281

EQUATION - 197
Directions (Q. 1-5): Two equations (I) and (II) are given in each question. On the basis of these
equations you have to decide the relation between x and y and give answer.
(1) if x > y
(2) if x < y
(3) if x y
(4) if x y
(5) if x = y or no relation can be established between x and y.
1.
I. 6x2 - 19x + 15 = 0
II. 10y2 - 29y + 21 = 0
2.
I. 12x2 + 11x - 56 = 0
II. 4y2 - 15y + 14 = 0
3.
I. 3x2 + 13x + 12 = 0
II. y2 + 9y + 20 = 0
2
4.
I. 8x - 15x + 7 = 0
II. 2y2 - 7y + 6 = 0
5.
I. 7x - 3y = 13
II. 5x + 4y = 40
Directions (Q. 6-10): In the following questions, two equations numbered I and II are given.
You have to solve both the equations and give answer
(1) if x > y
(2) if x < y
(3) if x y
(4) if x y
(5) if x = y or no relation can be established between x and
2
6.
I. 2x - 11x + 15 = 0
II. 21y2 - 23y + 6 = 0
7.
I. 5x2 - 16x + 11= 0
II. 5y2 - 3y - 2 = 0
2
8.
I. x + 11x + 28 = 0
II. 2y2 + 13y + 20 = 0
9.
I. 6x2 + 29x + 35 = 0
II. 3y2 + 19y + 30 = 0
I. 2x + 5y = 6
II. 5x + 11y = 9
Directions (Q. Nos. 11-15) In the following questions two equations numbered I and II are
given. You have to solve both the equations and
Give answer
(1) if x > y
(2) if x > y
(3) if x < y
(4) if x < y
(5) if x = y or the relationship cannot be established
10.

11.

I.

12.

I.

13.

I.

1225x 4900 0

18
x

6 12
8
2 2
x x
x

(2)5 (11)3
x3
6

II. (81)1/4 y + (343)1/3 = 0

II. y3 + 9.68 + 5.64 = 16.95

II. 4y3 = - (589 4) + 5y3

I. 12x2 + llx + 12 = 10x2+22x

II. 13y2 - 18y + 3 = 9y2 - 10y
I. (x7/5 9) = 169 y3/5
II. y1/4 y1/4 7 = 273 y1/2
Directions (Q. 16 - 20): Two equations (I) and (II) are given in each question. On the basis of
these equations you have to decide the relation between x and y and give answer
(1) if x > y
(2) if x < y
(3) if x y
(4) if x y
(5) if x = y, or no relation can be established between x and y.
14.
15.

II. 2y2 - 9y - 56 = 0

16.

I.

x=

17.
18.

I.
I.

5x2 + 3x - 14 = 0
8x2 + 31x + 21 = 0

II. 2y2 - 9y + 10 = 0
II. 5y2 + 11y - 36 = 0

19.

I.

3x - y = 12

II. y =

2401

1089

I. 15x + 68x + 77 = 0
II. 3y + 29y + 68 = 0
Directions (Q. 21-25): Two equations (I) and (II) are given in each question. On the basis of
these equations, you have to decide the relation between x and y and give answer
20.

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

282
(1) if x > y
(2) if x < y
(3) if x y
(4) if x y
(5) if x = y, or no relation can be established between x and y.
21.
I. 2x2 + x - 1 = 0
II. 6y2 - 13y + 5 = 0
22.
I. 21x2 - 122x + 165 = 0
II. 3y2 - 2y - 33 = 0
23.
I. 5x2 - 29x + 36 = 0
II. 10y2 - 3y - 27 = 0
24.
I. 7x + 4y = 3
II. 5x + 3y = 3
2
25.
I. 7x - 54x + 99 = 0
II. 4y2 - 16y + 15 = 0
Directions (Q. 26-30): Two equations (I) and (II) are given in each question. On the basis of
these equations, you have to decide the relation between x and y and give answer
(1) if x > y
(2) if x < y
(3) if x y
(4) if x y
(5) if x = y, or no relation can be established between x and y.
26.
I. 5x2 - 87x + 378 = 0
II. 3y2 - 49y + 200 = 0
27.
I. 10x2 - x - 24 = 0
II. y2 - 2y = 0
28.
I. x2 - 5x + 6 = 0
II. 2y2 - 15y + 27 = 0
29.
I. 3x + 2y = 301
II. 7x - 5y = 74
2
30.
I. 14x - 37x + 24 = 0
II. 28y2 - 53y + 24 = 0
Directions (Q. 31-35): In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or relationship between x and y cannot be established
31.
I. 11x + 5y = 117
II. 7x + 13y = 153
32.
I. 6x2 + 51x + 105 = 0
II. 2y2 + 25y + 78 = 0
33.
I. 6x + 7y = 52
II. 14x + 4y = 35
2
34.
I. x + 11x + 30 = 0
II. y2 + 12y + 36 = 0
35.
I. 2x2 + x - 1 = 0
II. 2y2 - 3y + l = 0
Directions (Q.36-40) In the following questions three equations numbered I, II and III are
given. You have to solve all the equations either together or separately, or two together and one
separately, or by any other method and give answer If
(1) x < y = z
(2) x < y < z
(3) x < y > z
(4) x = y > z
(5) x = y = z or if none of the above relationship is established
36.
I. 7x + 6y + 4z = 122 II.
4x + 5y + 3z = 88
III.
9x + 2y + z = 78
37.
I. 7x + 6y =110
II. 4x + 3y = 59
III. x + z = 15
38.

I.

x=

(36)1/2 (1296)1/4

II. 2y + 3z = 33

III. 6y + 5z = 71

I. 8x + 7y= 135
II. 5x + 6y = 99
III. 9y + 8z = 121
I. (x + y) 3= 1331
II. x - y + z = 0
III. xy = 28
Directions (Q. 41-45): In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or relationship between x and y cannot be established
41.
I. 7x2 - 9x + 2 = 0
II. y2 - 4y + 3 = 0
42.
I. x2 = 64
II. 2y2 + 25y + 72 = 0
43.
I. x2 + x - 20 = 0
II. 2y2 - 19y + 45 = 0
44.
I. 7x + 3y = 26
II. 2x + 17y = -41
2
45.
I. 3x - 20x + 33 = 0
II. 2y2 - 11y + 15 = 0
Directions (Q. 46-50): In each of these questions, two equations (I) and (II) are given. You
39.
40.

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

283
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or relationship between x and y cannot be established.
46.
I. 4x2 - 43x + 105 = 0
II. 7y2 - 29y + 30 = 0
47.
I. x2 + 13x + 40 = 0
II. y2 + 7y + 10 = 0
48.

I.

II. 2y2 - 54y + 364 = 0

x 3 2197

I. 5x2 - 27x + 36 = 0
II. y2 - 2y + 2 = 0
I. 13x - 8y + 81 = 0
II. 15x + 5y + 65 = 0
Directions (Q. 51-55): Two equations (I) and (II) are given in each question. On the basis of
these equations, you have to decide the relation between x and y and give answer
(1) if x > y
(2) if x < y
(3) if x y
(4) if x y
(5) if x = y, or no relation can be established between x and y.
51.
I. 15x2 - 19x + 6 = 0
II. 6y2 - 5y + 1 = 0
49.
50.

II. y2 - 29y + 210 = 0

x 172
53.
I. 3x2 - 20x + 32 = 0
II. 2y2 - 19y + 44 = 0
54.
I. 3x + 8y = -2
II. 4x + 18y = l
55.
I. 2x2 - 15x + 28 = 0
II. 10y2 - y - 119 = 0
Directions (Q. 56-70): In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
52.

I.

(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or relationship between x and y cannot be established.
56.

I.

676x2 - l = 0

II. y

1
13824

I. 8x + 13y = 62
II 13x - 17y + 128 = 0
2
I. 7x + 2x = 120
II. y2 + 11y + 30 = 0
I. x2 = 7x
II. (y + 7)2 = 0
I. 2x2 + 5x - 33 = 0
II. y2 - y - 6 = 0
Directions (Q. 61-65): In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or the relationship between x and y cannot be established.
61.
I. x2 + 12x + 36 = 0
II. y2 + 15y + 56 = 0
2
62.
I. x = 35
II. y2 + 13y + 42 = 0
57.
58.
59.
60.

63.

I.

2x2 - 3x - 35 = 0

II. y2 - 7y + 6 = 0

64.

I.

6x2 - 29x + 35 = 0

II. 2y2 - 19y + 35 = 0

I. 12x2 - 47x + 40 = 0 II.

4y2 + 3y - 10 = 0
Directions (Q. 66-70): In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or no relation can be established between x and y.
66.
I. x2 + 3x - 28 = 0
II. y2 - 11y + 28 = 0
67.
I. 6x2 - 17x + 12 = 0
II. 6y2 - 7y + 2 = 0
65.

68.

I.

256
576

II. 3y2 + y-2 = 0

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

284
I. x2 = 64
II. y2 = 9y
I. x2 + 6x - 7 = 0
II. 41y + 17 = 140
Directions (Q. 71-75): In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or a relationship between x and y cannot be established.
71.
I. x2 + 3x = 28
II. y2 + 16y + 63 = 0
69.
70.

72.

I.

x=

2197

II. y2 = 169

I. 8x2 - 49x + 45 = 0
II. 8y2 - y - 9 = 0
I. 42x - 17y = -67
II. 7x + 12y = -26
2
I. x - 8x + 15 = 0
II. 2y2 - 21y + 55 = 0
Directions (Q. 76-80): In each of these questions two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if p > q
(2) if p q
(3) if p < q
(4) if p q
(5) if p = q or no relation can be established between p and q.
76.
I. 2.3p - 20.01 = 0
II. 2.9q - p = 0
73.
74.
75.

77.

I.

p = 1764

II. q2 = 1764

I. p2 - 26p + 168 = 0
II. q2 - 25q + 156 = 0
I. p2 - 13p + 42 = 0
II. q2 + q - 42 = 0
I. 6p - 5q = -47
II. 5p + 3q = 11
Directions(Q. 81-85): In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or no relation can be established between x and y.
81.
I. 2x2 + 13x - 7 = 0
II. 2y2 - 5y + 3 = 0
82.
I. 2x2-15x + 28 = 0
II. 4y2 - 16y + 15 = 0
2
83.
I. x + 8x + 16 = 0
II. y2 = 16
2
84.
I. x - 2x - 24 = 0
II. y2 + 8y = 0
78.
79.
80.

85.

