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Sub : Maths
Hirav Trivedi’s Group TuiTions
CH : 3 Pair of linear equations
Batch : X
SECTION - A
1. The pair of equations 6x – 7y = 1 and 3x – 4y = 5 has
(a) a unique solution
(b)
two solutions
(c) infinitely many solutions
(d)
no solution
Sol:
2. The number of solutions of the pair of equations 2x – 5y = 10 and 6x + 15y – 30 = 0 is
(a)
0
(b)
1
(c)
2
(d)
infinite
3. The value of k for which the system of equations kx – y = 2, 6x – 2y = 3 has a unique solution is
21
1
4
(a)
(b)
(c)
6
(d)
4
21
6
4. The value of k for which the system of equations kx – y = 2, 6x – 2y = 3 has a unique solution is
(a)
=0
(b)
=3
(c)
0
(d)
3
5. If the system of equations 2x + 3y = 7 (a + b)x + (2a – b )y = 21, has infinitely many solutions, then
(a) a = 1, b = 5
(b)
a = -1, b = 5 (c)
a = 5, b = 1 (d)
a = 5, b = -1
6. If am  bl , then the system of equations ax + by = c, lx + my = n
(a) has a unique solution
(b)
has no solution
(c) has infinitely many solutions
(d)
may or may not have a solution
7. If 2x – 3y = 7 and (a + b)x – (a + b – 3)y = 4a + b represent coincident lines, then a and b satisfy the
equation
(a) a + 5b = 0
(b)
5a + b = 0
(c)
a – 5b = 0
(d)
5a – b = 0
8. The pair of equations x = a and y = b graphically represent lines which are
(a) parallel
(b)
intersecting at (b, a) (c)
coincident
(d)
intersecting at
9. If the lines given by 3x + 2ky = 2 and 2x + 5y + 1 = 0 are parallel, then the value of k is
5
3
2
15
(a)
(b)
(c)
(d)
4
5
2
4
10. A pair of linear equations which has a unique solution x = 3, y = -2 is
(a) x + y = -1
2x – 3y = 12
(b)
2x + 5y + 4 = 0 4x + 10y + 8 = 0
(c) 2x – y = 1 3x + 2y = 0
(d)
x – 4y = 14
5x – y = 13
11. Gunjan 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 number of rs 1, and rs 2 coins are respectively.
(a) 25 and 25 (b)
15 and 35
(c)
35 and 15
(d)
35 and 20
12. The sum of the digits of a two digit number is 12. If 18 is subtracted from it, the digits of the number
get reversed. The number is
(a) 57
(b)
75
(c)
84
(d)
48
13. The number of solutions of the pair of linear equations x + 3y – 4 = 0 and 2x + 6y = 7 is
(a)
0
(b)
1
(c)
2
(d)
infinite
14. A pair of linear equations which has x = 0, y = -5, as a solution is
(a) x + y + 5 = 0 2x + 3y = 10
(b)
x + y = 3 2x – y = 5
(c) 2x + y + 5 = 0 3y = x - 15
(d)
3x + 4y = -15 4x – 3y = -15
15. The value of k for which the lines (k+1)x + 3ky +15 + 0 and 5x + ky + 5 = 0 are coincident is
(a)
14
(b)
2
(c)
-14
(d)
-2
16. The value of γ for which the system of equations 5γx – 2y + 1 and 10x + y = 3 has a unique solution is
(a)
=4
(b)
4
(c)
= -4 (d)
 -4
Sub : Maths
Hirav Trivedi’s Group TuiTions
CH : 3 Pair of linear equations
Batch : X
17. The value of γ for which the system of equations 2x + y – 3 = 0 and 5x +ky +7 = 0 has no solution is
5
3
(a)
2
(b)
5
(c)
(d)
2
7
18. If the system of equations 4x + y = 3 and (2k – 1)x + (k – 1)y = 2k + 1 is inconsistent, then k =
2
3
2
3
(a)
(b)
(c)
(d)
2
2
3
3
19. If the system of equations 4x + 3y = 9 2ax + (a + b)y = 18 has infinitely many solutions, then
(a)
b = 2a
(b)
a = 2b
(c)
a +2b = 0
(d)
2a – b = 0
20. The value of k for which the system equation 2x + 3y = 7 and 8x + ( k + 4 )y - 28 = 0 has infinitely
many solution is
(a)
-8
(b)
8
(c)
3
(d)
-3
21. 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
22. A’s age is six times B’s age. Four years hence, the age of A will be four times B’s age. The present
ages, in years, of A and B are respectively
(a) 3 and 24
(b)
36 and 6
(c)
6 and 36
(d)
