Pair of linear equations in two variables Class 10

TOPIC WISE DPP
DPP-O1

1. In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions.

(i) 2x + 3y = 7                         (ii) 6x + 5y = 11                      (iii) -3x + 4y = 5

6x + 5y = 11                               9x +y = 21                       x – 6y +  = 0

2. For each of the following systems of equations determine the value of k for which the given system of equations has a unique solution:

(i) x-ky = 2                                          (ii) 2x – 3y = 1

3x + 2y = -5                                        kx + 5y = 7

(iii) 2x + 3y – 5 = 0                             (iv) 2x + ky = 1

kx – 6y – 8 = 0                                    5x – 7y = 5

3. For each of the following systems of equations determine the value of k for which the given system of equations has infinitely many solutions.

(i) 5x + 2y = k                                                 (ii) (k – 3) x + 3y = k

   10x + 4y = 3                                                      kx + ky = 12

(iii) kx + 3y = k – 3

     12x + ky = k                     

4. For each of the following system of equations determine the values of k for which the given system has no solution:

(i) 3x-4y + 7 = 0                                             (ii) 2x-ky + 3 = 0

     kx + 3y – 5 = 0                                               3x + 2y – 1 = 0

5 Find the value(s) of k for which the system of equations

  kx – y = 2

  6x – 2y = 3

has (i) a unique solution (ii) no solution.

Is there a value of k for which the system has infinitely many solutions?

6. For what value of k will the equations x + 2y + 7 = 0, 2x + ky + 14 = 0 represent coincident lines ?

7. For what value of k, will the following system of equations have infinitely many solutions?

2x + 3y = 4  ,     (k + 2) x + 6y = 3k + 2

8. For what value of k, will the following system of equations have infinitely many solutions?

2x + 3y = 4

(k + 2) x + 6y = 3k + 2

9. Determine the values of a and b for which the following system of linear equations has infinite solutions:

2x – (a – 4) y = 2b + 1

4x – (a -1) y = 5b -1

10. For what value of k will the following system of linear equations has no solution ?

3x + y = 1

(2k- 1)x + (k – 1)y = 2k + 1                                                                [NCERT, CBSE 2000]

11 Find the value of k for which the following system of linear equations has infinite solutions:

x + (k + 1) y = 5

(k + 1) x + 9y = 8k – 1                                                                        [CBSE 2002C]

12 Find the values of p and q for which the following system of equations has infinite number of solutions:

2x + 3y = 7

(p + q)x + (2p – q)y = 21                                                                                [CBSE 2001]

13. For what value of k, will the system of equations

x + 2y = 5

3x + ky -15 = 0.

has (i) unique solution? (ii) no solution ?                                                       [CBSE 2001]

14. Find the values of α and β for which the following system of linear equations has infinite number of solutions:

2x + 3y = 7

2ax + (α + β)y = 28

has (i)’a unique solution? (ii) no solution?                                                     [CBSE 2001]

15. Determine the values of m and n so that the following system of linear equations have infinite number of solutions:

(2m – 1)x + 3y -5 = 0

3x + (n – 1) y – 2 = 0

16. Determine the value of k so that the following linear equations have no solution:

(3k + 1) x + 3y – 2 = 0

(k²+ 1) x + (k – 2) y – 5 = 0                                                                            [CBSE 2001C]

17. Find the value of k for which each of the following systems of equations have infinitely many solution

 a. 2x + 3y = 2 ,    (k + 2) x + (2k +1) y = 2                                                   [CBSE 2000, 2003]

b. x + (k +1) y = 4    ,  (k +1) x + 9y = 5k + 2                                                [CBSE 2000C]

c. kx + 3y = 2k +1 ,     2(k + 1)x + 9y = 7k + 1                                  [CBSE 2000C]

d. 2x + (k – 2) y = k ,     6x + (2k-1)y = 2k + 5                                               [CBSE 2000C]

e. 2x + 3y = 7 ,    (k + 1) x + (2k – 1) y = 4k + 1                                [CBSE 2001]

19. 2x + 3y = k ,      (k-1)x + (k + 2)y = 3k                                        [CBSE 2001]

18. For what value of α, the system of equations

αx + 3y = a – 3  ,   12x + αy = α                                                         [CBSE 2003, 2009]

will have no solution?

19. Determine the values of a and b so that the following system of linear equations have infinitely many solutions:

(2a – 1) x + 3y – 5 = 0,      3x + (b-1) y – 2 = 0                                                [CBSE 2001C]

20. 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) x – (a + b – 3) y = 4a + b                                   [CBSE 2002]

21. Find the values of p and q for which the following system of linear equations has infinite number of solutions:

2x + 3y = 9 ,       (p + q) x + (2p – q) y = 3 (p + q + 1)                                   [CBSE 2002]

22. Find the values of α and b for which the following system of equations has infinitely many solutions:

(i) (2a – 1)x – 3y = 5                                        (ii) 2x – (2a + 5)y = 5

3x + (p – 2)y = 3         [CBSE 2002C]           (2b + 1)x – 9y = 15                 [CBSE 2002C]

(iii)(a – 1)x + 3y = 2                                        (iv) 3x + 4y = 12

6x + (1 -2b)y = 6        [CBSE2002C]            (a + b)x + 2(a -b)y = 5a -1            [CBSE2002C]

(v) 2x + 3y – 7 = 0,     (a – 1)x + (a+ 1)y = (3a – 1)        [CBSE 2010]

(vi) 2x + 3y = 7   ,      (a – 1)x + (a + 2)y = 3a            [CBSE 2010]