I.

x2 + 4x = 0

II. y2 + 10y + 25 = 0

Directions (Q. 86-90): In each of these questions two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or no relation can be established between x and y
86.
I. 2x2 + x 1 = 0
II. 2y2 + 13y + 15 = 0
87.
I. x2 + 12x + 32 = 0
II. 2y2 + 15y + 27 = 0
2
88.
I. 6x 17x + 12 = 0
II. 7y2 13y + 6 = 0
2
89.
I. x 82x + 781 = 0
II. y2 = 5041
90.
I. 6x2 47x + 80 = 0
II. 2y2 9y + 10 = 0
Directions (Q. 91-95): In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or no relation between x and y can be established.
91.
I. 3x2 7x 20 = 0
II. y2 8y + 16 = 0
2
92.
I. x 72 = 0
II. y2 9y + 8 = 0
93.
I. 9x2 114x + 361 = 0
II. y2 = 36
94.
I. 13x + 17y = 107
II. x 11y = 41
LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

285
I. 9x2 + 18x + 9 = 0
II. y2 3y + 2 = 0
Directions (Q. 96-100) : In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer .
(l) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or no relation can be established between x and y.
96.
I. 4x + 7y = 42
II. 3x - 11y = l
97.
I. 9x2 29x + 22 = 0
II. y2 7y + 12 = 0
2
98.
I. 3x 4x 32 = 0
II. 2y2 17y + 36 = 0
2
99.
I. 3x 19x 14 = 0
II. 2y2 + 5y + 3 = 0
2
100. I. x + 14x + 49 = 0
II. y2 + 9y = 0
Directions (Q. 101-105): In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer ,
(1) if x < y
(2) if x y
(3) if x = y, or no relation can be established between x and y
(4) if x > y
(5) if x y
2
101. I. 9x = 1
II. 4y2 + 11y - 3 = 0
2
102. I. 3x + 5x - 2 = 0
II. 2y2 - 7y + 5 = 0
103. I. 6x2 + 13x + 5 = 0
II. 3y2 + 11y + 10 = 0
104. I. 7x - 4y = 29
II. 5x + 3y - 50 = 0
105. I. x2 - 5 = 0
II. 4y2 - 24y + 35 = 0
Directions (Q. 106-110) : In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or no relation can be established between x and y
106. I. 35x2 - 53x + 20 = 0
II. 56y2-97y + 42 = 0
95.

107.
108.

I. x = 3 4913
I. x2 - 5x - 14 = 0

II. 13y + 3x = 246

II. y2 + 7y + 10 = 0

109.

I. x2 - 3481 = 0

II. 3y2 =

216000
I. 5x + 2x - 3 = 0
II. 2y + 7y + 6 = 0
Directions (Q. 111-115) : In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or no relationship can be established.
111. I. 20x2 - 67x + 56 = 0
II. 56y2 - 67y + 20 = 0
2

110.

112.

I.

x4 = 65536

II. y =

113.

I.

2x + 11x - 40 = 0

4096
II. 4y - 27y + 44 = 0

114.

I.

7x = 4y + 85

II. y =

115.

I.

x2 = 14641

II.

116.

I.

x2 + 42 = 13x

II. y 4 1296

117.
118.
119.
120.

I.
I.
I.
I.

x2 + x - 2 = 0
3x2 - 23x + 40 = 0
15x2 - 46x + 35 = 0
x2 + 5x - 6 = 0

II.
II.
II.
II.

17576

y = 14641
Directions (Q. 116-120): In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or if there is no relation between x and y.

y2 + 7y + 12 = 0
2y2 - 23y + 66 = 0
4y2 - 15y + 14 = 0
2y2 - 11y + 15 = 0

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

286
Directions (Q. 121-125) : In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or If there is no relation between x and y.
121. I. 2x2 21x + 54 = 0
II. y2 14y + 49 = 0
122. I. x2 19x + 70 = 0
II.2y2 17y + 35 = 0
2
123. I. 3x + 5x 8 = 0
II. y2 4y + 3=0
2
124. I. 12x 16x + 5 = 0
II. 18y2 45y + 25 = 0
2
125. I. 3x + 11x + 8 = 0
II. 3y2 + 20y + 32 = 0
Directions (Q. 126-130) : In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(l) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or no relationship can be established between x and y.
I. x = 3 357911
II. y = 5041
I. 5x + 7y = -43
II. 9x 17y = 41
I. x2 + 11x + 30 = 0
II. y2 + 9y + 20 = 0
2
I. 4x + 3x l = 0
II. 6y2 5y + l = 0
2
I. 3x + 15x + 18 = 0
II. 2y2 + 15y + 27 = 0
Directions (Q. 131-135) : In the following questions, two equations numbered I and H are
given. You have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or relationship cannot be established
131. I. 4x + 3y = (1600)1/2 II.
6x 5y = (484)1/2
126.
127.
128.
129.
130.

132.

I.

2x 2 (4 13)x 2 13 0

II. 10y 2 (18 5 13)y 9 13 0

133.

I.

(6x2 + l7) (3x2 + 20) = 0

II. (5y2 12) (9y2 16) = 0

134.

I.

(169)1/2 x 289 134

II. (361)1/2 y 2 270 1269

135.

I.

82lx2 757x2 = 256

II.

196 y 3 12y 3 16

Directions (Q. 136-140) : In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(l) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or no relation can be established between x and y.
136. I. 5x - 7y = -24
II. 13x + 3y = 86
137. I. x2 - 13x + 40 = 0
II. y2 + 3y - 40 = 0
2
138. I. 8x - 26x+15 = 0
II. 2y2 - 17y + 30 = 0
2
139. I. x = 484
II. y2 - 45y + 506 = 0
140.

I. 13x - 21=200 - 4x

II. y =

2197
Directions (Q. 141-145) : In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if p > q
(2) if p q
(3) if p < q
(4) if p q
(5) if p = q or there is no relation between p and q.
141. I. (p + q) 2 = 3136
II. q + 2513 = 2569
142. I. 4p2 - 16p +15 = 0
II. 2q2 + 5q - 7 = 0
2
143. I. p = 49
II. q2 +15q + 56 = 0
2
2
144. I. 2p + 5p - 12 = 0 II. 2q - q - 1 = 0
145. I. p2 - 12p + 35 = 0 II. q2 - 25 = 0
Directions (Q. 146150) : In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x < y
(3) if x y
(4) if x y
(5) if x = y or there is no relation between x and y.

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

287
I. 3x2 + 7x + 2 = 0
II. 2y2 + 9y + 10 = 0
2
I. x + x 2 = 0
II. y2 3y + 2 = 0
2
I. 20x 51x + 27 = 0
II. 15y2 16y + 4 = 0
2
I. 7x + 16x 15 = 0
II. y2 6y 7 = 0
I. x2 = 729
II. y2 + 58y + 840 = 0
Directions (Q. 151-155) : In the following questions two equations numbered I and II are
given. You have to solve both the equations and give answer if
(l) x > y
(2) x y
(3) x < y
(4) x y
(5) x = y or the relationship between x and y cannot be established.
146.
147.
148.
149.
150.

15

(x)2

151.

I.

152.

I.

x
x
5x + 2y = 96

153.

I.

(441)2 x 2 111 (15)2

II. y10 - (36)5 = 0

II. 3(7x + 5y) = 489

II.

121y 2 (6)3 260

I. 17x = (13)2 + 196 + (5) 2 + 4x

II. 9y - 345 = 4y - 260
I. 3x2 - 13x + 14 = 0
II. y2 - 7y + 12 = 0
Directions (Q. 156-160) : In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or no relation can be established between x and y.
156. I. 2x2 15x + 28 = 0
II. 2y2 + 3y-35 = 0
157. I. 7x 5y = 24
II. 4x + 3y = 43
154.
155.

II. y = 487
2744
159. I. x 9x + 8 = 0
II. 2y2 11y + 5 = 0
160. I. 2x2 + 3x + 1 = 0
II. 6y2 + 17y + 12 = 0
Directions (Q. 161-165) : In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or no relation can be established between x and y.
161. I. 3x2 29x + 56 = 0
II. 3y2 5y 8 = 0
2
162. I. 5x + 26x 24 = 0
II. 5y2 34y + 24 = 0
163. I. x2 7x = 0
II. 2y2 + 5y + 3 = 0
164. I. 7x 4y = 40
II. 8x + 8y = 8
165. I. 15x2 41x + 14 = 0
II. 2y2 13y + 20 = 0
Directions (Q. 166-170) : In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer >
158.

I.

x=

(1) if x > y
(2) if x y
(3) if x < y
(4) if x y
(5) if x = y or no relation can be established between x and y.
166.

I. x 2 8 3x 45 0

II. y 2 2y 24 0

167.

I.

II. y 5 2y 12 0

x 7 2x 24 0

I. 12x - 17x + 6 = 0
II. 20y2 - 31y + 12 = 0
I. 3x2 - 8x + 4 = 0
II. 4y2 - 15y + 9 = 0
I. x2 -16x + 63 = 0
II. y2 - 2y - 35 = 0
Directions (Q. 171-175): In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2)if x y
(3) if x < y
(4) if x y
(5) if x = y or no relation can be established between x and y.
168.
169.
170.

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

288
171.

I. 63x 94 x 35

II. 32y 52 y 21 0

172.

II. y 2 5 5y 30 0

173.

I. x 2 7 3x 35 15 5 5x
I. 14x2 + 11x - 15 = 0

174.

I.

25x 16y 41

II. 16x 25y 40

I.

(18) 2
x
0
x2

II. 20y2 - 31y + 12 = 0

15

175.

II.

(19)2
y
0
y

Directions (Q. 176-180) : In each of these questions, two equations (I) and (II) are given.
Solve both the equations and give answer
(1) if x > y
(2) if x < y
(3) if x y
(4) if x y
(5) if x = y or no relation can be established between x and y.
176.

I.

II. 99y 255 y 150 0

I.

63x 194 x 143 0

16x2 40x 39 = 0

177.
178.

I.

x 7 3x 36 0

II. y 12 2y 70 0

II. 12y2 113y + 255 = 0

II. y 2 5 5y 30 0
x 2 7 7x 84 0
180. I. 10x 6y = 13
II. 45x + 24y = 56
Directions (Q. 181-185) : In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x > y
(2) if x y
(3) if x < y
(4) I x y
(5) if x = y or no relation can be established between x and y.
181. I. x2 - 2x -15 = 0
II. y2 + 5y + 6 = 0
182. I. x2 - x - 12 = 0
II. y2 - 3y + 2 = 0
179.

I.

183.

I.

x-

184.

I.

x2 - 32 = 112

II. y2 - 169 = 0

169 = 0

II. y -

256 = 0

I. x - 25 = 0
II. y - 9y + 20 = 0
Directions (Q. 186-190): In the following questions, three equations numbered I, II and III
are given. You have to solve all the equations either together or separately, or two together and
one separately or by any other method and give answer
(1) if x = y > z
(2) if x < y = z
(3) if x < y > z
(4) if x = y = z or if none of the above relationship can be established.
(5) if x y < z
186. I. 3x + 5y = 69
II. 9x + 4y = 108
III. x + z = 12
185.

187.

I.

y (729)3 (6541)4

II. 2x + 5z = 54

III. 6x + 4z = 74

I. 2x + 3y + 4z = 66
II. 2x + y + 3z = 42
III. 3x + 2y + 4z = 63
3
I. (x + z) = 1728
II. 2x + 3y = 35
III. x - z = 2
I. 4x + 5y = 37
II. x + z = 8
III. 7x + 3y = 36
Directions (Q. 191-194): In each of these questions, two equations (I) and (II) are given. You
have to solve both the equations and give answer
(1) if x < y
(2) if x y
(3) if x = y or no relation can be established
(4) if x > y
(5) if x y
188.
189.
190.

191.

I. 7x + 3y = 77

II. 2x + 5y = (2601) 2

192.

I. 3x 2 (6 17)x 2 17 0

II. 10y 2 (18 5 17)y 9 17 0

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

289
1
2

193.

I.

(289) x 324 203

II. (484)2 y 225 183

194.

I.

679x2 - 168x2 = 3066

II.

144y 3 9y 3 1536

Directions (Q. 195-197): In the following questions two equations numbered I and II are
given. Solve both the equations and give answer
(1) if x < y
(2) if x y
(3) if x y
(4) if x > y
(5) if x = y or no relationship can be established
1

3x + 4y = (1681) 2

II. 3x + 2y = (961) 2

195.

I.

196.

I. 3x 2 (6 17)x 2 17 0

197.