4 and 24
23. The sum of the digits of a two digit number is 14. If 18 is added to the number, the digit get reversed.
The number is
(a) 95
(b)
59
(c)
68
(d)
86
24. Two numbers are in ratio 1:3. If 5 is added to both the numbers, the ratio becomes 1:2. The numbers
are
(a) 4 and 12
(b)
5 and 15
(c)
6 and 18
(d)
7 and 21
25. A pair of linear equations in two variables cannot have
(a) a unique solution
(b)
no solution
(c) infinitely many solutions
(d)
exactly two solutions
26. The pair of equations 3x _ 2y = 5 and 6x _ y = 3 have
(a) no solution
(b)
a unique solution
(c) two solutions
(d)
infinitely many solutions
27. If a pair of linear equations is inconsistent, then the lines representing them will be
(a) parallel
(b)
always coincident
(c) intersecting or coincident
(d)
always intersecting
28. If a pair of linear equations has infinitely many solutions, then the lines representing them will be
(a) parallel
(b)
intersecting or coincident
(c) always intersecting
(d)
always coincident
29. The pair of equations 4x – 3y + 5 = 0 and 8x – 6y – 10 = 0 graphically represents two lines in which
are
(a) coincident
(b)
parallel
(c) intersecting at exactly one point (d)
intersecting at exactly two points
30. The pair of equations y = a and y = b graphically represents lines which are
(a) intersecting at (a, b) (b) intersecting at (b, a)
(c)
parallel
(d)
coincident
31. The pair of equations x = 2 and y = 3 has
(a) one solution (b) one solution
(c)
parallel
(d)
coincident
32. The value of k for which the pair of equations kx + y = 3 and 3x + 6y = 5 has a unique solution is
1
(a) 
(b)
2
(c)
-2
(d)
all the above
2
Sub : Maths
Hirav Trivedi’s Group TuiTions
CH : 3 Pair of linear equations
Batch : X
33. If the pines given by 3x + 2ky = 2 and 2x + 5y + 1 = 0 are parallel, then the value of k is
15
3
2
5
(a)
(b)
(c)
(d)

5
4
2
4
34. One equation of a pair of dependent linear equations is 3x – 4y = 7. The second equation can be
(a) - 6x + 8y = 14
(b)
-6x + 8y + 14 = 0
(c) 6x + 8y = 14
(d)
-6x – 8y – 14 = 0
35. If x = a and y = b is the solution of the equations x + y = 5 and x – y = 7, then values of a and b are
respectively
(a) 1 and 4
(b)
6 and - 1
(c)
- 6 and 1
(d)
- 1 and – 6
36. A pair of linear equations which has a unique solution x = - 1, y = - 2 is
(a) x – y = 1 ; 2x + 3y = 5
(b)
2x – 3y = 4 ; x – 5y = 9
(c) x + y – 3 = 0 ; x – y = 1
(d)
x + y +3 = 0 ; 2x – 3y + 5 = 0
37. Sanya’s age is three times her sister’s age. Five years hence, her age will be twice her sister’s age. The
present ages (in years) of sanya and her sister are respectively
(a) 12 and 4
(b)
15 and 5
(c)
5 and 15
(d)
4 and 12
38. The sum of the digits number is 8. If 18 is added to it, the digits of the number get reversed. The
number is
(a) 53
(b)
35
(c)
62
(d)
26
39. Divya has only rs 2 and rs 5 coins with her. If the total number of coins that she has is 25 and the
amount of money with her is rs 80, then the number of rs 2 and rs 5 coins are, respectively
(a) 15 and 10
(b)
10 and 15
(c)
12 and 10
(d)
13 and 12
