RD Sharma Class 10 Solutions Chapter 3 Exercise 3.5 (Updated for 2023-24)

x – 3y = 3
3x – 9y = 2 (C.B.S.E. 1994)
Solution:

Question 2.
2x + y = 5
4x + 2y = 10 (C.B.S.E. 1995C)
Solution:

Question 3.
3x – 5y = 20
6x – 10y = 40 (C.B.S.E. 1993)
Solution:

Question 4.
x – 2y = 8
5x – 10y = 10 (C.B.S.E. 1993)
Solution:

Find the value of k for which the following system of equations has a unique solution: (5 – 8)
Question 5.
kx + 2y = 5
3x + y = 1 (C.B.S.E. 1990C, 92C)
Solution:

Question 6.
4x + ky + 8 = 0
2x + 2y + 2 = 0 [NCERT]
Solution:

Question 7.
4x – 5y = k
2x – 3y = 12
Solution:

Question 8.
x + 2y = 3
5x + ky + 7 = 0
Solution:

Find the value of k for which each of the following systems of equations has infinitely many solutions: (9 – 19)
Question 9.
2x + 3y – 5 = 0
6x + ky – 15 = 0 (C.B.S.E. 1991)
Solution:

Question 10.
4x + 5y = 3
kx + 15y = 9 (C.B.S.E. 1990C)
Solution:

Question 11.
kx – 2y + 6 = 0
4x – 3y + 9 = 0
Solution:

Question 12.
8x + 5y = 9
kx + 10y = 18 (C.B.S.E. 1999)
Solution:

Question 13.
2x – 3y = 7
(k + 2) x + (2k + 1) y = 3 (2k – 1) (C.B.S.E. 1999)
Solution:

Question 14.
2x + 3y = 2
(k + 2)x + (2k + 1) y = 2 (k – 1) (C.B.S.E. 2000, 2003)
Solution:

Question 15.
x + (k + 1) y = 4
(k + 1) x + 9y = 5k + 2 (C.B.S.E. 2000C)
Solution:

Question 16.
kx + 3y = 2k + 1
2(k+ 1) x + 9y = 7k + 1 (C.B.S.E. 2000C)
Solution:

Question 17.
2x + (k – 2) y = k
6x + (2k – 1) y = 2k + 5 (C.B.S.E. 2000C)
Solution:

Question 18.
2x + 3y = 7
(k + 1) x + (2k – 1)y = 4k + 1 (C.B.S.E. 2001)
Solution:

Question 19.
2x + 3y = k
(k – 1) x + (k + 2) y = 3k (C.B.S.E. 2001)
Solution:

Find the value of k for which the following system of equations has no solution : (20 – 25) :
Question 20.
kx – 5y = 2
6x + 2y = 1 (C.B.S.E. 1994C)
Solution:

Question 21.
x + 2y = 0
2x + ky = 5 (C.B.S.E. 1993C)
Solution:

Question 22.
3x – 4y + 7 = 0
kx + 3y – 5 = 0 (C.B.S.E. 1996)
Solution:

Question 23.
2x – ky + 3 = 0
3x + 2y – 1 = 0 (C.B.S.E. 1996)
Solution:

Question 24.
2x + ky = 11
5x – 7y = 5 (C.B.S.E. 1995)
Solution:

Question 25.
kx + 3y = 3
12x + ky = 6
Solution:

Question 26.
For what value of k, the following system of equations will be inconsistent?
4x + 6y = 11
2x + ky = 1 (C.B.S.E. 1994C)
Solution:

Question 27.
For what value of a system of equations
αx + 3y = α – 3
12x + αy = α
will have no solution. (C.B.S.E. 2003)
Solution:

Question 28.
Find the value of k for which the system
kx + 2y = 5
3x + y = 1
has (i) a unique solution, and (ii) no solution.
Solution:

k = 6

Question 29.
Prove that there is a value of c (≠ 0) for which the system
6x + 3y = c – 3
12x + cy = c
has infinitely many solutions. Find this value.
Solution:

Question 30.
Find the values of k for which the system
2x + k y = 1
3x – 5y = 7
will have (i) a unique solution, and (ii) no solution.
Is there a value of k for which the system has infinitely many solutions?
Solution:

Question 31.
For what value of k, the following system of equations will represent the coincident lines?
x + 2y + 7 = 0
2x + ky + 14 = 0 (C.B.S.E. 1992)
Solution:

Question 32.
Obtain the condition for the following system of linear equations to have a unique solution
ax + by = c
lx + my = n (C.B.S.E. 1991C)
Solution:

Question 33.
Determine the values of a and b so that the following system of linear equations has infinitely many solutions.
(2a – 1) x + 3y – 5 = 0
3x + (b – 1) y – 2 = 0 (C.B.S.E. 2001C)
Solution:

Question 34.
Find the values of a and b for which the following system of linear equations has an infinite number of solutions :
2x – 3y = 7
(a + b) x – (a + b – 3) y = 4a + b (C.B.S.E. 2002)
Solution:

Question 35.
Find the values of p and q for which the following system of linear equations has an infinite number of solutions:
2x + 3y = 9
(p + q) x + (2p – q) y = 3 (p + q + 1)
Solution:

Question 36.
Find the value of a and b for which the following system of equations has infinitely many solutions :
(i) (2a – 1) x – 3y = 5
3x + (b – 2) y = 3 (C.B.S.E. 2002C)
(ii) 2x – (2a + 5) y = 5
(2b + 1) x – 9y = 15 (C.B.S.E. 2002C)
(iii) (a – 1) x + 3y = 2
6x + (1 – 2b) y = 6 (C.B.S.E. 2002C)
(iv) 3x + 4y = 12
(a + b) x + 2 (a – b) y = 5a – 1 (C.B.S.E. 2002C)
(v) 2x + 3y = 7
(a – b) x + (a + b) y = 3a + b – 2
(vi) 2x + 3y – 7 = 0 [CBSE 2010]
(a – 1) x + (a + 1) y = (3a – 1)
(vii) 2x + 3y = 7
(a – 1) x + (a + 2) y = 3a [CBSE 2010]
(viii) x + 2y = 1
(a – b) x + (a + b) y = a + b – 2 [NCERT Exemplar]
(ix) 2x + 3y = 7
2ax + ay = 28 – by [NCERT Exemplar]
Solution:

Question 37.
For which value(s) of λ, do the pair of linear equations λx + y = λ² and x + λy = 1 have
(i) no solution?
(ii) infinitely many solutions?
(iii) unique solutions? [NCERT Exemplar]
Solution:

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