I.

x2 - 16x + 63 = 0

II. 10y 2 (15 17) 3 17 0

II. y2 - 2y - 35 = 0

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

290

SHORT ANSWER
1.
9.
17.
25.
33.
41.
49.
57.
65.
73.
81.
89.
97.
105.
113.
121.
129.
137.
145.
153.
161.
169.
177.
185.
193.

(3)
(1)
(2)
(1)
(3)
(4)
(1)
(3)
(2)
(2)
(3)
(5)
(3)
(1)
(3)
(3)
(3)
(2)
(2)
(1)
(2)
(5)
(2)
(5)
(4)

2.
10.
18.
26.
34.
42.
50.
58.
66.
74.
82.
90.
98.
106.
114.
122.
130.
138.
146.
154.
162.
170.
178.
186.
194.

(4)
(2)
(5)
(1)
(2)
(5)
(3)
(1)
(4)
(3)
(1)
(2)
(4)
(3)
(1)
(2)
(2)
(4)
(3)
(3)
(4)
(1)
(2)
(3)
(1)

3.
11.
19.
27.
35.
43.
51.
59.
67.
75.
83.
91.
99.
107.
115.
123.
131.
139.
147.
155.
163.
171.
179.
187.
195.

(1)
(1)
(2)
(5)
(4)
(3)
(1)
(1)
(1)
(4)
(4)
(4)
(1)
(1)
(4)
(4)
(1)
(4)
(4)
(3)
(1)
(5)
(1)
(3)
(4)

4.
12.
20.
28.
36.
44.
52.
60.
68.
76.
84.
92.
100.
108.
116.
124.
132.
140.
148.
156.
164.
172.
180.
188.
196.

(2)
(5)
(1)
(4)
(1)
(1)
(2)
(5)
(2)
(1)
(5)
(5)
(5)
(2)
(2)
(4)
(2)
(5)
(1)
(2)
(1)
(1)
(2)
(2)
(5)

5.
13.
21.
29.
37.
45.
53.
61.
69.
77.
85.
93.
101.
109.
117.
125.
133.
141.
149.
157.
165.
173.
181.
189.
197.

(2)
(1)
(4)
(2)
(3)
(2)
(4)
(1)
(5)
(2)
(1)
(1)
(3)
(5)
(1)
(2)
(5)
(3)
(5)
(1)
(3)
(3)
(2)
(1)
(2)

6.
14.
22.
30.
38.
46.
54.
62.
70.
78.
86.
94.
102.
110.
118.
126.
134.
142.
150.
158.
166.
174.
182.
190.

(1)
(2)
(5)
(3)
(2)
(1)
(2)
(1)
(3)
(5)
(1)
(3)
(1)
(1)
(3)
(5)
(2)
(1)
(1)
(3)
(5)
(1)
(5)
(2)

7.
15.
23.
31.
39.
47.
55.
63.
71.
79.
87.
95.
103.
111.
119.
127.
135.
143.
151.
159.
167.
175.
183.
191.

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

(3)
(4)
(3)
(3)
(4)
(4)
(3)
(5)
(2)
(2)
(5)
(3)
(5)
(1)
(3)
(1)
(4)
(2)
(2)
(5)
(2)
(3)
(2)
(4)

8.
16.
24.
32.
40.
48.
56.
64.
72.
80.
88.
96.
104.
112.
120.
128.
136.
144.
152.
160.
168.
176.
184.
192.

(4)
(5)
(2)
(1)
(5)
(4)
(3)
(4)
(2)
(3)
(1)
(1)
(4)
(4)
(3)
(4)
(3)
(5)
(1)
(1)
(4)
(5)
(3)
(3)

291

DETAIL - EXPLANATIONS
01.

3; I.

6x2 - 9x - 10x + 15 = 0

or, 3x(2x - 3) - 5(2x - 3) = 0

x < y
6.

1; I.

or, (3x - 5) (2x - 3) = 0

2x2 - 6x - 5x + 15 = 0

or,2x(x - 3) - 5(x - 3) = 0
or, (2x - 5) (x - 3) = 0

5 3
,
3 2

x 3,

II. 10y - 15y - 14y + 21 = 0

or, 5y(2y - 3) - 7(2y - 3) = 0

II. 21y2 - 14y - 9y + 6 = 0

or, (5y - 7) (2y - 3) = 0

or,7y(3y - 2) - 3 (3y - 2) = 0

or,(7y - 3)(3y - 2) = 0

7 3
,
5 2

x y
2.

4; I. 12x2 + 32x - 21x - 56 = 0

7.

3; I.

5x2 - 5x - 11x + 11 = 0

or,5x(x - 1) - 11(x - 1) = 0
or,(x - 1) (5x - 11) = 0

7 8
,
4 3

x = 1,

11
5

x > y

II. 4y2 - 8y - 7y + 14 = 0

II. 5y2 - 5y + 2y - 2 = 0

or, 4y(y - 2) - 7(y - 2) = 0

or, 5y (y - 1) + 2(y - 1) = 0

or, (4y - 7) (y - 2) = 0

or, (5y + 2)(y - 1) = 0

7
4

y = 1, -

x y

8.

2
5

x > y

4; I. x2 + 4x + 7x + 28 = 0

1; I. 3x2 + 9x + 4x + 12 = 0

or, x(x + 4) +7(x + 7) = 0

or, 3x(x + 3) + 4(x + 3) = 0

or, (x + 4) (x + 7) = 0

or, (3x + 4) (x + 3) = 0

x = - 4, - 7

II. 2y2 + 8y + 5y + 20 = 0

4
, 3
3

or, 2y(y + 4) + 5(y + 4) = 0

or, (y + 4) (2y + 5) = 0

II. y2 + 5y + 4y + 20 = 0
or, y(y + 5) + 4(y + 5) = 0

y = -4,

or, (y + 4) (y + 5) = 0
y = - 4, - 5

9.

or,(3x + 7) (2x + 5) = 0

or, 8x(x - 1) -7(x - 1) = 0

or, (8x - 7) (x - 1) = 0

7
,1
8

7 5
,
3 2

II. 3y2 + 9y + 10y + 30 = 0

or, 3y(y + 3) +10(y + 3) = 0

II. 2y2 - 4y - 3y + 6 = 0

or,(3y + 10) (y + 3) = 0

or, 2y(y - 2) -3(y - 2) = 0

y 3,

or, (y - 2) (2y - 3) = 0

y 2,

3
2

10.

2;

10
3

x > y

eqn (I) 5 - eqn (II) 2

10x + 25y = 30

2; Eqn (I) 4 + Eqn (II) 3

10x 22y = 18

28x - 12y = 52

15x + 12y = 120

43x

x < y

or, 3x(2x + 5) + 7(2x + 5) = 0

2; I. 8x2 - 8x - 7x + 7 = 0

5
2

1; I. 6x2 + 15x + 14x + 35 = 0

x > y

5.

x > y

or, (4x - 7) (3x + 8) = 0

y 2,

4.

3 2
,
7 3

or, 4x(3x + 8) - 7(3x + 8) = 0

3.

5
2

3y = 12

= 172

x = 4 and y = 5

y = 4 and x = -7
11.

1; I.

1225x 4900 0

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

y > x

292
or, 35x + 70 = 0

or, x =

II. 3y + 7 = 0
:.
12.

5; I.

or y =

or, (2y - 5)(y - 2) = 0

70
2
35

or, y = 2,

7
3

x> y

x < y
18.

5;

18 6x 12
8
2
x2
x

or, (x + 3) (8x + 7) = 0
x = - 3,

II. y2 = 16.95 - 9.68 - 5.64 = 1.63

:. y = 1.277
13.

1; I.

x3

or y3 =
14.

2; I.

or, 5y(y + 4) - 9(y + 4) = 0

or, (y + 4) (5y - 9) = 0

589
4

y = -4,

:. x > y
19.

2;

II.

II. 4y2 - 8y + 3 = 0

3 1
,
2 2

:. x > y

7
5

4; I. x 9 169 x
7
5

or, x

73
5

1;

or, y

5;

I. x =

7 11
,
3 5

II. 3y2 + 29y + 68 = 0

1
2

or, 3y2 + 12y + 17y + 68 = 0

or, 3y(y + 4) + 17(y + 4) = 0
or, (y + 4) (3y + 17) = 0

39

or, y = 39
x < y
16.

15x2 + 68x + 77 = 0

273
y y y
7
1 1 1

4 4 2

I.

or, (5x + 11) (3x + 7) = 0

x = 39
1
4

12 y 12 33 45

15
3
3
3

or, 5x(3x + 7) + 11(3x + 7) = 0

or, x2 = 1521

1521

or, 15x2 + 35x + 33x + 77 = 0

3
5

3
5

1
4

y 1089

x < y
20.

or, x x 169 9

II.

I.

or, y = 33

3
or, x = 4,
2

15.

y = -4,
x = 7

2401

21.

4;

I.

2x2 + 2x - x - 1 = 0

(2y + 7) (y - 8) = 0
x = - l,

7
2

1
2

II. 6y2 - 3y - 10y + 5 = 0

or, 3y(2y - 1) - 5(2y - 1) = 0

Hence, no relation exists between x and y.

or, (3y - 5)(2y - 1) = 0

2; I. 5x + 10x - 7x - 14 = 0
or, 5x(x + 2) - 7(x + 2) = 0

y = - 3,

or, (x + 2) (5x - 7) = 0
x = - 2,

x > y

or, (x + 1) (2x - 1) = 0

2y(y - 8) + 7(y - 8) = 0

17.

17
3

or, 2x(x + 1) - 1(x + 1) = 0

II. 2y 2 - 16y + 7y - 56 = 0

y 8,

9
5

Hence, no relation exists between x and y.

589
4

2x2 - llx + 12 = 0

7
8

II. 5y2 + 20y - 9y - 36 = 0

32 1331 1363

6
6

II. 5y3 - 4y3 =

I. 8x2 + 24x + 7x + 21 = 0
or, 8x(x + 3) + 7(x + 3) = 0

1
= .333
3

or, x =

5
2

7
5

11
3

x y
22.

5;

I. 21x2 - 45x - 77x + 165 = 0

II. 2y2 - 4y - 5y + 10 = 0

or, 3x(7x - 15) - 11 (7x - 15) = 0

or, 2y(y - 2) - 5(y - 2) = 0

or, (3x - 11) (7x - 15) = 0

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

293
x

or, 2x(5x - 8) + 3(5x - 8) = 0

or, (2x + 3) (5x - 8) = 0

11 15
,
3 7

II. 3y2 + 9y - 11y - 33 = 0

or,3y(y + 3) - 11(y + 3) = 0
or,(3y - 11) (y + 3) = 0

11
3

y = - 3,

Hence, no relation can be established between x

and y.
23.

3;

28.

I. 5x2 - 20x - 9x + 36 = 0
or, 5x(x - 4) - 9(x - 4) = 0
or,(x - 4) (5x - 9) = 0
x = 4,

9
5

3 8
,
2 5

II. y 2 - 2y = 0
or, y(y - 2) = 0
y = 0, 2
ie no relationship exists between x and y.
4; I.
x2 - 2x - 3x + 6 = 0
or, x(x - 2) - 3(x - 2) = 0
or, (x - 2) (x - 3) = 0
x = 2, 3
II. 2y2 - 6y - 9y + 27 = 0
or, 2y(y - 3) - 9(y - 3) = 0
or, (y - 3) (2y - 9) = 0
y = 3,

II. 10y2 + 15y - 18y - 27 = 0

or, 5y(2y + 3) - 9(2y + 3) = 0
or, (2y + 3) (5y - 9) = 0

29.