40. The pair of equations 6x – 4y + 9 = 0 and 3x – 2y + 9 = 0 and 3x – 2y + 10 = 0 has
(a) a unique solution (b) no solution (c) exactly two solutions (d) infinitely many solutions
41. The pair of equations x = a and y = b graphically represents line which are
(a) coincident
(b) parallel
(c) intersecting at (a, b) (d) intersecting at (b, a)
42. If the lines given by 2x – 5y + 10 = 0 and kx + 15y – 30 = 0 are coincident, then the values of k is
1
1
(a)
-6
(b)
6
(c)
(d)
3
3
43. If x = a, y = b is the solution of the equation x + y = 3 and x – y = 5, then the values of a and b are,
respectively
(a)
4 and -1
(b)
1 and 2
(c)
-1 and 4
(d) 2 and 3
1
1
44. If we add 1 to the numerator and denominator of a fraction, it becomes . It becomes if we only
3
2
add 1 to the denominator. The fraction is
1
3
3
2
(a)
(b)
(c)
(d)
5
5
4
4
45. If a pair of linear equations is consistent, then the lines will be
(a) always intersecting
(b) always coincident
(c) intersecting or coincident
(d) parallel
46. The pair of equations x + 2y – 3 = 0 and 4x + 5y = 8 has
(a) no solution
(b) infinitely many solutions
(c) a unique solution
(d) exactly two solutions
47. The value of c for which the pair of equations 4x – 5y + 7 = 0 and2cx – 10y + 8 = 0 has no solution is
(a) 8
(b)
-8
(c)
4
(d)
-4
48. A pair of linear equations which has a unique solution x = 1, y = - 3 is
(a) x – y = 1 ; 2x + 3y = 5
(b)
2x – y = - 5 ; 5x – 2y = 11
(c) 3x + y = 0 ; x + 2y = - 5
(d)
x + y = -2 ; 4x + 3y = 5
Hirav Trivedi’s Group TuiTions
CH : 3 Pair of linear equations
Batch : X
Sub : Maths
49. Anmol’s age is six times his son’s age. Four years hence, the age of Anmol will be four times his son’s
age. The present age in years, of the father and the son are respectively
(a)
24 and 4
(b)
30 and 5
(c)
36 and 6
(d) 24 and 3
SECTION - B
50. The value of k for which the system of equations
kx – y = 2
6x – 2y = 3
has a unique solution, is
(a) = 3
(b) ≠ 3
(c) ≠ 0
(d) = 0
51. The value of k for which the system of equations 2x + 3y = 5 4x + ky = 10 has infinite number of
solutions, is
(a) 1
(b) 3
(c) 6
(d) 0
52. The value of k for which the system of equations x + 2y- 3 = 0 and 5x + ky + 7 = 0 has no solution, is
(a) 10
(b) 6
(c)3
(d) 1
53. The value of k for which the system of equations 3x + 5y = 0 and kx + 10y = 0 has a non-zero
solution, is
(a) 0
(b) 2
(c) 6
(d) 8
54. If the system of equations
2x + 3y = 7
2ax + (a + b) y = 28
has infinitely many solutions, then
(a) a = 2b
(b) b = 2a
(c) a + 2b = 0
(d) 2a + b = 0
55. If the system of equations 2x + 3y = 5, 4x + ky = 10 has infinitely many solutions, then k =
(a) 1
(b) ½
(c) 3
(d) 6
SECTION – C
56. If 2x + 3y = 7 and 3x + 2y = 3, then x – y = …………..
(a)
4
(b)
–4
(c)
2
(d)
–2
57. If the pair of linear equations ax + 2y = 7 and 2x + 3y = 8 has a unique solution, then a ≠……….
(a)
3
4
(b) -
3
4
(c)
4
3
(c) -
4
3
Sub : Maths
Hirav Trivedi’s Group TuiTions
CH : 3 Pair of linear equations
Batch : X
58. In a two- digit number, the digit at tens place is x and the sum of the digit is 8 times the digit at unit
place. Then the number is …………..
