 m and width = 16 m

2. Draw the graphs of the equations x – y+1=0 and 3x+2y – 12=0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.
Ans. For equation x – y + 1 = 0, we have following points which lie on the line

For equation 3x + 2y – 12 = 0, we have following points which lie on the line

We can see from the graphs that points of intersection of the lines with the x–axis are (–1, 0), (2, 3) and (4, 0).

3. The age of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju.
Ans. Let the age of Ani and Biju be x years and y years respectively.
Age of Dharam = 2x years and Age of Cathy =  years
According to question,
x – y = 3 … (1)
And 
 4x – y = 60    … (2)
Subtracting (1) from (2), we obtain:
3x = 60 – 3 = 57
 x = Age of Ani = 19 years
Age of Biju = 19 – 3= 16 years
Again, According to question,   y – x= 3 … (3)
And 
 4x – y = 60 … (4)
Adding (3) and (4), we obtain:
3x = 63
 x = 21
Age of Ani = 21 years
Age of Biju = 21 + 3 = 24 years

4. One says, “Give me a hundred, friend! I shall then become twice as rich as you.” The other replies, “If you give me ten, I shall be six times as rich as you.” Tell me what is the amount of their (respective) capital? [From the Bijaganita of Bhaskara II]
Ans. Let the money with the first person and second person be Rs x and Rs y respectively.
According to the question,
x + 100 = 2(y – 100)
 x + 100 = 2y – 200
 x – 2y = 300 … (1)
Again,   6(x – 10) = (y + 10)
 6x – 60 = y + 10
 6x – y = 70 … (2)
Multiplying equation (2) by 2, we obtain:
12x – 2y = 140 … (3)
Subtracting equation (1) from equation (3), we obtain:
11x = 140 + 300
 11x = 440
 x = 40
Putting the value of x in equation (1), we obtain:
40 – 2y = –300
 40 + 300 = 2y
 2y = 340
 y = 170
Thus, the two friends had Rs 40 and Rs 170 with them.

5. The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.
Ans. Let the number of rows be x and number of students in a row be y.
Total number of students in the class = Number of rows x Number of students in a row = xy
According to the question,
Total number of students = (x – 1) (y + 3)
 xy = (x – 1) (y + 3)
 xy= xy – y + 3x – 3
 3x – y – 3 = 0
 3x – y = 3 … (1)
Total number of students = (x + 2) (y – 3)
 xy = xy + 2y – 3x – 6
 3x – 2y = –6 … (2)
Subtracting equation (2) from (1), we obtain:
y = 9
Substituting the value of y in equation (1), we obtain:
3x – 9 = 3
 3x = 9 + 3 = 12
 x = 4
Number of rows = x = 4
Number of students in a row = y = 9
Hence, Total number of students in a class = xy = 4 x 9 = 36

6. Find the values of and for which the following system of linear equations has infinite number of solutions, 2x + 3y = 7, 2x + y = 28.
Ans. 

7. Find the condition for which the system of equations and bx + ay = 4ab (a, b0) is inconsistent.
Ans. 

8. Find the value of so that the following linear equations have no solution

Ans. No solution

9. Solve for x and y: ax + by = a – b and bx – ay = a + b
Ans. 

10. The path of a train A is given by the equation x + 2y – 4 = 0 and the path of another train B is given by the equation 2x + 4y – 12 = 0 represent this situation graphically.
Ans. 

11. For what value of ‘’the system of linear equations .x + 3y =– 3, 12x +y = has no solution.
Ans. 

12. Find the values of ‘a’ and ‘b’ for which the following system of linear equations has infinite number of solutions. 2x + 3y = 7, (a + b + 1) x +(a + 2b + 2) y = 4 (a + b) +1
Ans. 

13. Solve for ‘x’ and ‘y’ where x + y = a – b, ax – by = a² + b²
Ans. 

14. The cost of two kg of apples and 1 kg of grapes on a day was found to be Rs. 160. After a month the cost of 4 kg apples and 2 kg grapes is Rs. 300. Represent the situation algebraically and graphically.
Ans. Let the cost of one Kg of apple is x and one Kg of grapes is y.
According to question,
2x + y = 160 and 4x + 2y = 300
2x + y = 160

4x + 2y = 300

15. Find the value of ‘k’ for which the system of equation kx + 3y = k – 3 and 12x + ky = k will have no solution.
Ans. 

16. ABCD is a rectangle, find the values of x and y.

Ans. 

17. Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variable such that the geometrical representation of the pair so formed is
(a) intersecting lines
(b) parallel lines
(c) overlapping
Ans. 2x + 3y – 8 = 0 another linear equation representing.

18. Find the value of ‘k’ for which the system of equation has infinitely many solutions 2x + (k – 2) y = k and 6x + (2k – 1) y = 2k + 5
Ans. 

19. Find the relation between a, b, c and d for which the equations ax + by = c and cx + dy = a have a unique solution. 
Ans. 

20. Solve for ‘x’ and ‘y’:
(a – b) x + (a + b) y = a² – b² – 2ab
(a + b) (x + y) = a² + b²
Ans.