9 3
,
5 2

y =

x y
24.

21x + 12y = 9
30.

20x + 12y = 12
-

x y
2; I.
eqn (I) 5 + eqn (II) 2
15x + 10y = 1505
14x - 10y = 148
29x
= 1653
x =

2; eqn (I) 3 - eqn (II) 4

=- 3

and y = 6
25.

x =

1; I. 7x - 21x - 33x + 99 = 0
or, 7x(x - 3) - 33(x - 3) = 0

II.
or,
or,
or,

or, (x - 3) (7x - 33) = 0

x = 3,

33
7

or, 2y(2y - 3) - 5(2y - 3) = 0

x

or, (2y - 3)(2y - 5) = 0

31.

3 5
y= ,
2 2
26.

8 3
,
7 4
y

3; eqn (I) 7
77x + 35y = 819
- 77x 143y = 1683

1; I.
5x - 45x - 42x + 378 = 0
or, 5x(x - 9) - 42(x - 9) = 0
or, (5x - 42) (x - 9) = 0
x = 9,

42
5

II. 3y2 - 24y - 25y + 200 = 0

or, 3y(y - 8) - 25(y - 8) = 0
or, (y - 8) (3y - 25) = 0

27. 5;

3 8
,
2 7

eqn (II) 11

y 8,

57

28y2 - 53y + 24 = 0
28y2 - 21y - 32y + 24 = 0
7y(4y - 3) - 8(4y - 3) = 0
(7y - 8) (4y - 3) = 0

II. 4y2 - 6y - 10y + 15 = 0

1653
=
29

and y = 65
x < y
3; I. 14x2 - 37x + 24 = 0
or, 14x2 - 21x - 16x + 24 = 0
or, 7x(2x - 3) - 8(2x - 3) = 0
or, (2x - 3) (7x - 8) = 0

x < y
2

9
2

25
3

- 108y = - 864
y = 8, x = 7
32.

ie x < y

1; I. 6x2 + 21x + 30x + 105 = 0

or, 3x(2x + 7) + 15(2x + 7) = 0
or, (3x + 15) (2x + 7) = 0
x = -5,

7
2

II. 2y 2 + 12y + 13y + 78 = 0

or, 2y(y + 6) + 13(y + 6) = 0

x > y
I.
10x2 - 16x + 15x - 24 = 0

or, (2y + 13) (y + 6) = 0

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

294
y =

13
, 6
2

x < y
33.

3; eqn (I) 4

39.

eqn (II) 7
24x + 28y = 208
- 98x 28y = 245
- 74x = - 37
x =

1
,
2

y = 7
40.

x < y
34.

2; I.

x + 5x + 6x + 30 = 0

or, x(x + 5) + 6(x + 5) = 0

or, (x + 5) (x + 6) = 0
x = - 5, - 6
II. y 2 + 12y + 36 = 0
or, (y + 6)2 = 0
or, y + 6 = 0
y = - 6
ie x y
35.

4;

I.

2x2 + 2x - x - 1 = 0

or, 2x(x + 1) - 1(x + 1) = 0

41.

or, 7x(x - 1) - 2(x - 1) = 0

or, (2x - 1) (x + 1) = 0

(7x - 2) (x - 1) = 0

1
x , 1
2

or, x =

II. 2y2 - 2y - y + 1 = 0

or, y(y - 1) - 3(y - 1) = 0

or, (2y - 1)(y - 1) = 0

or, (y - 3) (y - 1) = 0

1
y ,1
2

y = 1, 3
x y
42.

ie x y

37.

38.

1; 7x + 6y + 4z = 122 ... (i)

4x + 5y + 3z = 88 ... (ii)
9x + 2y + z = 78 ... (iii)
From (i) and (ii)
5x - 2y = 14... (iv)
From (ii) and (iii)
23x + y = 146 ... (v)
From (iv) and (v),
x = 6, y = 8
Putting the value of x and y in eqn (i), we get
z= 8
:. x < y = z
3; 7x + 6y = 110 ... (i)
4x + 3y = 59 ... (ii)
x + z = 15 ... (iii)
From eqn (i) and (ii), x = 8, y = 9
Put the value of x in eqn (iii).
Then, z = 7
x < y > z
2;

2
,1
7

II. y2 - y - 3y + 3 = 0

or, 2y(y - 1) - 1(y - 1) = 0

36.

2y + 3z = 33 ... (ii)
6y + 5z = 71 ... (iii)
From eqn (ii) and (iii),
y = 6 and z = 7
x = y , z
4; 8x + 7y = 135 ... (i)
5x + 6y = 99 ... (ii)
9y + 8z = 121 ... (iii)
From eqn (i) and (ii),
x = 9, and y = 9
Putting the value of y in eqn (iii),
z= 5
:. x = y > z
5; (x + y)3 = 1331
or, x + y = 11 ... (i)
(x + y)2 = 121
(x - y)2 + 4xy = 121
x - y = 3... (ii)
[value of xy from eqn (iii)]
From eqn (i) and (ii), x = 7, y = 4
Put the value x and y in the eqn
x - y + z = 0
7 -y + z= 0
3 + z = 0
z = -3
4; I. 7x2 - 7x - 2x + 2 = 0

5; I.

x2 = 64
x = 8

II. 2y2 + 9y + 16y + 72 = 0

or, y(2y + 9) + 8(2y + 9) = 0
or, (y + 8) (2y + 9) = 0

y 8,

9
2

ie, no relation between x and y.

43.

3; I.

x2 + x - 20 = 0

or, x2 + 5x - 4x - 20 = 0
or, x(x + 5) - 4(x + 5) = 0
or, (x - 4) (x + 5) = 0
x = 4, - 5
II. 2y2 - 10y - 9y + 45 = 0
or, 2y(y - 5) - 9(y - 5) = 0
or, (y - 5) (2y - 9) = 0

y 5,

x (62 )1/2 (64 )1/4

9
2

x < y

6 6 6 ... (i)

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

295
44.

1; Eqn (I) 2

Eqn (II) 7
14x + 6y = 52

II. y 2 - y - 2y + 2 = 0

14x + 119y = - 287

-

or, y(y - 1) - 2(y - 1) = 0

or, (y - 1)(y - 2) = 0

- 113y = 339

y = 1, 2

y = - 3 and x = 5, ie x > y
45.

2; I.

12
,3
5

x > y

3x2 - 9x - 11x + 33 = 0

or, 3x(x - 3) - 11(x - 3) = 0

50.

3; eqn (I) 5 + eqn (II) 8

65x - 40y + 405 = 0

or, (3x - 11) (x - 3) = 0

120x + 40y + 520 = 0

x = 3,

11
3

185x + 0

II. 2y2 - 6y - 5y + 15 = 0
or, 2y(y - 3) - 5(y - 3) = 0
or, (y - 3) (2y - 5) = 0

y 3,

5
2

x y
46.

1; I. 4x2 - 28x - 15x + 105 = 0

or, 4x(x - 7) - 15(x - 7) = 0
or, (x - 7) (4x - 15) = 0

51.

15
4

x = 7,

15
7

1 1
,
3 2

x > y

x > y
2

4; I. x + 8x + 5x + 40 = 0

52.

or, x(x + 8) + 5(x + 8) = 0

or, (x + 5) (x + 8) = 0
x = - 5, - 8
II. y 2 + 2y + 5y + 10 = 0
or, y(y + 2) + 5(y + 2) = 0
or, (y + 2)(y + 5) = 0

53.

y = - 2, - 5
x y
4; I. x =

3 2
,
5 3

II. 6y2 - 3y - 2y + 1 = 0
or, 3y(2y - 1) -1(2y - 1) = 0
or, (3y - 1)(2y - 1) = 0

or, (y - 2)(7y - 15) = 0

48.

13x 81
8

x < y
1; I. 15x2 - 10x - 9x + 6 = 0
or, 5x(3x - 2) -3(3x - 2) = 0
or, (5x - 3) (3x - 2) = 0

or, 7y(y - 2) - 15(y - 2) = 0

47.

925
5
185

65 81 16

2
8
8

II. 7y2 - 14y - 15y + 30 = 0

y = 2,

+ 925 = 0

2197

2; I.

x 172

x = 13.11
II. y2 - 14y - 15y + 210 = 0
or, y(y - 14) - 15(y - 14) = 0
or, (y - 14) (y - 15) = 0
y = 14, 15
x < y
4; I. 3x2 -12x - 8x + 32 = 0
or, 3x(x - 4) - 8(x - 4) = 0
or, (x - 4) (3x - 8) = 0
x = 4,

x = 13

8
3

II. 2y2 - 8y - 11y + 44 = 0

or, 2y(y - 4) -11(y - 4) = 0
or, (y - 4) (2y - 11) = 0

II. 2y2 - 28y - 26y + 364 = 0

or, 2y(y - 14) - 26(y - 14) = 0
or, (2y - 26) (y - 14) = 0
y = 14, 13

y 4,

x y
49.

1; I. 5x2 - 15x - 12x + 36 = 0

or, 5x(x - 3) - 12(x - 3) = 0
or, (5x - 12) (x - 3) = 0

54.

x
2; 4
12x
12x
-

11
2

y
eqn (I) - 3 eqn (II),
+ 32y = -8
+ 54y = 3
.

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

296
or, (x + 6)2 = 0

-22y = -11
y =

55.

or, x + 6 = 0

1
and x = -2
2

or, x = - 6
II. y 2 + 15y + 56 = 0

x < y
3; I. 2x2 - 8x - 7x + 28 = 0
or, 2x(x - 4) - 7(x - 4) = 0
or, (x - 4) (2x - 7) = 0
x = 4,

or, y2 + 7y + 8y + 56 = 0
or, y(y + 7) + 8(y + 7) = 0
or, (y + 7) (y + 8) = 0
y = -7, -8

7
2

x > y

II. 10y - 35y + 34y - 119 = 0

or, 5y(2y - 7) + 17(2y - 7) = 0
or, (2y - 7)(5y + 17)

62.

1;

I. x2 = 35
x =

35

56.

II.

58.

1
=
676

60.

or, y(y + 6) + 7(y + 6) = 0

or, (y + 6) (y + 7) = 0

1
x
26

1
y 3
13824

x > y
63.

5; I. 2x2 - 3x - 35 = 0
or, 2x2 - 10x + 7x - 35 = 0

1
y
24

or, 2x(x - 5) + 7(x - 5) = 0

or, (2x + 7) (x - 5) = 0
x =

7
,
2

II. y2 - 7y + 6 = 0
or, y2 - y - 6y + 6 = 0
or, y(y - 1) - 6(y - 1)
or, (y - 1)(y - 6) = 0

30
7

y = 1, 6
No relation can be established between x and y.

II. y 2 + 6y + 5y + 30 = 0
or, y(y + 6) + 5(y + 6) = 0
or, (y + 5) (y + 6) = 0
y = -5, - 6
ie, x > y
1; I. x2 = 7x
or, x2 - 7x = 0
or, x(x - 7) = 0
x = 0, 7
II. (y + 7)2 = 0
or, (y + 7) = 0
y = -7
ie, x > y
5; I. 2x2 - 6x + 11x - 33 = 0
or, 2x(x - 3) + 11(x - 3) = 0
or, (2x + 11) (x - 3) = 0
x = 3,

64.