(a)
70
(b)
71
(c)
17
(d)
78
59. 3 years ago, the sum of the ages of a father and his son was 40 years. After 2 years the sum of ages of
the father and his son will be ……………
(a)
40
(b)
46
(c)
50
(d)
60
60. If pair of equations a1 x + b2  b1 y  c1  0 and a2 x  b2 y  c1  0 if …………….., then solution set
is infinite.
a
a
a
a1 b1 c1
b
b
b
c
c
c
(a) 1  1  1
(b) 1  1  1
(c) 1  1  1
(d)


a 2 b2 c 2
a 2 b2 c 2
a 2 b2 c 2
a 2 b2 c2
3x 3 y 3
61. The solution set of a pair of equations 5x – 5y = - 5 and

  0 is…………
2
2 2
 3 
 3 
(a)  5, 
(b)
(c)
infinite set
(d)
empty set
 ,5 
 2 
 2 
x 6
62. If   3 , then x + y = ………….
2 y
(a)
2
(b)
4
(c)
6
(d)
8
63. The solution of pair of equations x + 2y = 5 and 3x + 5y = 13 is ……..
(a)
x = 1, y = 2 (b)
x = 2, y = 1 (c)
x = -1, y = -2 (d)
x = 2, y = -1
3 2
2 3
1 1
64. For the pair of equations   17 and   13 ,   ………….
x y
x y
x y
(a)
5
(b)
30
(c)
20
(d)
6
65. From the pair of equations x + 2y = 8…(1) and 3x – 4y = -6…..(2) first equation should multiplied by
……. To eliminate y.
(a)
1
(b)
3
(c)
2
(d)
4
5 4
4 5
1 1
66. If   4 and   5 , then
  ………………(x, y  0 )
x y
x y
x y
(a)
5
(b)
4
(c)
9
(d)
1
x y
x y
67. If
 6 , then x = ………..
 2 and
xy
xy
1
1
(a)
4
(b)
(c)
2
(d)
4
2
68. If a two – digit number, the digit at unit’s place is x and the digit at ten’s place is y. The number
obtained by interchanging the digits is ………..
(a)
10y + x
(b)
10x + y
(c)
x+y
(d)
x-y
69. The sum of the two – digit number and the number obtained by interchanging the digits is always
divisible by ………..
(a)
9
(b)
10
(c)
11
(d)
12
70. If the sum and the product of the digits of two – digits of two – digit number is the same, then the
number is ………..
(a)
11
(b)
23
(c)
10
(d)
22
71. Four years ago, the sum of the ages of mother and her two daughters was x years, then after two years,
the sum of their ages becomes …… years.
(a)
x+6
(b)
x + 12
(c)
x + 18
(d)
x + 24
Sub : Maths
Hirav Trivedi’s Group TuiTions
CH : 3 Pair of linear equations
Batch : X
72. The present age of Ramesh is x years. Mahesh is 5 years elder than Ramesh and three years younger
than Suresh. The present age of Suresh is ………
(a)
x+3
(b)
x+5
(c)
x+8
(d)
x + 24
73. Five years ago, the sum of the ages of a father and two son was x years, then after five years, the sum
of the ages of all will be……… years.
(a)
x+5
(b)
5x + 15
(c)
x + 30
(d)
x + 25
74. Five years ago, the sum of the ages of four persons was 60 years. The sum of the present ages of that
four persons is …. Years.
(a)
100
(b)
70
(c)
65
(d)
80
75. y years ago, the sum of the ages of five friends was x years, then the sum of the present ages of them is
……
(a)
5x + y
(b)
5x - y
(c)
x – 5y
(d)
x + 5y
76. x years ago, the age of Mahesh was y years than after z years the age will be ……. Years.
(a)
x–y+z
(b)
x+y+z
(c)
x–y-z
(d)
x+y-z
77. “The two digit’s number, whose the digit at unit’s place is x and the digit at ten’s place is y, then the
new number is………
(a) x + 10y = 3y
(b)
10x + y = 3( x + y )
(c) 10y + x = 3( x + y )
(d)
3( 10y + x ) = x + y
78. The average age of Ina, Mina and Dika is x years, then after y years the average age of them will be
……… years.
(a)
3x + 3y
(b)
x + 3y
(c)
y + 3x
(d)
x+y
79. The solution set of the pair of linear equations 2x – 3y = 4 and 6x – 9y = 12 is ……..
(a) Singleton set
(b)
(c)
Infinite set
(d)
two solutions

x y
80. The standard form of the equation   1 is …….
3 5
(a) 3x + 5y – 15 = 0
(b) 5x + 3y – 15 = 0 (c)
5x + 3y = 1 (d)
3x + 5y = 1
81. For which value of a, does the pair od linear equations x – ay = 2 and 3x + 2y = -5 have unique
solution?
2
3
3
2
(a)
a=
(b)
(c)
a=
(d)
a
a
3
2
2
3
x y
x y
82. If
 6 , then y = …..
 2 and
xy
xy
1
1
1
1
(a)
(b)
(c)
(d)


4
3
2
4
x x
83. The solution of a pair of equations   2 and ax – by = a 2  b 2 is …………
a b
(a) x = a, y = -b
(b) x = -a, y = - b
(c) x = a, y = b
(d) x = -a, y = b
84. The sum of the numerator and the denominator of a fraction is 12. If 3 is added to denominator, then
1
the fraction will be . The fraction is …….
2
7
3
5
9
(a)
(b)
(c)
(d)
5
9
7
3
85. The graph of a pair of linear equations 5x – 3y + 9 = 0 and 10x – 6y + 18 = 0 represents…….. lines.
(a)
parallel
(b)
coincident
(c)
intersecting (d)
mutually perpendicular
86. The equation of line parallel to 2x – 3y + 5 = 0 from the graph is ……..
(a) 4x + 5y – 15 = 0
(b) 6x - 9y + 15 = 0 (c) 8x - 12y + 20 = 0 (d)
4x - 6y + 15 = 0
Sub : Maths
Hirav Trivedi’s Group TuiTions
CH : 3 Pair of linear equations
Batch : X
SECTION - D
87. For what value of k, the pair of linear equations 2x – y – 3 = 0, 2kx + 7y – 5 = 0 has a unique solution
x = 1, y = -1?
(a)
3
(b)
4
(c)
6
(d)
-6
88. If the pair of linear equations x – y = 1, x + ky = 5 has a unique solution x = 2, y = 1, then the value of
k is
(a)
-2
(b)
3
(c)
-2
(d)
4
89. The pair of linear equations 2kx + 5y = 7, 6x – 5y = 11 has a unique solution if
2
(a)
k  -3
(b)
(c)
k 5
(d)
k  5
k
3
2
90. The pair of linear equations 2x + ky – 3 = 0, 6 x  y  7  0 has a unique solution if
3
2
2
2
(a)
(b)
(c)
k 5
(d)
k
k
k
9
3
3
91. The pair of linear equations 3x + 4y = k, 9x + 12y = 6 has infinitely many solution if
(a)
k=2
(b)
k=6
(c)
k 6
(d)
k=3
92. The pair of linear equations 13x + ky = k and kx + 15y = 18 has infinite many solutions if
(a)
k=1
(b)
k=2
(c)
k=4
(d)
k=6
93. If the sum of the ages of a father and his son in years is 65 and twice the difference of their ages in
years is 50, then the age of the father is years is
(a)
45
(b)
40
(c)
50
(d)
55
94. Three chairs and two tables cost rs. 1850. Five chairs and three tables cost rs. 2850. Then the total cost
of one chair and one table is
(a)
rs. 800
(b)
rs. 850
(c)
rs. 900
(d)
rs. 950
95. Six years hence, a man’s age will be three times the age of his son and three years ago he was nine
times as old as his son. The present age of the man is
(a)
28
(b)
30
(c)
32
(d)
34
2
2
ab
a b
a b
96. The solution of the pair of equations   0 and

 a 2  b 2 is
x y
x
y
(a)
x=a
(b)
y=b
(c)
y=-b
(d)