4; I. 6x2 - 29x + 35 = 0
or, 6x2- 15x - 14x + 35 = 0
or, 3x(2x - 5) -7(2x - 5) = 0
or, (3x - 7) (2x - 5) = 0

7 5
,
3 2

II. 2y 2 - 19y + 35 = 0
or, 2y2 - 14y - 5y + 35 = 0
or, 2y(y - 7) -5 (y - 7) = 0
or, (2y - 5)(y - 7) = 0

y=

5
,7
2

x y
65.

2; I.

12x2 - 47x + 40 = 0

or, 12x2 - 32x - 15x + 40 = 0

11
2

or, 4x(3x - 8) -5(3x - 8) = 0

II. y2 - 3y + 2y - 6 = 0
or, y(y - 3) + 2(y - 3) = 0
or, (y + 2)(y - 3) = 0
y = - 2, 3 i.e no relation exists between x and y
61.

y = -6, - 7

ie, x < y
3; On solving these two equations, we get
x = -2, y = 6
ie, x < y
1; I. 7x2 - 28x + 30x - 120 = 0
or, 7x(x - 4) + 30(x - 4) = 0
or, (x - 4) (7x + 30) = 0
x = 4,

59.

or, y2 + 6y + 7y + 42 = 0

x y
3; I. 676x2 - 1 = 0
or x2

57.

II. y + 13y + 42 = 0

7 17
,
2 5

1; I. x2 + 12x + 36 = 0

or, (4x - 5) (3x - 8) = 0

x =

5 8
,
4 3

II. 4y2 + 3y - 10 = 0

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

297
or, 4y2 + 8y - 5y - 10 = 0

or, x2 + 7x - x - 7 = 0

or, 4y(y + 2) -5(y + 2) = 0

or, x(x + 7) -1 (x + 7)= 0

or, (4y - 5) (y + 2) = 0

or, (x - 1) (x + 7) = 0
x = 1, -7

5
, 2
4

y=

II. 41y + 17 = 140

or, 41y = 140 - 17 = 123

x y
66.

4; I. x2 + 7x - 4x - 28 = 0

123
3
41

y =

or, x(x + 7) - 4 (x + 7) = 0
or, (x - 4)(x + 7) = 0
x = 4, - 7

x < y
71.

II. y 2 - 11y + 28 = 0

or, x2 + 7x - 4x - 28 = 0

67.

or, y - 7y - 4y + 28 = 0

or, x (x + 7) - 4 (x + 7) = 0

or, y (y - 7) -4(y - 7) = 0

or, (x - 4) (x + 7) = 0

or, (y - 4) (y - 7) = 0

or, x = 4, -7

y = 4, 7

II. y 2 + 9y + 7y + 63 = 0

x y

or, y(y + 9) + 7(y + 9) = 0

1; I. 6x2 - 17x + 12 = 0

or, (y + 7)(y + 9) = 0

or, 6x2 - 9x - 8x + 12 = 0

or, y = -7, -9

or, 3x (2x - 3) - 4 (2x - 3) = 0

x y

or, (3x - 4) (2x - 3) = 0

72.

4 3
x ,
3 2

y = 13
x y

or, 3y (2y - 1) - 2 (2y - 1) =0

or, (3y - 2) (2y - 1) = 0

73.

or, (8x - 9) (x - 5) = 0
or, x = 5, 9/8

x > y

2; I.

II. 8y2 + 8y - 9y -9 = 0
or, 8y (y + 1) -9 (y + 1) = 0

256
576

or, (8y - 9) (y + 1) = 0

16 2

24 3
74.

or, 3y (y + 1) - 2(y + 1) = 0

eqn (II) 6
.

-89y = 89

or, (3y - 2) (y + 1) = 0

2
, 1
3

89
1 and x 2
89

x < y

x y
75.

5; I. x2 = 64

4; I. x2 - 8x + 15 = 0

x = 8

or, x2 - 3x - 5x + 15 = 0

II. y 2 = 9y

or, x (x - 3) -5 (x - 3) = 0

or, y - 9y = 0

or, (x - 3) (x - 5) = 0

or, y (y - 9) = 0

or, x = 3, 5

y = 0, 9

II. 2y2 - 10y + 55 = 0

no relationship can be established between x

and y.

or, 2y (y - 5) -11 (y - 5) = 0

70.

3; 42x - 17y = -67

42x + 72y = -156

or, 3y2 + 3y - 2y - 2 = 0

69.

9
, 1
8

x y

II. 3y2 + y - 2 = 0

y =

2; I. 8x2 - 40x - 9x + 45 = 0
or, 8x (x - 5) -9 (x - 5) = 0

2 1
,
3 2

2197

II. y 2 = 169

II. 6y - 3y - 4y + 2 = 0

2; I. x =

x = 13

68.

2; I. x2 + 3x - 28 = 0

or, (y - 5)(2y - 11) = 0

3; I. x2 + 6x - 7 = 0

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

298
or, y = 5,

76.

or 2y(y - 1) - 3(y - 1) = 0
or (2y - 3) (y - 1) = 0

11
2

y 1, 3

x y
1; I. 2.3p - 20.01 = 0

20.01
8.7
2.3

82.

II. 2.9q - p = 0
q

or, p = 2.9q

x 4, 7

8.7
8.7
2.9

78.

79.

80.

2; I. p =

II. 4y - 16y + 15 = 0
or 4y2 - 6y - 10y + 15 = 0
or 2y (2y - 3) - 5(2y - 3) = 0
or (2y - 5) (2y - 3) = 0

81.

1764

p = 42
II. q2 = 1764
q = + 42
ie p > q
5; I. p2 - 26p + 168 = 0
p2 - 12p - 14p + 168 = 0
p(p - 12) - 14(p - 12) = 0
(p - 12) (p - 14) = 0
p = 12, 14
II. q2 - 25q + 156 = 0
q2 - 13q - 12q + 156 = 0
q(q - 13) - 12(q - 13) = 0
(q - 12) (q - 13) = 0
q = 12, 13
Hence, no relation can be established between p
and q
2; I. p2 - l3q + 42 = 0
p2 - 6p - 7p + 42 = 0
p(p - 6) - 7(p - 6) = 0
(p - 6) (p - 7) = 0
p = 6, 7
II. q2 + q - 42 = 0
q2 + 7q - 6p - 42 = 0
q(q + 7) - 6(q + 7) = 0
(q - 6)(q + 7) = 0
q = 6, - 7 ie p q
3; eqn(I) 3
18p - 15q = -141
eqn (II) 5 25p + 15q = 55
43p
= -86

ie p > q
77.

Hence x < y
1; I. 2x2 - 8x - 7x + 28 = 0
or 2x (x - 4) - 7(x - 4) = 0
or (2x - 7) (x - 4) = 0

y 5 , 3
2 2
83.

84.

85.

86.

86
2
43

5p + 3q = 11
3q = 11 - 5p
3q = 11 + 10
3q = 21
q = 7 ie p < q
3; I. 2x2 + 13x - 7 = 0
or 2x2 + 14x - x - 7 = 0
or 2x (x + 7) - 1 (x + 7) = 0
or (2x - 1) (x + 7) = 0

x 1 , 7
2
II. 2y2 - 5y + 3 = 0
or 2y2 - 2y - 3y + 3 = 0

Hence x > y
4; I. x2 + 8x + 16 = 0
or (x + 4)2 = 0
or x + 4 = 0 x = -4
II. y 2 = 16 y = 4
Hence, x y
5; I. x2 - 2x - 24 = 0
or x2 + 4x - 6x - 24 = 0
or x(x + 4) - 6(x + 4) = 0
or (x - 6) (x + 4) = 0
x = 6, - 4
II. y 2 + 8y = 0
or y(y + 8) = 0
y = 0, - 8 ie No relation can be established
between x and y.
1; I. x2 + 4x = 0
or x(x + 4) = 0 x = 0, - 4
II. y 2 + 10y + 25 = 0
or (y + 5)2 = 0
or y + 5 = 0
y = - 5
x > y
1; I. 2x2 + 2x x 1 = 0
or 2x(x + 1) 1(x + 1) = 0
or (2x 1) (x + 1) = 0

x 1,

1
2

II. 2y2 + 3y + 10y + 15 = 0

or y(2y + 3) + 5(2y + 3) = 0
or (y + 5) (2y + 3) = 0

y 5,

87.

3
2

x > y
5; I. x2 + 4x + 8x + 32 = 0
or x(x + 4) + 8(x + 4) = 0
or (x + 4) (x + 8) = 0
x = 4, 8
II. 2y2 + 6y + 9y + 27 = 0
or 2y(y + 3) + 9(y + 3) = 0
or (2y + 9) (y + 3) = 0

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

299
9
y , 3
2
No relation can be established between x and y.
88.

1; I. 6x2 9x 8x + 12 = 0
or 3x(2x 3) 4(2x 3) = 0
or (2x 3) (3x 4) = 0

II. 7y 2 7y 6y + 6 = 0
or 7y(y 1) 6(y 1) = 0
or (7y 6) (y 1) = 0

x > y
No relation between x and y.
5; I. x2 11x 71x + 781 =0
or x(x 11) 71(x 11) = 0
or(x 11)(x 71) = 0
x = 11, 71
II. y 2 = 5041
y = 71
2; I. 6x2 15x 32x + 80 = 0
or 3x(2x 5) 16(2x 5) = 0
or (3x 16) (2x 5) = 0

95.

6
7

y 1,

90.

3 4
,
2 3

89.

94.

97.

5
2

x y

98.

4; I. 3x2 - 12x + 5x - 20 = 0
or 3x(x - 4) + 5(x - 4) = 0
or (3x + 5) (x - 4) = 0

5
x ,4
3

92.

93.

x = 44 - 41 = 3
x < y
3; I. 9x2 + 18x + 9 = 0
or x2 + 2x + 1 = 0
or (x + l)2 = 0
x + 1 = 0, or x = -1
II. y 2 - y - 2y + 2 = 0
or y(y - 1) -2(y - 1) = 0
or (y - 1) (y - 2) = 0

1; eqn
12x
12x
-

(I) 3 - eqn (II) 4

+ 21y = 126
- 44y = -4
+
+
.
65y = 130
y = 2
and x = 7
3; I. 9x2 - 18x - 1 lx + 22 = 0
or 9x(x - 2)- 11(x - 2) = 0
or (x - 2)(9x - 11) = 0

11
9

II. y2 - 3y - 4y + 12 - 0
or y(y - 3) - 4(y - 3) = 0
or (y - 3) (y - 4) = 0
y = 3, 4
x < y
4; I. 3x2 - 4x - 32 = 0
or 3x2 - 12x + 8x - 32 = 0
or 3x(x - 4) + 8(x - 4) = 0
or (3x + 8) (x - 4) = 0

x 4,

II. y 2 - 8y + 16 = 0
or (y - 4)2 = 0
(y - 4) = 0
or y = 4
x y
5; I. x2 - 72 = 0
or x2 = 72
x = + 8.485
II. y 2 - y - 8y + 8 = 0
or y(y - 1) - 8(y - 1) = 0
or (y - 1) (y - 8) = 0
y = 1, 8
1; I. 9x2- l14x + 361 =0
or (3x - 19)2 = 0
3x - 19 = 0
x =

640
4 and x=11y-41
160

x 2,

II. 2y2 4y 5y + 10 = 0
or 2y(y 2) 5(y 2) = 0
or (y 2) (2y 5) = 0

91.

19
= 6.33
3

eqn (II) 13

x < y
96.

16 5
,
3 2

y 2,

II. y 2 = 36
y = 6
x > y
3; I. 13x + 17y = 107
13x 143y = 533
160y = 640

8
3

II. 2y2 - 8y - 9y + 36 = 0
or 2y(y - 4) - 9(y - 4) = 0
or (2y - 9) (y - 4) = 0
or (2y - 9) (y - 4) = 0

y 4,

99.

9
2

x y
1; I. 3x2 - 21x + 2x - 14 = 0
or 3x(x - 7) + 2(x - 7) = 0
or (3x + 2) (x - 7) = 0

x 7,
II.
or
or
or

2
3

2y2 + 5y + 3 = 0
2y2 + 2y + 3y + 3 = 0
2y(y + 1) + 3(y + 1) = 0
(2y + 3) (y + 1) = 0

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

300
21x - 12y = 87

3
, 1
2

20x + 12y = 200

x > y
100. 5; I. x2 + 14x + 49 = 0
or (x + 7)2 = 0
x + 7 = 0
or, x = -7
II. y2 + 9y = 0
or y(y + 9) = 0
y = 0, -9
ie no relation between x and y.
101. 3; I. 9x2 = 1

x2

41x = 287
x = 7
Putting the value of x in (I), we get
y = 5
Hence, x > y
105. 1; I. x2 = 5

x 5 2.236
II. 4y2 - 24y + 35 = 0
or, 4y2 - 14y - 10y + 35 = 0

1
9

or, 2y(2y - 7) - 5(2y - 7) = 0

or, (2y - 5) (2y - 7) = 0

1
x
3

II. 4y 2 + 11y - 3 = 0
or, 4y2 + 12y - y - 3 = 0
or, 4y(y + 3) - 1(y + 3) = 0

1
, 3
4

Hence, x < y
106. 3; I. 35x2 - 28x - 25x + 20 = 0
or 7x(5x - 4) - 5(5x - 4) = 0
or (7x - 5) (5x - 4) = 0

Hence, there is no relation between x and y.

or, 3x2 + 6x - x - 2 = 0
or, 3x(x + 2) - 1(x + 2) = 0

or, (3x - 1) (x + 2) = 0

1
3

107. l ;

II. 2y - 2y - 5y + 5 = 0
or, 2y(y - 1) - 5(y - 1) = 0

5
2

Hence, x < y
103. 5; I. 6x2 + 13x + 5 = 0
or, 6x2 + 3x + 10x + 5 = 0
or, 3x(2x + 1) + 5(2x + 1) = 0
or, (3x + 5) (2x + 1) = 0

5
1
x ,
3
2
II. 3y2 + 11y + 10 = 0
or, 3y2 + 6y + 5y + 10 = 0
or, 3y(y + 2) + 5(y + 2) = 0
or, (3y + 5) (y + 2) = 0

5
y , 2
3
Hence, x y
104. 4; I.

7x - 4y = 29

II. 5x + 3y = 50
(I) 3 + (II) 4

7 6
,
8 7

x < y

y 1,

5 4
,
7 5

II. 56y2 - 48y - 49y + 42 = 0

or 8y(7y - 6) - 7(7y - 6) = 0
or (8y - 7) (7y - 6) = 0

102. 1; I. 3x2 + 5x - 2 = 0

x 2,

5 7
, 2.5, 3.5
2 2

I. x =

4913
x = 17
II. 13y = 246 - 3x
or 13y = 246 - 51 = 195
y = 15
x > y
108. 2; I. x2 - 7x + 2x - 14 = 0
or x(x - 7) + 2(x - 7) = 0
(x + 2) (x - 7) = 0
x = -2, 7
II. y 2 + 5y + 2y + 10 = 0
or y(y + 5) + 2(y + 5) = 0
or (y + 2) (y + 5) = 0
y = -2, -5
x y
109. 5; I. x2 = 3481
x = 59

II. 3y2 =

216000

3y = 60
y = 20
No relation
110. 1; I. 5x2 + 5x - 3x - 3 = 0
or 5x (x + 1) - 3(x + 1) = 0
or (5x - 3) (x + 1) = 0

3
, 1
5

II. 2y2 + 4x + 3y + 6 = 0
or 2y(y + 2) + 3(y + 2) = 0

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

301
or (2y + 3) (y + 2) = 0

3
, 2
2

ie x > y
111. 1; I. 20x2 - 35x - 32x + 56 = 0
or 5x(4x - 7) - 8(4x - 7) = 0
or (5x - 8) (4x - 7) = 0

or (y + 3) (y + 4) = 0
y = -3, -4
x > y
118. 3; I. 3x2 - 23x + 40 = 0
or 3x2 - 15x - 8x + 40 = 0
or 3x(x - 5) - 8(x - 5) = 0
or (3x - 8) (x - 5) = 0

x 5, 8

x = 8,7
5 4
II. 56y2 - 32y - 35y + 20 = 0
or 8y(7y - 4) - 5(7y - 4) = 0
or (8y - 5) (7y - 4) = 0
y

5 4
,
8 7

II. y =

4096
y = 16
x y
113. 3; I. 2x2 + 16x - 5x - 40 = 0
or 2x(x + 8) - 5(x + 8) = 0
or (2x - 5) (x + 8) = 0

x = 5 , 8
2
II. 4y2 - 16y - 11y + 44 = 0
or 4y(y - 4) - 11(y - 4) = 0
or(4y - 11) (y - 4) = 0
y = 4,

11
4

x < y

114. 1; I. 7x = 4y + 85
or 7x = 4 26 + 85 (Put y = 26)
x =
II. y =

189
= 27
7
3

17576

y 6, 11

x > y

112. 4; I. x4 = 65536
x = +16

II. y =

14641
y = 121
x y
116. 2; I. x2 + 42 = 13x
or x2 - 13x + 42 = 0
or x2 - 7x - 6x + 42 = 0
or x(x - 7) - 6(x - 7) = 0
or (x - 6) (x - 7) = 0
x = 6, 7

x 7 , 5
5 3
II. 4y 2 - 8y - 7y + 14 = 0
or 4y(y - 2) - 7(y - 2) = 0
or (4y - 7) (y - 2) = 0
y = 2, 7 4
x < y
120. 3; I. x2 - x + 6x -6 = 0
or x(x - 1) + 6(x - 1) = 0
or (x - 1) (x + 6) = 0
x = 1, -6
II. 2y 2 - 6y - 5y + 15 = 0
or 2y(y - 3) - 5(y - 3) = 0
or (y - 3) (2y - 5) = 0
y = 3, 5 2

121. 3; I.
or
or
or

x < y
2x2 - 21x + 54 = 0
2x2 - 12x - 9x + 54 = 0
2x(x - 6) - 9(x - 6) = 0
(x - 6) (2x - 9) = 0

x 6,

122. 2;

II. y 1296
y = 6
x y
117. 1; I. x2 + x - 2 = 0
or x2 + 2x - x - 2 = 0
or x(x + 2) - 1(x + 2) = 0
or (x - 1) (x + 2) = 0
x = 1, - 2
II. y2 + 7y + 12 = 0
or y2 + 3y + 4y + 12 = 0
or y(y + 3) + 4(y + 3) = 0

x < y
119. 3; I. 15x2 - 25x - 21x + 35 = 0
or 5x(3x - 5) - 7(3x - 5) = 0
or (5x - 7) (3x - 5) = 0

y = 26

x > y
115. 4; I. x2 = 14641
x = 121

II. 2y2 - 23y + 66 = 0

or 2y2 - 12y - 11y + 66 = 0
or 2y (y - 6) -11 (y - 6) = 0
or (y - 6)(2y - 11) = 0

9
2

II. y2 - 14y + 49 = 0
or (y - 7)2 = 0
or y - 7 = 0
y = 7
Hence x < y
I. x2 - 19x + 70 = 0
or x2 - 5x - 14x + 70 = 0
or x(x - 5) - 14(x - 5) = 0
or (x - 5) (x - 14) = 0
x = 5, 14
II. 2y2 - 10y - 7y + 35 = 0
or 2y(y - 5) - 7(y - 5) = 0
or (y - 5) (2y - 7) = 0
y = 5,

123. 4;

7
Hence x y
2

I. 3x2 + 5x - 8 = 0
or 3x2 - 3x + 8x - 8 = 0
or 3x(x - 1) + 8(x - 1) = 0

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

302
or (x - 1) (3x + 8) = 0

x 1,

124. 4;

or (3y - 1) (2y - 1) = 0

8
3

II. y2 - 4y + 3 = 0
or y2 - y - 3y + 3 = 0
or y(y - 1) - 3(y - 1) = 0
or(y - l)(y - 3) = 0
y = 1, 3 Hence, x y
I. 12x2 - 16x + 5 = 0
or 12x2 - 6x - 10x + 5 = 0
or 6x(2x - 1) - 5(2x - 1) = 0
or (6x - 5) (2x - 1) = 0

x
130. 2; I.
or
or
x
II.
or
or

125. 2;

I.
or
or
or

5 5
,
Hence, x y
3 6

3x2 + 11x + 8 = 0
3x2 + 3x + 8x + 8 = 0
3x(x+ 1) + 8(x + 1) = 0
(x + 1) (3x + 8) = 0

x 1,
II.
or
or
or

126. 5; I. x =

8
3
132. 2;

II. y =

x = -l,

1
4

2x2 4x 13x 2 13 0 ...(i)

x 2,

13
2

10y 2 18y 5 13y 9 13 0 ...(ii)

or, 2y 5y 9 13(5y 9) 0

or, (2y 13)(5y 9) 0

y

9
,
5

13
2

Hence, x y.
133. 5; 6x2 + 17 - 3x2 - 20 = 0 ... (i)
or, 3x2 = 3
x l
5y 2 - 12 - 9y2 + 16 = 0 .... (ii)
or, 4y2 = 4
y 1
Hence x = y.
134. 2; 13x + 17 = 134 .... (i) .

117
9.
13

(36l) 1/2y2 - 270 = 1269

or, 19y2 = 1629 + 270 = 1539

y2

II. 6y2 - 3y - 2y + 1 = 0
or 3y(2y - 1) - l(2y - 1) = 0

or, x 2 2x 13 0

x = 71

y = 71
5041
x = y
127. 1; Eqn(l) 9 - Eqn (II) 5
45x + 63y = -387
45x - 85y = 205
+
.
148y = -592
y = -4 and x = -3
x > y
128. 4; I.
x2 + 11x + 30 = 0
or x(x + 5) + 6(x + 5) = 0
or (x + 5) (x + 6) = 0
x = -5, -6
II. y2 + 4y + 5y + 20 = 0
or y(y + 4) + 5(y + 4) = 0
or (y + 4) (y + 5) = 0
y = -4,-5
x y
129. 3; I.
4x2 + 4x - x - l = 0
or 4x(x+ 1)- l(x + 1) = 0
or (4x - 1) (x + 1) = 0

152
4
38

or, 2x x 2 13 x 2 0

8
Hence, x y
3

357911

Putting the value of y in equation (i), we have

4x + 3 x 4 = 40
or, 4x = 40 12 = 28
x = 7
Hence, x > y.