x = -a
97. Which of the following is/ are solutions of the pair of equations 3x – 2y = 4 and 6x – 4y = 8?
(a) x = 2, y = 1
(b) x = 4, y = 4
(c) x = 6, y = 7(d) x = 5, y = 2
98. Find the values of a and b so that the following system of linear equations have infinitely many
solutions, 2x – 3y = 7, ( a + b )x – (a + b – 3)y = 4a + b
(a)
4, -3
(b)
-1, 4
(c)
8, 12
(d)
-5, -1
99. Find the value of p and q for which the following system of linear equations has infinitely number of
solutions 2x + 3y = 7, ( p + q )x + (2p – q)y = 3( p + q + 1)
5
10
(a)
-1,
(b)
5, 1
(c)
(d)
4, -3
 ,8
2
3
100. In a two digit number, the units digit is twice the ten’s digit. If 27 is added to the number, the digits
interchange there places. Find the number.
(a)
40
(b)
52
(c)
28
(d)
36
Hirav Trivedi’s Group TuiTions
CH : 3 Pair of linear equations
Batch : X
Sub : Maths
101. The sum of two digit number and the number formed by interchanging and digit is 132. If 12 is
added to the number, the new number becomes 5 times the sum of the digits. Find the number
(a)
132
(b)
39
(c)
48
(d)
88
102. The sum of a two digit number and the number obtained by reversing the order of its digits is 121
and the two digits differ by 3. Find the number.
(a) 62 and 26
(b) 47 and 74
(c)
41 and 14
(d)
87 and 87
103. Rohit can row downstream 4 km in 1 hours. Find h is speed of rowing in still water and speed of
the current.
(a) 6 km/hr, 2 km/hr
(b) 4 km/hr, 4 km/hr (c) 20 km/hr, 4 km/hr (d)
4 km/hr, 2 km/hr
ANSWER KEY
QUE
1
2
3
4
5
6
ANS
QUE
ANS
QUE
ANS
QUE
ANS
QUE
ANS
QUE
ANS
QUE
ANS
QUE
ANS
QUE
ANS
QUE
ANS
QUE
ANS
a
11
a
21
c
31
a
41
c
51
c
61
c
71
c
81
b
91
a
101
c
d
12
b
22
b
32
d
42
a
52
a
62
d
72
c
82
a
92
b
102
b
c
13
a
23
c
33
b
43
a
53
c
63
a
73
c
83
c
93
b
103
a
d
14
c
24
b
34
b
44
b
54
b
64
d
74
d
84
c
94
b
c
15
a
25
D
35
b
45
c
55
d
65
c
75
d
85
b
95
b
a
16
d
26
b
36
b
46
c
56
b
66
d
76
b
86
d
96
a,b
7
8
9
10
c
17
c
27
a
37
b
47
c
57
c
67
d
77
c
87
c
97
a,b,c
d
18
d
28
d
38
b
48
c
58
b
68
b
78
d
88
b
98
d
c
19
b
29
b
39
a
49
c
59
c
69
c
79
c
89
d
99
b
b
20
B
30
c
40
b
50
B
60
C
70
d
80
b
90
a
100
d