3y2 + 20y + 32 = 0
3y2 + 12y + 8y + 32 = 0
3y(y + 4) + 8(y + 4) = 0
(3y + 8) (y + 4) = 0

y 4,

9
2

x y
131. 1; 4x + 3y = 40 .........(i) 6
6x - 5y = 22 .........(ii) 4
24x +18y = 240
24x - 20y = 88
+
38y = 152

18y2 - 45y + 25 = 0
18y2 - 30y - 15y + 25 = 0
6y(3y - 5) - 5(3y - 5) = 0
(3y - 5) (6y - 5) = 0

< y
3x2 + 9x + 6x + 18 = 0
3x(x + 3) + 6(x + 3) = 0
(x + 3)(3x + 6) = 0
= -3, -2
2y2 + 6y + 9y+ 27 = 0
2y(y + 3) + 9(y + 3) = 0
(2y + 9)(y + 3) = 0

y 3,

1 5
x ,
2 6
II.
or
or
or

1 1
,
2 3

1539
81
19

y 9

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

303
135. 4;

136. 3;

137. 2;

138. 4;

Hence, x y.
64x2 = 256 .... (i)
or, x2 = 4
x = 2
14y3 - 12y3 = 16 .... (ii)
or, 2y3 = 16
y3 = 8
y = 2
Hence x y.
15x - 21y = -72
91x + 21y = 602
106x
= 530
x = 5,
y = 7
x < y
I.
x2 - 13x + 40 = 0
or x2 - 5x - 8x + 40 = 0
or x(x - 5) -8(x - 5) = 0
or (x - 5)(x - 8) = 0
x = 5, 8
II. y2 + 3y - 40 = 0
or y2 - 5y + 8y - 40 = 0
or y(y - 5) + 8(y - 5) = 0
or (y - 5) (y + 8) = 0
y = 5, -8
Hence, x y
I. 8x2 -26x + 15 = 0
or 8x2 - 20x - 6x + 15 = 0
or 4x(2x - 5) - 3(2x - 5) = 0
or (4x - 3) (2x - 5) = 0
x =

3 5
,
4 2

II.
or
or
or

2y2-17y + 30 = 0
2y2 - 12y - 5y + 30 - 0
2y(y - 6) - 5(y - 6) = 0
(2y - 5) (y - 6) = 0

y=

5
,6
2

x y
139. 4; I.
x2 = 484

x = + 22
II. y2 - 45y + 506 = 0
or y2 - 22y - 23y + 506 = 0
or
y(y - 22) - 23(y - 22) = 0
or
(y - 22) (y - 23) = 0
y = 22, 23
x y
140. 5; I. 13x -21 = 200- 4x
or 13x + 4x = 200 + 21

II. y =

or, 2p(2p - 5) - 3(2p - 5) = 0

or, (2p - 3) (2p - 5) = 0
p =

3 5
,
2 2

II. 2q2 + 5q - 7 = 0
or, 2q2 + 7q - 2q - 7 = 0
or, q(2q + 7) - 1(2q + 7) = 0
or, (q - 1) (2q + 7) = 0
q = 1,

7
2

143. 2; I. p2 = 49
P = 7
II. q2 + 15q + 56 = 0
or, q2 + 8q + 7q + 56 = 0
or, q(q + 8) + 7(q + 8) = 0
or, (q + 7) (q + 8) = 0
q = -7, -8
p q
144. 5; I. 2p2 + 5p - 12 = 0
or, 2p2 + 8p - 3p - 12 = 0
or, 2p(p + 4) - 3(p + 4) = 0
or, (2p - 3) (p + 4) = 0
p =

3
, 4
2

II. 2q2 - q - 1 = 0
or, 2q2 - 2q + q - 1 = 0
or, 2q(q - 1) + 1(q - 1) = 0
or, (2q + 1) (q - 1) = 0
q = 1,

1
2

No reation between p
145. 2; I. p2 - 12p + 35 = 0
or p2 - 5p - 7p + 35 =
or p(p - 5) - 7(p - 5) =
or (p - 7) (p - 5) = 0
p = 5, 7
II. q2 - 25 = 0
or, q2 = 25
q = +5
p q
146. 3; I. 3x2 + 6x + x + 2 =
or 3x(x + 2) + 1(x + 2)
or (x + 2) (3x + 1) = 0
x = 2,

and q.
0
0

0
= 0

1
3

II. 2y2 + 4y + 5y + 10 = 0
or 2y(y + 2) + 5(y + 2) = 0
or (2y + 5) (y + 2) = 0

221
13
17
3

2197
y = 13
x = y
141. 3; I. (p + q)2 = 3136
p + q = +56
II. q + 2513 = 2569
or, q = 2569 - 2513 = 56
Putting the value of q in (I) we have,
p = 0. -112
p < q
142. 1; I. 4p2 - 16p + 15 = 0
or, 4p2 - 10p - 6p + 15 = 0

y = 2,

5
2

x y
147. 4; I. x2 + 2x x 2 = 0
or x(x + 2) 1(x + 2) = 0
or (x 1) (x + 2) = 0
x = 1, 2
II. y2 y 2y + 2 = 0
or y(y 1) 2(y 1) = 0
or (y 1)(y 2)
y = l, 2
x y

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

304
148. 1; I. 20x2 15x 36x + 27 = 0
or 5x(4x 3) 9(4x 3) = 0
or (5x 9) (4x 3) = 0

x =

II. 121y 2 63 260

9 3
,
5 4

x =

or, 11y2 + 63 = 260

or, 11y2 = 260 - 216 = 44
or, y2 = 4
y = +2
x > y
154. 3; I. 17x = 169+ 14 + 25 + 4x
or, 13x = 208

II. 5y2 10y 6y + 4 = 0

or 5y(3y 2) 2(3y 2) = 0
or (5y 2) (3y 2)

2 2
,
5 3

x > y
149. 5; I. 7x2 + 21x 5x 15 = 0
or 7x(x + 3) 5(x + 3) = 0
or (x + 3) (7x 5) = 0

x 3,
II.
or
or

150. 1; I.

II.
or
or
or

x =

5
7

y2 7y + y 7 = 0
y(y 7) + 1(y 7) = 0
(y + 1) (y 7)
y = 1, 7
no relation between x and y.
x2 = 729
x = 27
y2 + 58y + 840 = 0
y2 + 28y + 30y + 840 = 0
y(y + 28) + 30(y + 28) = 0
(y + 30) (y + 28) = 0
y = 30, 28
x > y

151. 2; I.

1
15
9

(x)2
x
x

or, x = 4
II. y10 - (36)5 = 0
or, y10 = (36)5
or y = (36)

36

x =

152. 1;

x > y
1
2

2
2
153. 1; I. (441) x 111 (15)
2

7
2

x 4,

II. 2y2 + 10y - 7y - 35 = 0

or 2y(y + 5) - 7(y + 5) = 0
or (2y - 7)(y + 5) = 0

x = 4

36 6
x y
5x + 2y = 96
... (i)
21x + 15y = 489
... (ii)
Now, eqn (i) 15 and eqn (ii) 2
75x + 30y = 1440
42x + 30y = 978
.
33x
= 462
x = 14
Putting the value of x in eqn (i), we get
5 14 + 2y = 96 or, 2y = 96 - 70 = 26

26
or, y =
= 13
2

7
,2
3

II. y2 - 7y + 12 = 0
or, y2 - 4y - 3y + 12 = 0
or, y(y - 4) -3(y - 4) = 0
or,(y -3)(y - 4) = 0
y = 4, 3
x < y
156. 2; I. 2x2 - 8x - 7x + 28 = 0
or 2x(x - 4) - 7(x - 4) = 0
or (2x - 7) (x - 4) = 0

1
2

y=

208
16
13

II. 9y - 4y = 345 - 260 = 85

or, 5y = 85
y = 17
x < y
155. 3; I. 3x2 - 13x + 14 = 0
or, 3x2 - 7x - 6x + 14 = 0
or,3x(x - 2) -7(x - 2) = 0
or, (3x - 7) (x - 2) = 0

1
15 9
or,
x2 x
x

5
10

336
16
21

157. 1;

158.

7
, 5
2

x y
28x - 20y = 96
28x + 21y = 301
.
-41y = -205
y = 5 and x = 7
x > y
3; I. x =

2744 14

II. y =

487 = 22

484 32
159.

x < y
5; I. x2 - x - 8x + 8 = 0
or x(x - 1) - 8(x - 1) = 0
or (x - 1) (x - 8) = 0
x = 1, 8
II. 2y2 - y - 10y + 5 = 0
or y(2y - 1) - 5(2y - 1) = 0
or (y - 5) (2y - 1) = 0

2 2
or, (21) x 225 111 336
2
or, 21x = 336

y = 5,

1
2

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

305
160.

1; I. 2x2 + 2x + x + 1 = 0
or 2x(x + 1) + 1(x + 1) = 0
or (x + 1)(2x + 1) = 0

x 1,

II.
or
or
or

1
0.5
2

y = 4,

II. 6y2 + 9y + 8y + 12 = 0
or 3y(2y + 3) + 4(2y + 3) = 0
or (3y + 4) (2y + 3) = 0

or,x(x 5 3) 3 3(x 5 3) 0
or,(x 3 3)(x 5 3) 0
x 3 3, 5 3
II.

x = 8 ,7
3
II. 3y2 - 5y - 8 = 0
or 3y2 + 3y - 8y - 8 = 0
or 3y(y + 1) - 8(y + 1) = 0
or (3y - 8) (y + 1) = 0
or (3y - 8) (y + 1) = 0

or (y 4 2y)(y 3 2) 0
or (y 3 2)(y 4 2)
y 3 2,4 2
Hence relation cannot be established between x
and y.

x y
162. 4; I.
5x2 + 26x - 24 = 0
or 5x2 + 30x - 4x - 24 = 0
or 5x(x + 6) - 4(x + 6) = 0
or (5x - 4) (x + 6) = 0

167. 2; I.

or

x( x 4 2) 3 2( x 4 2) 0

or ( x 3 2)( x 4 2) 0

4
, 6
5

Now, if

then

x 3 2 0

x 3 2

x = 9 2 = 18

4
,6
y =
5

If x 4 2 0

then

x y
163. 1; I. x2 - 7x = 0
or x (x - 7) = 0
x = 0, 7
II. 2y2 + 5y + 3 = 0
or 2y2 + 2y + 3y + 3 = 0
or 2y(y + 1) + 3(y + 1) = 0
or (2y + 3) (y + 1) = 0

x 4 2

x = 16 2 = 32
II.

y 5 2y 12 0

or y 3 2y 2 2y 12 0
or

y( y 3 2) 2 2( y 3 2) 0

or ( y 2 2)( y 3 2) 0

3
2

If ( y 2 2) 0

x > y
164. l ; 7x - 4y = 40
and 8x + 8y = 8
or x + y = 1
Solving (i) and (ii), we have
x = 4, y = -3
x > y
165. 3; I. 15x2 - 4!x + 14 = 0
or 15x2 - 6x - 35x + 14 = 0
or 3x(5x - 2) - 7(5x - 2) = 0
or (3x - 7)(5x - 2) = 0

7 2
,
3 5

x 7 2x 24 0

or x 4 2x 3 2x 24 0

II. 5y2 - 30y - 4y + 24 = 0

or 5y(y - 6) - 4(y - 6) = 0
or (5y - 4) (y - 6) = 0 4

x =

y 2 2y 24 0

or y 2 4 2y 3 2y 24 0

8
y = 1,
3

y = 1,

x 2 8 3x 45 0

or, x2 5 3x 3 3x 45 0

x > y
3x2 - 29x + 56 = 0
3x2 - 21x - 8x + 56 = 0
3x(x - 7) - 8(x - 7) = 0
(3x - 8) (x - 7) = 0

x =

5
2

x < y
166. 5; I.

4
3
y , 1.33, 1.5
3
2

161. 2; I.
or
or
or

2y2 - 13y + 20 = 0
2y2 - 8y - 5y + 20 = 0
2y(y - 4) - 5(y - 4) = 0
(2y - 5) (y - 4) = 0

...(i)

then

y 2 2

y = 4 2 = 8
...(ii)

if y 3 2 0
then

y 3 2

y = 9 2= 18
x y
168. 4; I. 12x2 - 17x + 6 = 0
or 12x2 - 9x - 8x + 6 = 0
or 3x(4x - 3) - 2(4x - 3) = 0
or (3x - 2) (4x - 3) = 0

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

306
If 3x - 2 = 0
then 3x = 2

x 7 3, 5 5
II. y 2 5 5y 30 0

2
3

or, y 2 3 5y 2 5y 30 0

If 4x - 3 = 0
then x =

or, (y 3 5), (y 2 5) 0

3
4

y 3 5, 2 5

II. 20y2 - 31y + 12 = 0

or 20y2 - 16y - 15y + 12 = 0
or 4y(5y - 4) - 3(5y - 4) = 0
or (4y - 3) (5y - 4) = 0

169. 5;

173. 3; I.
14x2 + 11x - 15 = 0
or (7x - 5) (2x + 3) = 0

3 4
,
4 5

II. 20y2 - 31y + 12 = 0

or (4y - 3), (5y - 4) = 0

Hence x y
I. 3x2 - 8x + 4 = 0
or 3x2 - 6x - 2x + 4 = 0
or (3x - 2) (x - 2) = 0

2
x 2,
3

174. 1;

II. 4y2 - 15y + 9 = 0

or 4y2 - 12y - 3y + 9 = 0
or 4y(y - 3) - 3(y - 3) = 0
or (4y - 3) (y - 3) = 0

5
3
,
7
2

3 4
,
4 5

x < y
I. 5x + 4y = 41
II. 4x + 5y = 40
On solving both equations, we have
x = 5 and y = 4
x > y

... (i)
... (ii)

15

3
,3
4

175. 3; I.

Relation cannot be established between x and y.

170. 1; I. x2 - 16x + 63 = 0
or x2 - 9x - 7x + 63 = 0
or x(x - 9) - 7(x - 9) = 0
or (x - 7) (x - 9) = 0
x = 7, 9
II. y 2 - 2y - 35 = 0
or y2 - 17y + 5y - 35 = 0
or y(y - 7) + 5(y - 7) = 0
or (y + 5) (y - 7) = 0
y = -5, 7
Hence, x y
171. 5; I.

63x 94 x 35 0

or, or,63x 49 x 45 x 35 0

or, (9 x 7)(7 x 5) 0
x

49 25
,
81 49

or, 32y 28 y 24 y 21 0
or,(4 y 3).(8 y 7) 0

9 49
,
16 64

172. 1; I.

x 2 7 3x 35 15 5 5x

or, x2 5 5x 7 3x 35 15 0
or, (x 7 3)(x 5 5) 0

15

x = (18)3
9

II.

(19)2
0
y

or y 2 (19)2

y = (19)3
x < y
176. 5; I. 63x 194 x 143 0

or 63x 117 x 77 x 143 0

or (7 x 13)(9 x 11) 0
169 121
,
49 81

II. 99y 225 y 150 0

or 99y 90 y 165 y 150 0

or (11 y 10)(9 y 15) 0

Therefore relation cant be established between x

and y.

(18) 2
0
x2

or x 2 (18) 2

II. 32y 52 y 21 0

100 225
,
121 81

Therefore relation cannot be established between

x and y.
177. 2; I. 16x2 - 40x - 39 = 0
or 16x2 - 52x + 12x - 39 = 0
or (4x- 13) (4x + 3)

13 3
,
4
4

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

307
II. 12y2 - 113y + 255 = 0
or 12y2 - 45y - 68y + 255 = 0
or (4y - 15) (3y - 17) = 0

or, y(y - 2) - 1 (y - 2) = 0

15 17
,
y
4 3

Hence, no relation can be established.

or, (y - 2)(y - 1) = 0
y = 2, l

183. 2; I.

Therefore y > x
or, x < y
178. 2; I. x 7 3x 36 0

or x 7 3. x 36 0

x 169 0

or, x 169
x 13
II.

y2 - 169 = 0

or, y2 = 169

or x 3 3. x 4 3. x 36 0
or, y =

or ( x 3 3)( x 4 3) 0
x = 27, 48

169

y = 13
Hence, x

II. y 5 2y 7 2y 70 0

or y 5 2. y 7 2. y 70 0

y = 50, 98
x < y
179. 1; I.

or (x 4 7)(x 3 7) 0

144

x = 12
II. y -

x 2 7 7x 84 0

or, x2= 112 + 32 = 144

or, x =

or ( y 5 2)( y 7 2) 0

184. 3; I. x2 - 32 = 112

256 = 0

or, y =

256

y = 16
Hence, x < y

x 4 7,3 7

185. 5; I.

II. y 5 5y 30 0

or (y 2 5)(y 3 5) 0

x2 - 25 = 0

or, x2 = 25
or, x =

25

x = 5

y 2 5, 3 5

II. y2 - 9y + 20 = 0

x > y
180. 2; I. 10x + 6y = 13
II. 45x + 24y = 56

4
5
On solving both eqns, x = , y
5
6
x < y
181. 2; I.
x2 - 2x - 15 = 0
or,x2 - 5x + 3x - 15 = 0
or, x(x - 5) + 3(x - 5) = 0
or,(x - 5) (x + 3) = 0
x = 5, -3
II.

y2 + 5y + 6 = 0

or, y2 + 3y + 2y + 6 = 0
or, y(y + 3) + 2(y + 3) = 0
or,(y + 3)(y + 2) = 0

or, y2 - 5y - 4y + 20 = 0
or, y(y - 5) - 4(y - 5) = 0
or, (y - 5) (y - 4) = 0
y = 5, 4
Hence, no relation can be established.
186. 3; 3x + 5y = 69
... (i)
9x + 4y = 108
... (ii)
x + z = 12
... (iii)
Now, from (i) and (ii), we have
3x + 5y = 69
... (i) 4
9x + 4y = 108
... (ii) 5
12x + 20y = 276
45x + 20y = 540
- 33x
= - 264
On subtracting, we get
or, 33x = 264

y = -3, -2
x
182. 5; I.

x =

x2 - x - 12 = 0

or, x2 - 4x + 3x - 12 = 0
or, x(x - 4) + 3(x - 4) = 0

II.

y =

y2-3y + 2 = 0

or, y2 - 2y - y + 2 = 0

Putting the value of x in equation (i), we get 3 8

+ 5y = 69
or, 5y = 69 - 24 = 45

or, (x - 4) (x + 3) = 0
x = 4, -3

264
=
33

45
5

= 9

Again, putting the value of x in equation (iii),

we get

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

308
x + z = 12
or, z = 12 - 8 = 4
Hence, x < y > z

187. 3; I.

y 9

1
3

or,

1
4

9 9 9 ...(I)

II. 2x + 5z = 54 .. (ii)
III. 6x + 4z = 74
or, 3x + 2z = 37
... (iii)
From equation (ii) 2 - (iii) 5, we get
4x + 10z = 108
15x + 10z = 185
.
- 11x
= - 77
or, 11x = 77
x = 7
Putting the value of x in equation (ii), we get
2 7 + 5z = 54
or, 5z = 40
z = 8
Hence, x < y > z
188. 2; I. 2x + 3y + 4z = 66
... (i)
II. 2x + y + 3z = 42
... (ii)
III.3x + 2y + 4z = 63
... (iii)
From (iii) and (i),
x - y = - 3
...(iv)
From equation (i) 3 - equation (ii) 4
6x + 9y + 12z = 198
8x + 4y + 12z = 168
.
2x + 5y
= 30
... (v)
Solving equation (iv) and (v), we get
x = 5, y = 8
Now, on putting the value of x and y in equation
(i),
10 + 24 + 4z = 66
or, 4z = 32

z
189. 1;

32
8
4

Hence, x < y = z
I. (x + z)3 = 1728 = 123
or, x + z = 12
...(i)
II. 2x + 3y = 35
... (ii)
III. x - z = 2
...(iii)
Now, equation (i) and (ii),
x = 7, z = 5
Putting the value x in question (ii) we have,
2 7 + 3y = 35
or, 3y = 35 - 14 = 21
or, y =

21
3

12x + 15y =
35x + 15y =
-23x = - 69
x = 3
Putting the value of
4 3 + 5y =
or, y =

111
180
.

x in equation (i)
37

25
5
5

Now, putting the value of x in equation (ii)

z = 5. Hence, x < y = z
191. 4; I. 7x + 3y = 77 ... (i)
1

II. 2x + 5y = (2601) 2 = 51
Now,
or,

7x + 3y = 77 ... (i) 5
2x + 5y = 51 ... (ii) 3
35x + 15y = 385
6x + 15y = 153
.
29x = 232

232
8
29

Putting the value of x in equation (i), we have

7 8 + 3y = 77
or, 3y = 77 - 56 = 21

21
3

or, y =

= 7

Hence, x > y
192. 3; I.

3x2 - 6x -

or, 3x(x - 2) -

17x 2 17
17

(x - 2) = 0

17
3

x = 2,

II. 10y2 - 18y or, 2y(5y - 9) -

5 17y 9 17y 0
17

or, (2y -

17

or, y

17 9
,
2
5

(5y - 9) = 0

) (5y - 9) = 0

Hence, x = y > z
190. 2; 4x + 5y = 37
... (i)
x + z = 8
... (ii)
7x + 3y = 36
... (iii)
From equation (i) and (iii),
4x + 5y = 37 ... (i) 3
7x + 3y = 36 ... (ii) 5

= 0

or, (3x 17)(x 2) 0

193. 4; I.

= 7

...(ii)

(289)2 x 324 203

or, 17x - 18 = 203

or, 17x = 221
x =

221
17

= 13

II.

(484)2 y 225 183

or, 22y - 15 = 183

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

309
or, 22y = 198

+ 4y = 41
or, 4y = 41 - 21 = 20

198
9
22

or, y =

Hence, x > y
194. 1; I. 511x2 = 3066

or, x 2

196. 5; I. 3x2 - 6x -

3066
6
511

17x 2 17 0

or, 3x(x - 2) or, (3x -

II. 12y3 - 9y3 = 1536

or, 3y 3 = 1536

1536
=
3

= 5

Hence, x > y

x 6

or, y3 =

20
4

17

17

(x - 2) = 0

)(x - 2) = 0

17
3

or, x = 2,

512 = 83
II. 10y2 - 15y +

17y 3 17 0

y = 8 Hence, x < y
or, 5y(y - 3) +

17

195. 4, I. 3x + 4y = (4681) 2 = 41
or, 3x + 4y = 41

... (i)

or, (5y +

17

II. 3x + 2y = (961) 2
3x + 2y = 31
...(ii)
Solving (i) and (ii), we get
3x + 4y = 41
...(i) 2
3x + 2y = 31 ...(ii) 4
6x + 8y = 82
12x + 8y = 124
.
- 6x = - 42

...(i)

42
7
6

Putting the value of x in equation (i), we get 3 7

y 3,

(y - 3) = 0

) (y - 3) = 0

17
5

197. 2; I. x2 - 16x + 63 = 0
or, x2 - 9x - 7x + 63 = 0
or, x(x - 9) - 7(x - 9) = 0
or, (x - 7) (x - 9) = 0
x = 7, 9
II. y2 - 2y - 35 = 0
or, y2 - 7y + 5y - 35 = 0
or, y(y - 7) + 5(y - 7) = 0
or, (y - 7) (y + 5) = 0
y = 7, - 5
Hence, x y

LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)